Legendrian Weaves: N-graph Calculus, Flag Moduli and Applications
LLEGENDRIAN WEAVES – N-GRAPH CALCULUS, FLAG MODULI AND APPLICATIONS –
ROGER CASALS AND ERIC ZASLOW
Abstract.
We study a class of Legendrian surfaces in contact five-folds by encoding theirwavefronts via planar combinatorial structures. We refer to these surfaces as Legendrianweaves, and to the combinatorial objects as N -graphs. First, we develop a diagrammaticcalculus which encodes contact geometric operations on Legendrian surfaces as multi-coloredplanar combinatorics. Second, we present an algebraic-geometric characterization for themoduli space of microlocal constructible sheaves associated to these Legendrian surfaces.Then we use these N -graphs and the flag moduli description of these Legendrian invariantsfor several new applications to contact and symplectic topology.Applications include showing that any finite group can be realized as a subfactor of a 3-dimensional Lagrangian concordance monoid for a Legendrian surface in ( J S , ξ st ), a newconstruction of infinitely many exact Lagrangian fillings for Legendrian links in ( S , ξ st ),and performing F q -rational point counts that distinguish Legendrian surfaces in ( R , ξ st ).In addition, the manuscript develops the notion of Legendrian mutation, studying microlocalmonodromies and their transformations. The appendix illustrates the connection betweenour N -graph calculus for Lagrangian cobordisms and Elias-Khovanov-Williamson’s SoergelCalculus. Contents
1. Introduction N -graphs and Legendrian Weaves N -graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.2. Singularities of wavefronts . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.1. The A germ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.2. The A germ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2.3. The D − germ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3. Legendrian Weaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4. Smooth Topology of Weaves . . . . . . . . . . . . . . . . . . . . . . . . . 172.5. Combinatorial Homology . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3. Combinatorial Constructions N -Triangulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2. Local Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 Mathematics Subject Classification.
Primary: 53D10. Secondary: 53D15, 57R17. a r X i v : . [ m a t h . S G ] J u l .3. Global Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4. Bicubic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4. Diagrammatic Calculus For Legendrian Weaves N -Graph Stabilization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.8. Legendrian Mutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.9. Diagrammatic Rules for N -graph Mutations . . . . . . . . . . . . . . . . 554.10. Sufficiency For Stabilized Legendrians . . . . . . . . . . . . . . . . . . . . 59
5. Flag Moduli Spaces N -graph . . . . . . . . . . . . 625.3. Sheaf Description of Flag Moduli and Invariance . . . . . . . . . . . . . . 635.4. Local Flag Moduli Computations . . . . . . . . . . . . . . . . . . . . . . . 665.5. Flag Moduli under Legendrian Surgeries . . . . . . . . . . . . . . . . . . . 695.6. Non-characterstic Property of Stabilization . . . . . . . . . . . . . . . . . 70
6. Applications and Vexillary Computations
7. Microlocal Monodromies and Lagrangian Fillings N -Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.1.3. Explicit Examples of Lagrangian Fillings . . . . . . . . . . . . . . 857.2. Microlocal monodromies and cluster structures . . . . . . . . . . . . . . . 877.2.1. Microlocal monodromies as cluster coordinates . . . . . . . . . . . 897.2.2. Legendrian Mutations are cluster transformations . . . . . . . . . 917.3. N-graph Realization of Quiver Mutations . . . . . . . . . . . . . . . . . . 94
8. Moduli Space for N -triangles and Non-Abelianization .1. Flag moduli space of the N -triangle . . . . . . . . . . . . . . . . . . . . . 1008.1.1. Tetrahedral Triangulations at N = 3 and N = 4 . . . . . . . . . . 1038.2. A Computation of the Non-Abelianization Map . . . . . . . . . . . . . . . 104Example: Tetrahedron with 3-Triangulation. . . . . . . . . . . . . . . . . 105 Appendix A. Soergel Calculus and Legendrian Weaves
References N -graphs, and algebraic geometry, by studying moduli space of simple sheaves microlocallysupported at a front as incidence problems for flags in projective space. This yields newconnections with combinatorics, algebraic geometry and cluster algebras and we use them toobtain new results in contact and symplectic topology.1. Introduction
Legendrian knots in contact 3–manifolds [Etn05, Gei08] are central to the study of 3–dimensional contact geometry [Ben83, Eli93, Gom98]. The study of Legendrian knot invari-ants makes extensive use of their planar front projections, both in the context of Floer theory[EGH00, Che02, Ng03] and microlocal analysis [KS85, GKS12, STZ17]. Higher-dimensionalLegendrian submanifolds have proven equally instrumental in the study of higher-dimensionalsymplectic and contact topology, including the development of Legendrian Kirby Calculus[Eli90, Gom98, CMP19] and Lagrangian skeleta [RSTZ14, Nad17a, Sta18].In the case of 6–dimensional symplectic manifolds and their 5–dimensional contact boundaries[CE12, CM19], spatial front projections for Legendrian surfaces are available [Ad90, AdG01].First, this article develops a multi-colored planar diagrammatic calculus for the manipulationof such Legendrian surfaces in 5–dimensional contact manifolds and their Lagrangian projec-tions in 4-dimensional symplectic manifolds. This diagrammatic calculus is first used for theefficient computation of microlocal Legendrian isotopy invariants, as we prove and illustratethroughout the manuscript. Then we provide several new applications, including new re-sults in higher-dimensional contact geometry and low-dimensional symplectic topology. Wealso expect that this concrete description will prove itself useful for further results, such asthe computation of symplectic invariants of Weinstein manifolds [GPS19a, Section 6.4] andhomological mirror symmetry [Nad17b, TZ18], see also [CM19, Section 4.4] and Remark 6.1.1.1.
Summary of Contributions.
Let G be an N -graph drawn on a smooth surface C .The notion of an N -graph, combinatorial in nature, is first defined in Section 2. In a nutshell,our main contributions are as follows:A. Diagrammatic Calculus and Legendrian Weaves . The construction of a Leg-endrian surface Λ( G ) in the five-dimensional jet space ( J C, ξ st ) associated to the N -graph G , along with a description of Legendrian Surface Reidemeister moves in Informally, an N -graph G ⊆ C is a collection of trivalent graphs on C decorated with labels i ∈ [1 , N ] suchthat graphs with successive labels can only intersect at hexavalent vertices, where the six radiating half-edgeson the surface must interlace. See Definition 2.2 for details, and note that a 2-graph is simply an embeddedtrivalent graph. erms of combinatorial N -graphs moves. Likewise, we show that Legendrian surgeriesand Legendrian mutations, which we introduce, can be reflected by the diagrammat-ics of N -graphs. This is part of a general calculus of multi-colored planar diagramsthat, as we show, captures Legendrian surfaces and 3-dimensional Lagrangian cobor-disms between them. The translation from five-dimensional contact topology to suchplanar diagrammatics allows us to study contact topology through combinatorics andgraph theory. In fact, we use this combinatorial perspective to construct Lagrangianand Legendrian surfaces that prove new results in contact topology.B. The Microlocal Sheaf Theory of N -Graphs . A Legendrian surface Λ ⊆ ( J C, ξ st )specifies a category of constructible sheaves on C × R with singular support con-strained by Λ. When Λ = Λ( G ) for an N -graph G , we show that the moduli stackof objects M ( G ) has a combinatorial description in terms of flag varieties, which weintroduce in Section 5. This space solves an incidence moduli problem for flags ofsubspaces in an N -dimensional k -vector space V , with k a field, as dictated by the N -graph G . This stack is typically an algebraic variety and can be studied by algebraicgeometric and representation-theoretic techniques. Following [GKS12, STZ17, TZ18],this space is shown to be a Legendrian invariant for surfaces Λ( G ) and can be used todistinguish Legendrian isotopy types. In addition, we explicitly give formulas for themicrolocal monodromies along certain cycles of H (Λ( G ) , Z ) in terms of generalizedcross-ratios of flags, and their transformation under Legendrian mutations.C. Applications of N -Graph Calculus . First, in Section 6 we use the diagrammaticsin (A) to study the flag moduli spaces M ( G ) in (B), including their rational pointcounts over finite fields F q . This allows us to distinguish many Legendrian surfaces,up to Legendrian isotopy and, independently, show that for any finite group G , thereexists a Legendrian surface in ( R , ξ ) whose 3-dimensional Lagrangian concordancemonoid has G as a subfactor. Second, Section 7 explains how to apply N -graphcalculus to systematically study Lagrangian fillings of Legendrian links in ( S , ξ st ).In particular, we use Legendrian mutations to give new families of Legendrian linkswhich admit infinitely many Lagrangian fillings.Finally, given N -triangulations ( C, τ ) of the smooth surface C , we construct N -graphs G ( τ ) such that the Lagrangian projections of the Legendrian surfaces Λ( G ( τ ))relate to the Goncharov-Kenyon conjugate surfaces [GK13, Section 1.1.1] associatedto an N -triangulation. In Section 3 we provide the construction of G ( τ ). In Sec-tion 8, we provide an example of how Hitchin’s non-Abelianization map is describedfrom this viewpoint. This provides a context for the symplectic study of the clusterstructures associated to moduli spaces of framed local systems of Fock-Goncharov[FG06a, FG06b], and certain classes of Gaiotto-Moore-Neitzke’s spectral networks[GMN10, GMN13, Nei14, GMN14]. In particular, the microlocal sheaf theory ofΛ( G ( τ )) connects, through the moduli space M ( G ( τ )) in (B), with their spaces offlag configurations [Gon17, Section 3].1.2. Main Results.
We now elaborate upon these topics and state our results.
Diagrammatic Calculus and Legendrian Weaves . Weinstein manifolds [CE12, CE14,CM19], the symplectic counterpart of Stein manifolds, place Legendrian submanifolds atthe forefront of higher-dimensional contact and symplectic topology. In this manuscript, we See also [Gon17, Section 2.1], and [STWZ19, Section 4.2] describes the conjugate surface as a Lagrangian. (23)(12)(34) Figure 1. S . These cor-respond to Legendrian surfaces in the contact 5-space ( J S , ξ st ), respectivelyof genus 3 and 4.define and study a new class of Legendrian surfaces Λ( G ) in contact 5-manifolds, associatedto an N -graph G , building on our previous works [CM18, TZ18]. Prior work on Legendriansurfaces [ENS18, She19] has focused on the class of Legendrian tori Λ K ⊆ ( T ∞ S , ξ std ) arisingas the conormal torus of a smooth knot K ⊆ S . The Legendrian surfaces Λ( G ) we studyprovide a second infinite family of Legendrian submanifolds whose contact topology and sheafinvariants can be understood. Their geometry is governed by the combinatorial data of the N -graph G . Figure 1 depicts two examples of N -graphs, representing Legendrian surfaces ofgenus 3 (left) and 4 (right).We study three geometric operations for Legendrian surfaces in 5-dimensional contact man-ifolds. These are Legendrian isotopies [Ad90, CE12, Gei08], exact Lagrangian cobordisms[Ad76, BST15, EHK16], and Legendrian mutations, which we define in Section 4. Lagrangiancobordisms of indices 1 and 2 correspond to Legendrian 0- and 1-surgeries. We establish acorrespondence between each of these three types of geometric operations and the combi-natorics of N -graphs. In addition, we describe a combinatorial stabilization of an N -graph,which can be understood as a five-dimensional analogue of the Markov stabilization of aLegendrian braid [Rol76, PS97]. Part of these results are summarized in the following twotheorems (see Section 4 for details), which are developed in the text: Theorem 1.1 (Diagrammatics for Legendrian Weave Calculus I) . Let G be a local N -graph.The combinatorial moves in Figures 2 and 3 are Legendrian isotopies for Λ( G ) . (cid:3) Theorem 1.2 (Diagrammatics for Legendrian Weave Calculus II) . Let G be a local N -graph. The combinatorial moves in Figure 4 are Legendrian surgeries, of indices 0, 1 and 2,Legendrian mutations and connected sums with the standard and Clifford tori. (cid:3) Theorems 1.1 and 1.2 provide an efficient diagrammatic calculus to manipulate the Legen-drian surfaces Λ( G ) associated to N -graphs G . We refer to the Legendrian surfaces Λ( G ) as Legendrian weaves , due to the resemble of their Legendrian fronts to a weaving pattern – seeDefinition 2.7. Theorems 1.1 and 1.2 are geometric in nature and are proven by manipulatingLegendrian fronts for Legendrian surfaces in five dimensions. This is the content of Section4, as part of our study of generic three-dimensional front singularities and their homotopies.In addition, Section 3 provides several combinatorial constructions of Legendrian surfacesΛ ⊆ ( S , ξ st ) which are used in our applications in Sections 6, 7 and 8. igure 2. Combinatorial Moves for Legendrian Isotopies of Surfaces Λ( G ).Moves I–V are local Legendrian isotopies in the 1-jet space ( J R , ξ st ). MoveS in the lower right is local in ( J S , ξ st ) after satelliting to the Legendrianunknot Λ ⊆ ( R , ξ st ). Figure 3.
Combinatorial Moves for Legendrian Isotopies of Surfaces Λ( G ).These are homotopies of spatial wavefronts involving A -swallowtail singular-ities. Remark 1.3.
The Legendrian weaves Λ( G ) ⊆ ( J C, ξ st ) associated to an N -graph G ⊆ C admit spatial wavefronts π (Λ( G )) ⊆ C × R with front singularities solely of types A , The A -singularity corresponds to a crossing, and the A -singularity is given by three planes intersectingtransversely at a point. The A -singularity corresponds to a simple cusp, A -singularities are swallowtails,and A A -singularities are obtained by intersecting a cusp with a linear space. igure 4. Table of Combinatorial Moves for Surfaces Λ( G ) corresponding toLegendrian Surgeries, mutations and tori connected sums. (cid:3) A and D − , following V.I. Arnol’d’s notation [Ad76, Ad90]. That said, their satellites ι (Λ( G )) ⊆ ( Y, ξ st ) typically acquire A , A A and A singularities. For instance, the satel-lite of Λ( G ) along the standard Legendrian unknot Λ ⊆ ( R , ξ st ) necessarily develops A -singularities. In addition, the standard 5-dimensional Legendrian Reidemeister surface movesinclude the creation of A singularities, and the interaction of A and A singularities yielda D +4 singularity. These Legendrian singularities and 3-dimensional Reidemeister moves willalso be discussed in Section 4. (cid:3) The Microlocal Sheaf Theory of N -Graphs . The relationship between sheaf theoryand contact and symplectic geometry [NZ09, Nad09, GKS12, GS14] provides invariants ofLagrangian and Legendrian submanifolds up to Hamiltonian and contact isotopies [STZ17,STWZ19, CG20]. These invariants are an alternative to the more analytical Floer-theoreticmethods [EES05b, EENS13a, EENS13b], and have recently been shown to contain equivalentdata [GPS19a, GPS19b, GPS19c].Let G be an N -graph on C , Λ( G ) ⊆ J ( C ) its Legendrian surface, and C ( G ) the categoryof simple constructible sheaves on C × R microlocally supported along Λ( G ) . In Section5, we describe the moduli space of objects in C ( G ) in terms of the combinatorics of G .Specifically, we define the flag moduli space M ( G ) of an N -graph G ⊆ C , an algebraic stack– often a variety – as being described by explicit relations among elements in the flag varietyGL( N, k ) /B , where B is the Borel subgroup of upper triangular matrices. Already when N = 2 , the number of rational F q -points of M ( G ), for a finite field F q is, up to a factor, thechromatic polynomial of the dual graph evaluated at q + 1 = | ( GL (2 , F q ) /B )( F q ) | [TZ18],and hence the moduli stack M ( G ) geometrizes a familiar graph-theoretic construction.For general N , this algebraic space M ( G ) is the moduli space of an incidence problembetween flags and their stabilizing monodromies. It has two particular virtues. First, M ( G )changes explicitly under certain combinatorial moves of the N -graph G — thus, each timewe can simplify G with our moves from Theorems 1.1 and 1.2, we get closer to solving themoduli problem via purely diagrammatic techniques. Second, M ( G ) is an invariant of the egendrian isotopy class of Λ( G ) ⊆ J ( C ). In short, M ( G ) is defined purely in terms of thecombinatorics of the N -graph G , in a manner we understand, and we show it geometricallydescribes the following invariant: Theorem 1.4.
Let C be a closed, smooth surface and G ⊆ C an N -graph. The flag modulispace M ( G ) is isomorphic to the moduli space of microlocal rank-one sheaves on C × R microlocally supported along Λ( G ) ⊆ ( J C, ξ st ) . (cid:3) After the work of Guillermou-Kashiwara-Schapira [GKS12], which constructs an equivalenceof sheaf categories from a Legendrian isotopy, we conclude that the algebraic isomorphismtype of the moduli stack M ( G ) is a Legendrian isotopy invariant of the Legendrian surfaceΛ( G ) ⊆ ( J C, ξ st ). In fact, it will remain a Legendrian isotopy invariant for certain satel-lites along C ⊆ ( R , ξ st ), yielding a Legendrian invariant for Λ( G ) ⊆ ( R , ξ st ). Theorem 1.4,proven in Section 5, is a generalization to N ≥ R is ex-pressed in algebraic combinatorial terms. Applications of N -Graph Calculus . Sections 6, 7 and 8 exhibit a gallery of computationsand uses of the flag moduli space M ( G ), including the study of M ( G ) as a complex varietyand its finite F q -counts. For instance, our techniques readily prove the following sampleresult: Theorem 1.5 (Flag Moduli for Ladder Graphs) . Let L n ⊆ S be the (2 n ) -runged ladder3-graph of Figure 5, and let F q a finite field. Then the flag moduli space M ( L n ) has orbifoldpoint count |M ( L n )( F q ) | = q n − − q n − + q n − + q − q − In particular, the Legendrian 3-links of 2-spheres Λ( L n ) and Λ( L m ) are Legendrian isotopicif and only if n = m . (cid:3) The infinitely many Legendrian surfaces Λ( L n ) in Theorem 1.5, n ∈ N , are pairwise smoothlyisotopic. The distinct finite F q -counts of their flag moduli space M (Λ( L n ))( F q ) give a directproof that they are not Legendrian isotopic as Legendrian surfaces in ( R , ξ st ). Also, addingthe ladder 3-graphs in Theorem 1.5 into a face of an arbitrary N -graph G typically changesthe flag moduli space of M ( G ) and thus produces another Legendrian surface, smoothlyisotopic but not Legendrian isotopic to Λ( G ).In general, the computation of these Legendrian invariants translates into an incidence moduliproblem, which can itself be simplified with our diagrammatic techniques, and then possiblysolved with methods from algebraic geometry. In particular, we will understand the effectof combinatorial moves for N -graphs G on the Legendrian invariants M (Λ( G )). This willfrequently allow for the computation of this moduli stack and distinguish Legendrian weavesup to Legendrian isotopy. This yields a wide range of results in the vein of Theorem 1.5, aswe will illustrate. From this perspective, Legendrian weaves, which are in general surfaces ofany genus, constitute an attractive complement to the family of knot conormals. Microlocal rank-one sheaves are also called microlocally simple or just simple [KS85, Chapter 7]. It should be emphasized that the Legendrian invariants M ( G ) are significantly easier to compute thantheir Floer-theoretic counterparts, such as the Legendrian DGA [Che02, EES05b]. igure 5. The bipartite Ladder 3-Graph L n , where the right and left sidesare identified after 2 n rungs.We now illustrate a second application of our flag moduli stacks, detailed in Section 6. LetΛ ⊂ ( S , ξ st ) be an embedded Legendrian surface and let L (Λ) be the space of embedded Leg-endrian surfaces which are Legendrian isotopic Λ, with base point Λ. Let L (Λ) be the monoidof 3-dimensional exact Lagrangian concordances in the symplectization ( S × R ( t ) , e t λ st ), upto Hamiltonian isotopy, based at Λ. The flag moduli spaces M ( G ) will be used to show thefollowing result: Theorem 1.6.
Let G be an arbitrary finite group. Then there exists a Legendrian surface Λ G ⊆ ( S , ξ st ) such that (i) G is a subfactor of the fundamental group π ( L (Λ G )) , (ii) G is a subfactor of the 3-dimensional Lagrangian concordance monoid L (Λ G ) .In fact, the latter is the image of the former via the graph map gr : π ( L (Λ)) −→ L (Λ) . Theorem 1.6 essentially states that the study of the 3-dimensional Lagrangian concordancemonoid can be as complicated as any finite group. The proof of Theorem 1.6 will exhibitthe advantage of using combinatorial constructions on an N -graph G to extract contactand symplectic information in 5- and 6-dimensions. Note that for 1-dimensional max-tbLegendrian torus links, T. K´alm´an provided finite cyclic subgroups of the 2-dimensionalLagrangian concordance monoid [K´05], and J. Sabloff and M. Sullivan provided finite cyclicsubgroups of the 3-dimensional Lagrangian monoid for certain Legendrian surfaces [SS16].Sections 5, 6 and 7 contain several computations and applications of the flag moduli spaces M ( G ). Remark 1.7.
Even though the Legendrian DGA of a Legendrian knot in ( R , ξ st ) can becomputed algorithmically, the Floer-theoretic invariants of general Legendrian submanifoldsin higher-dimensions represent a challenge [DR11, EES05a, EES05b] — see [RS19a, RS19b]for progress in this direction. The class of Legendrian 2-tori arising as knot conormals issheaf-theoretically understood [ENS18, Ng11, She19] and our results, in line with Theorem1.5 and Theorem 1.6, aim at achieving both a geometric and sheaf-theoretic understandingfor the class of Legendrian weaves Λ( G ). (cid:3) For a third class of applications, consider an N -graph G ⊆ D with boundary. The La-grangian projection of the Legendrian weave Λ( G ) yields an exact Lagrangian filling of aLegendrian link in ( S , ξ st ), associated to ∂G . In Section 7 we will construct different N -graphs G , G with ∂G = ∂G , and explain how microlocal monodromies can be used toshow that the Lagrangian projections of the Legendrian weaves Λ( G ) and Λ( G ) are not Hamiltonian isotopic relative to their 1-dimensional Legendrian boundaries. In fact, N -graphcalculus, in combination with Legendrian mutations, allows us to construct infinitely manydistinct embedded Lagrangian fillings for certain Legendrian knots. The following family ofLegendrian links is studied in detail in Subsection 7.3: The results of [SS16] are stronger in higher-dimensions, but for Legendrian surfaces the only finite sub-groups of the special orthogonal group SO (2) must be cyclic – see [SS16, Remark 4.7]. A combinatorial criterion for embeddedness, which will be useful, is described in Lemma 7.4. heorem 1.8. Let Λ s,t = Λ( β s,t ) ⊆ ( S , ξ st ) be the Legendrian link given by the standardsatellite of the positive braid β s,t = ( σ σ )( σ σ ) s σ σ ( σ σ ) t ( σ σ )( σ t +12 σ σ s +22 ) , s, t ∈ N , s, t ≥ . Then Λ s,t ⊆ ( S , ξ st ) admits infinitely many embedded exact Lagrangian fillings in ( D , λ st ) realized as -graphs G s,t ⊆ D and their Legendrian mutations. The 3-graphs representing the infinitely many Lagrangian fillings in Theorem 1.8 are dia-grammatically interesting, with their complexity increasing as we geometrically realize theiterates in an infinite sequence of quiver mutations. For instance, Figure 6 depicts an ex-ample of a Lagrangian filling associated to such a 3-graph, obtained after five mutations.Fortunately, the local mutations rules that we develop in Section 4.9 will allow us to controlcertain infinite sequences of N -graphs mutations and construct infinite sequences of pairwisedistinct Lagrangian fillings.
12 3 4 56 (x3)(x2) (x2)(x2)(x2)
Figure 6.
The 3-graph for one of the infinitely many Lagrangian fillings ofthe Legendrian link Λ , ⊆ ( S , ξ st ), as featured in Theorem 1.8. Iterative3-graph mutations will yield new 3-graphs G ⊆ D representing pairwise non-Hamiltonian isotopic Lagrangian fillings of Λ , .Theorem 1.8 is an appropriate complement to the recent results [CG20], as the constructionof the infinitely many Lagrangian fillings in Theorem 1.8 is obtained directly by Legendrianmutations. In more generality, Section 7 develops the relation between the cluster algebraassociated to the intersection quiver of a Lagrangian filling and the Legendrian mutationsfrom Section 4.8. In particular, N -graph calculus can serve as an effective tool to show thata given Legendrian link admits infinitely many Lagrangian fillings, in case the quiver is ofinfinite mutation type and its vertices are represented by mutable 1-cycles in the N -graph G . In fact, any Legendrian link Λ( β ) ⊆ ( S , ξ st ) associated to a positive braid β ∈ Br + N In contrast, the construction for torus links given by the first author in [CG20] uses Lagrangian concor-dances of infinite order. This is generically the case. dmits a Lagrangian filling – oftentimes many – given by an N -graph G ⊆ D .A final application of N -graph calculus for Legendrian weaves develops the connection ofsymplectic topology to V. Fock and A. Goncharov’s cluster varieties of framed local systems[FG06b] (see also [Gon17, STWZ19]), and should relate to the spectral networks of Gaiotto-Moore-Neitzke [GMN10, GMN13, GMN14]. For that, consider N ∈ N and τ an ideal N -triangulation of the smooth punctured surface C . In Section 3, we present a new constructionthat associates an N -graph G ( τ ) to an ideal N -triangulation ( C, τ ). In particular, each ideal N -triangulation τ yields a Legendrian surface Λ( G ( τ )) ⊆ ( J C, ξ st ). In general, different N -triangulations lead to smoothly isotopic Legendrian surfaces which are not Legendrianisotopic, and they are distinguished by their flag moduli space M ( G ( τ )). This also relies onthe connection between microlocal monodromies and cluster algebras. Figure 7.
The Legendrian weave associated to a 4-triangle (left) and to a5-triangle (right). The open Legendrian surface for the 4-triangle has genusone and two boundary components. The Legendrian surface for the 5-trianglehas genus two and three boundary components. (cid:3)
The N -graph G ( τ ) and the Legendrian weave Λ( G ( τ )) are both constructed with a localmodel on an N -triangle. Figure 7 depicts a Legendrian weave associated to the 4- and 5-graphs dual to 4- and 5-triangles. We will prove that their local flag moduli space is a complextorus by using Theorem 1.1 and the flag moduli space results from Section 5. The precisestatement, proven in Section 8, reads as follows: Theorem 1.9.
Let G ( t N ) be the N -graph associated to an N -triangle t N , and let k a field.The flag moduli space of G ( t N ) is a (cid:0) N − (cid:1) -dimensional complex torus, i.e. M ( t N , G ( t N ); k ) ∼ = ( k ∗ )( N − ) . The combinatorial number (cid:0) N − (cid:1) appears geometrically as the rank of the first homologyclass of the Legendrian weave Λ( G ( t N )). Now, the class of Legendrian weaves Λ( G ( τ )) aris-ing from ideal N -triangulations τ of punctured surfaces is of central interest in the studyof moduli spaces of framed local systems for the Lie group GL( N, C ) [FG06b]. Indeed, theLegendrian surface Λ( G ( τ )) is a compactification of the Legendrian lift of the Goncharov-Kenyon Lagrangian conjugate surface L τ ⊆ ( T ∗ C, λ st ), see [Gon17, STWZ19]. Thus, thenon-Abelianization technique, expressing higher-rank local systems in S in terms of rank-onelocal systems on L τ , can also be recovered by studying these Legendrian weaves Λ( τ ) – seeSection 8.2 for an explicit computation. In particular, the set of Legendrian surfaces { Λ( τ ) } τ rovides a symplectic geometric realization of the set of cluster charts in this moduli spacesof framed local systems. This parallels the work of [STWZ19] on conjugate surfaces. SeeSection 8 for details. Basic Notation and Color Code.
The germs of singularities of caustics and wavefrontsare referred to according to the classical notation from the theory of singularities, followingV.I. Arnol’d [Ad90]. Given a subset X ⊆ Y of a smooth manifold Y , we denote by O p ( X )an arbitrarily small but fixed open neighborhood of it, following M. Gromov [Gro86].Regarding colors, the two colors blue and red are associated to edges with adjacent transposi-tions, i.e. edges with consecutive transpositions ( i − , i ) , ( i, i + 1), for a choice 2 ≤ i ≤ N − three colors blue, red and yel-low together denote edges labeled by three consecutive transpositions ( i − , i ) , ( i, i + 1) and( i + 1 , i + 2), respectively, for a choice 2 ≤ i ≤ N −
2. In a diagram with the two colorsblue and yellow, without red, these two colors denote any edges with disjoint transpositions.The color orange will exclusively be used to denote cusp edges, corresponding to edges of A -singularities. Finally, we use purple dots (or black dots) for D − singularities, yellow dotsfor A singularities and orange dots for A -swallowtail singularities. (cid:3) Acknowledgements.
We thank Honghao Gao for his thorough reading of the initial versionof this manuscript, and Honghao Gao, Eugene Gorsky and Harold Williams for valuable com-ments. We are grateful to Dylan Thurston for providing key examples of quiver mutations,and to Ben Elias for discussions on Soergel calculus. We also thank J. Etnyre, O. Lazarev,I. Le, E. Murphy, L. Ng, J. Sabloff, L. Traynor and D. Treumann for discussions, questionsand interest in this work. R. Casals is supported by the NSF grant DMS-1841913, a BBVAResearch Fellowship and the Alfred P. Sloan Foundation. E. Zaslow is supported by the NSFgrant DMS-1708503. (cid:3) N -graphs and Legendrian Weaves In this section we introduce the notion of an N -graph G and construct the Legendrian surfaceΛ( G ) associated to it. The interaction between the combinatorics of G and the contactgeometric invariants of Λ( G ) is the starting focus of this article. The reader is referred to[BM08, Die17] for introductory material on graph theory and to [Etn05, Gei08] for the basicsof contact topology.2.1. N -graphs. Let C be a smooth surface and N ∈ N a natural number. An embeddedgraph G ⊆ C is said to be trivalent if all its vertices have degree three. Such a vertex isdepicted on the left in Figure 8. Figure 8.
Trivalent vertex (left) and Hexagonal Point (right). efinition 2.1. Let J and K be two trivalent graphs embedded in C , having an isolatedintersection point at a common vertex v ∈ J ∩ K. The intersection v is said to be hexagonal if the six half-edges in C incident to v interlace, i.e. alternately belong to J and K . (cid:3) The right diagram in Figure 8 depicts a hexagonal vertex, where the graph J is labeled( i − , i ) in blue and K is labeled ( i, i + 1) in red. Definition 2.2. An N -graph G on a smooth surface C is a set G = { G i } ≤ i ≤ N − of N − G i ⊆ C , possibly empty or disconnected, such that G i is allowedto intersect G i +1 only at hexagonal points, 1 ≤ i ≤ N − . (cid:3) Two examples of N -graphs on the plane C = R are depicted in Figure 1. The (trivalent)vertices are depicted by purple or black dots and the hexagonal intersection points by yellow dots. Note that G i , G j ⊆ C are allowed to intersect (anywhere) if j (cid:54) = i, i ± Remark 2.3.
We can think of an N -graph as an immersed graph with colored edges, thecolor i corresponding to the graph G i , 1 ≤ i ≤ N −
1. Edges labeled by numbers differingby two or more may pass through one another (hence the immersed property, which ismet generically), but not at a vertex. In particular, a 3-graph is a bicolored graph withmonochromatic trivalent vertices and interlacing hexagonal vertices. (cid:3)
Consider τ ( N ) := { ( i, i + 1) ∈ S N : 1 ≤ i ≤ N − } ⊆ S N the subset of simple transpositionsand denote τ i := ( i, i + 1). We label the edges of an N -graph G = { G i } which belong to thegraph G i with the transposition τ i , as we have done in Figure 1. These edges will also bereferred to as τ i -edges, or i -edges. By definition, the trivalent vertices belonging to the graph G i have three incident τ i -edges. The hexagonal points in G i ∩ G i +1 have six edges incidentto it, alternately labeled with the transpositions τ i and τ i +1 in τ ( N ). Figure 8 depicts thelocal model for the trivalent vertices of the cubic graph G i − and a hexagonal intersectionpoint in G i ∩ G i − . Observe that a 2-graph is, by definition, an embedded trivalent graph.The study of N -graphs brings the combinatorial ingredients of the article, and we providein Section 3 several combinatorial constructions of N -graphs. For now, we introduce itsgeometric counterpart, the Legendrian surface associated to an N -graph.2.2. Singularities of wavefronts.
The Legendrian surface Λ( G ) associated to an N -graph G ⊆ C is an embedded Legendrian in the 1-jet space ( J C, ξ st ). The Legendrian surface Λ( G )is described by using germs of Legendrian wavefronts [Ad90, Section 3.1] in the Darboux chart( R , ξ st ), where the contact 4-distribution ξ st is defined as ξ st = ker α st , where α st := dz − y dx − y dx , and ( x , x , y , y , z ) ∈ R are Cartesian coordinates in R . This is the local model forany contact 4-distribution in the neighborhood of a point [Gei08, Theorem 2.5.1]. Since λ st = y dx + y dx is the Liouville form of the cotangent bundle ( T ∗ R , ω st ), this Darbouxchart ( R , ξ st ) is contactomorphic to the 1-jet space ( J R , ker { dz − λ st } ).The Legendrian fibration π : R −→ R , π ( x , x , y , y , z ) = ( x , x , z ) allows us to assigna smoothly embedded Legendrian surface Λ(Σ) ⊆ R in the domain of π to certain singularsurfaces Σ ⊆ R in its target. The coordinates ( y , y ) of the Legendrian Λ(Σ) assigned to Σare y = x -slope of the tangent plane T ( x ,x ,z ) Σ ,y = x -slope of the tangent plane T ( x ,x ,z ) Σ . In a local parametrization σ : R −→ R of Σ, σ ( u, v ) = ( u, v, z ( u, v )), this reads y = ∂ u z ( u, v ) , y = ∂ v z ( u, v ) . This assignment is dictated by the vanishing of the contact 1-form α st along Λ = Λ(Σ).The three-dimensional case is explained in detail in [Gei08, Section 3.2], the general case s discussed in [AdG01, Chapter 5], [EES05a, Section 3.2] and [CM19, Section 2]. Thegerms of singularities of Σ that lift to an embedded Legendrian Λ, and equivalently, thesingularities of the map π | Λ , are restricted. These are known as singularities of fronts, orequivalently, Legendrian singularities [AdG01]. By definition, singular surfaces Σ obtainedas the image of an embedded Legendrian submanifold via a Legendrian mapping are referredto as (wave)fronts. Remark 2.4.
The classification of generic singularities of spatial fronts Σ ⊆ R is stated in[AdG01, Theorem 3.1.1], and that of generic singularities of a 1-parametric family of spatialfronts Σ ⊆ R is explained in [AdG01, Theorem 3.4.2]. (cid:3) The main spatial wavefronts Σ that we use in the course of this article use three differentgerms of singularities of Legendrian fronts: A , A and D − , which we now describe. Weemphasize that these are singularities of the wavefront projections only: the correspondinglocal Legendrian surfaces are all smooth.2.2.1. The A germ. This germ is obtained as a product of a 2-dimensional planar fronttimes an interval. It is described by the germ of the singular surfaceΣ( A ) = { ( x , x , z ) ∈ R : ( x − z ) = 0 } at the origin. This wavefront is informally called an A - crossing , or a crossing , and theset of points { ( x , x , z ) ∈ Σ( A ) : x = 0 , z = 0 } is referred to as an edge, or segment,of A -crossings . This spatial front is depicted on the left in Figure 9. Its Legendrian liftΛ(Σ( A )) ⊆ ( R , ξ st ) consists of two disjoint embedded Legendrian 2-disks.2.2.2. The A germ. The wavefront A is given by the germ at the origin of the singularsurface Σ( A ) = { ( x , x , z ) ∈ R : ( x − z )( x + z )( z − x ) = 0 } ⊆ R , This spatial front is depicted in the center of Figure 9. Considered as a germ, the origin isthe A -wavefront singularity, and the codimension-1 singular strata consists of six half-linesof A singularities. The Legendrian lift Λ(Σ( A )) of the A germ to ( R , ξ st ) consists of threedisjoint embedded Legendrian 2-disks.2.2.3. The D − germ. The third germ Σ( D − ) = Im( δ − ) ⊆ R of a Legendrian singularitythat we use is given by the germ at the origin for the image of the map δ − : R −→ R , δ − ( x, y ) = (cid:18) x − y , xy,
23 ( x − xy ) (cid:19) . The D − -singularity of the spatial wavefront Im( δ − ) is at (0 , , ∈ R . The front Im( δ − )itself also has three half-lines of A -crossings, intersecting at the origin. This is depicted inthe right of Figure 9. The Legendrian lift Λ(Im( δ − )) ⊆ ( R , ξ st ) of the D − spatial front is anembedded Legendrian 2-disk. We refer the reader to [Ad90, TZ18] for more descriptions —see also Remark 2.5 below. Figure 9.
The A spatial front (left), the germ of the A Legendrian singu-larity (center) and the D − Legendrian wavefront (right). he connection of the above three Legendrian singularities with the Weyl groups, justifyingtheir nomenclature, can be found in [AdG01, Section 3.3]. It might be relevant to noticethat D − is not the germ of a singularity for a generic Legendrian wavefront, but still a validsingularity for a given spatial wavefront. In addition, it is known that the singularity D − isgeneric in 1-parameter families of Legendrian fronts [Ad90, Section 3.3]. As a result, mostof the Legendrians we construct are non-generic, in their isotopy class, with respect to thefixed Legendrian projection. This rigidification simplifes the analysis and combinatorics. Remark 2.5.
The D − Legendrian singularity has the property that its singular strata,excluding A singularities, is a point, which lies in real codimension 2. This is not the casefor the majority of Legendrian surface singularities, such as the Legendrian A -swallowtail,cusp-edges A A and the purse wavefront D +4 , the former two even being generic. (Thesesingularities feature in Section 4.) The geometric reason for this codimension-2 phenomenonis the existence of the holomorphic Legendrian surface singularity t : C −→ ( J ( C , C ) , ker { dw − w dw } ) , ( w , w , w ) = t ( w ) = (cid:18) w , w, w (cid:19) , whose real part is the real Legendrian singularity D − . This holomorphic map is the com-plexification of the real simple cusp singularities appearing in generic front projections ofembedded Legendrian knots in a Darboux chart ( R , ξ st ). (cid:3) We also use the A , A A front singularities, geometrically represented by a simple cuspin R times an interval, and its intersection with a 2-plane. These A -singularities do notdirectly arise from an N -graph G ⊆ C , but rather from satelliting the smooth surface C toa Legendrian surface in a contact 5-manifold ( Y, ξ ), typically ( S , ξ st ).2.3. Legendrian Weaves.
Let G ⊆ C be an N -graph, as introduced in Subsection 2.1above. The principle that associates a Legendrian Λ( G ) to the N -graph G is that G dictatesthe configuration of A singularities (crossings) of its Legendrian wavefront. This is possiblebecause the singularities introduced in Subsection 2.2 are uniquely determined by their A front singularities. Let us explain the construction in detail.First, we choose the ambient contact manifold, where the embedded Legendrian surface Λ( G )belongs, to be the 1-jet space of the smooth surface C . That is,Λ( G ) ⊆ ( J C, ξ st ) = ( { ( x, z ) ∈ T ∗ C × R } , ker { dz − λ st } ) , where λ st ∈ Ω ( T ∗ C ) is the Liouville form [Gei08, Section 1.4], and see [Ad90, Example 2]and [Gei08, Example 2.5.11] for details on the 1-jet space. The local germs described inSubsection 2.2 above and the Legendrian front projection π : ( J C, ξ st ) −→ C × R allow usto assign a Legendrian Λ(Σ) ⊆ ( J C, ξ st ) to a spatial wavefront Σ ⊆ C × R in the target, asfollows.The construction of the front Λ( G ) ⊆ ( J C, ξ st ) is obtained by gluing local wavefront modelsin U i × R , i ∈ I , U i ∼ = D , which are the targets of front projections in the Darboux charts( J U i , ξ st ) ∼ = ( J D , ξ st ), for i ∈ I . This is formalized in the following definition: Definition 2.6.
Let D N = D × { } ∪ . . . ∪ D × { N } ⊆ D × R . We consider D N as adisconnected, horizontal wavefront. Let P ⊆ D × { } be one of the following four localmodels of an N -graph G ⊆ D :1. A unique i -edge in D , as drawn at the bottom of the second column in Figure 10.2. A unique trivalent i -vertex, as shown at the bottom of the third column in Figure 10.3. A unique hexagonal ( i, i + 1)-point, depicted in the fourth column in Figure 10.4. The empty set. ere, recall that an i -edge is an edge belonging to the graph G i ⊆ G of the N -graph G ⊆ D ,for 1 ≤ i ≤ ( N − D N ( P ) ⊆ D × R associated to P is obtained as follows:- If P is a i -edge, insert an A -intersection along the two sheets D ×{ i } and D ×{ i +1 } of the wavefront D N . This A intersection must be inserted such that the image ofthe A singular locus coincides with P under the projection D × R −→ D onto thefirst factor.- If P is a trivalent i -vertex, introduce a D − -singularity between the two sheets D ×{ i } and D × { i + 1 } in the wavefront D N . This D − singularity must be introduced suchthat, under the projection D × R −→ D onto the first factor, the image of the A -crossings coincides with the three edges of P and the D − singular point is mappedto the unique trivalent vertex of P .- If P is a hexagonal ( i, i + 1)-point, insert an A -intersection along the three disjointsheets D × { i } , D × { i + 1 } and D × { i + 2 } of the wavefront D N . The pattern forthe A -wavefront must be inserted such that, under the projection D × R −→ D onto the first factor, the origin in the A -singularity maps to the unique vertex of P ,and the six half-lines of A -crossings map to the six edges emanating from the vertex.These wavefronts are depicted in Figure 10. For P empty we use the front D × { } ∪ . . . ∪ D × { N } ⊆ D × R . We refer to the wavefronts D N ( P ) as being obtained from the wavefront D N by weaving according to the pattern P . (cid:3) (i+1,i+2) (i,i+1)(i,i+1) (i,i+1) ...... ...... (i,i+1) ...... ...... i+2 (i,i+1)(i,i+1) (i,i+1) (i+1,i+2)(i+1,i+2) Figure 10.
The leftmost wavefront is D N , then from left to right we find D N ( P ) where P is an edge, a trivalent vertex and a hexagonal vertex.Definition 2.6 describes how to weave the wavefont D N ⊆ D × R , which we have fixed,according to a pattern P ⊆ D × { } . To glue models, let { U i } i ∈ I be a finite cover of C by open 2-disks U i ∼ = D , refined as necessary so that each U i contains no more than onenon-empty feature P of the N -graph G. Now, let us consider two 2-disks U , U ⊆ C and twocorresponding patterns P , P therein.Suppose that the patterns P and P coincide along the intersection U ∩ U . Then we saythat P ∪ P defines a pattern in U ∪ U . By definition, the wavefront Σ( P ∪ P ) associatedto P ∪ P is obtained by considering the set-theoretical union of D N ( P ) and D N ( P ) in( U ∪ U ) × R . For brevity of notation, we will say that Σ( P ∪ P ) is obtained by weaving D N ∪ D N ⊆ ( U × R ) ∪ ( U × R ) according to the pattern P ∪ P . Finally, the Legendriansurface associated to an N -graph is defined as follows: efinition 2.7. Let C be a smooth surface and G ⊆ C an N -graph, the Legendrian weaveΛ( G ) ⊆ ( J C, ξ st )is the embedded Legendrian surface whose wavefront Σ( G ) ⊆ C × R is obtained by weavingthe wavefront C × { } ∪ . . . ∪ C × { N } ⊆ C × R according to the pattern G ⊆ C . (cid:3) Let { ϕ t } t ∈ [0 , ⊆ Diff c ( C ), ϕ = Id, be a compactly supported isotopy of the smooth surface C . Then the Legendrian surfaces Λ( ϕ t ( G )) ⊆ ( J C, ξ st ), as described in Definition 2.7, areLegendrian isotopic, relative to their boundaries. Hence, for the purposes of this article, our N -graphs G ⊆ C are considered up to such planar isotopies. Similarly, Legendrian fronts in R are to be considered up to homotopy of fronts.Thanks to Definition 2.7, the wealth of contact topology invariants [EGH00, EES05a, GKS12,STZ17, CM19] can be used to define algebraic structures associated to N -graphs G ⊆ C . Forinstance, the articles [CM18, TZ18] show that the chromatic polynomial of (the dual of) atrivalent graph G – which is a 2-graph – is contained in the Floer-theoretical invariants of theLegendrian weave Λ( G ). Conversely, from a contact topology perspective, the connection tocombinatorics and algebraic geometry provides a new tool for computing contact invariants ofhigher-dimensional Legendrian submanifolds. This will be the focus of subsequent sections. Remark 2.8.
The one-dimensional analogue of a Legendrian weave is a Legendrian braid,i.e. a positive braid. Indeed, an N -graph in a one-manifold I is defined to be a set of points,each point labeled with a permutation in τ ( N ) ⊆ S N . The only planar front singularitythat we can use is A , corresponding to a crossing, necessarily positive. Thus, 1-dimensionalweaving consists of introducing positive crossings to the N strands D N = I × { } ∪ . . . ∪ I × { N } ⊆ I × R and concatenating them side by side. This is precisely the front for an N -strand positivebraid [PS97], which lifts to a Legendrian link in ( J S , ξ st ) [Gei08, Section 3.3.1]. TheLegendrian weaves introduced in Definition 2.7 are thus the Legendrian surface generalizationof Legendrian braids. (cid:3) Smooth Topology of Weaves.
Let G be an N -graph in a surface C , in this subsectionwe address the smooth topology of the Legendrian surface Λ( G ) ⊆ ( J C, ξ st ). The smoothinvariants of Λ( G ) are the first homology H (Λ( G ) , Z ), in particular its genus g (Λ( G )) ∈ N ,and the number of boundary components | ∂ Λ( G ) | . For simplicity, we assume that C is aclosed surface, and thus ∂ Λ( G ) = ∅ . We also assume that G is a connected N -graph, i.e. theunion of the graphs G i , i ∈ I , is a connected topological subspace of C .The surface Λ( G ) is a branched N -fold cover over C simply branched over the trivalentvertices of G . Indeed, the image of Λ( G ) by the projection J C −→ T ∗ C along the Reeb R -direction yields an immersed surface L ( G ) ⊆ T ∗ C , and the canonical projection T ∗ C −→ C restricts to L ( G ) as an N -fold branched cover. The branch set is the image of the set of D − singularities. As a result, the genus of Λ( G ) is provided by the Riemann-Hurwitz formula χ (Λ( G )) = N χ ( C ) − v ( G ) , i.e. g (Λ( G )) = 12 ( v ( G ) + 2 − N χ ( C ))where v ( G ) is the number of (trivalent) vertices of G . Remark 2.9.
If the surface C has boundary, each boundary component of ∂C contributesto a piece of the boundary ∂ Λ( G ) of the Legendrian surface Λ( G ) ⊆ ( J C, ξ st ). Let κ ∈ N bethe number of cycles in the (minimal length) factorization of the monodromy of the branched This is necessary for our applications, especially in the study of microlocal monodromies and Lagrangianfillings in Section 7 and the non-Abelianization map in Section 8. over along a given boundary component of ∂C . Then, that one boundary component of C contributes to κ distinct boundary components for the Legendrian surface Λ( G ). (cid:3) Example 2.10.
The Legendrian weaves Λ( G ) , Λ( G ) ⊆ ( J ( S ) , ξ st ) associated to the - and -graphs in Figure 1 are closed Legendrian surfaces of genus and , respectively. Shouldthe graphs G , G ⊆ R be considered in the 2-plane R , instead of the 2-sphere S , theLegendrian surfaces Λ( G ) , Λ( G ) ⊆ ( J ( R ) , ξ st ) have genus and , with and boundarycomponents, respectively. (cid:3) Now, the Z -monodromy of Λ( G ) along a non-trivial 1-cycle of the base C is trivial, and thusthe contributions of the graph G to H (Λ( G ) , Z ), as expressed by the above formula, canbe considered by studying planar pieces. Let us then assume that g ( C ) = 0 and construct1-cycles in H (Λ( G ) , Z ) in terms of the edges of the N -graph. (i,i+1) (i+1,i+2) (i,i+1)
23 2 1111 21
Figure 11.
Two combinatorial descriptions of 1-cycles in H (Λ( G ) , Z ).There are two direct descriptions of 1-cycles γ ∈ H (Λ( G ) , Z ):1. Each edge e of the graph G connecting two trivalent vertices defines a 1-cycle γ ( e ) ∈ H (Λ( G ) , Z ). The projection of this 1-cycle onto the pattern P with two trivalentvertices is depicted in orange on the left of Figure 11. In order to construct γ ( e ) fromthe orange curve, lift a point in the orange curve to the annulus Λ ( P ), to eitherone of the two sheets, and uniquely follow the lift along the orange curve. Sincethe lift is isotopic to one of the boundary components of the annulus, it generates H (Λ ( P ) , Z ) ∼ = Z . The 1-cycle γ ( e ) is drawn directly in the wavefront projectionin Figures 12. We refer to this type of 1-cycles as monochromatic edges or (short) I -cycles.
123 4561 2 3 4 5 6
Figure 12.
The first type of 1-cycle γ ( e ) drawn in the wavefront (left) andin a vertical slicing (right). Each slice on the left is labeled by a number. The1-cycle γ ( e ) appears as five-pointed stars in each slice as shown on the right. here is a simple extension of this construction, depicted in Figure 13. Considera trivalent vertex v ∈ G i and a linear chain of edges e , e , . . . , e k in G such that e connects v to a hexagonal vertex, e i connect two hexagonal vertices for 2 ≤ i ≤ k − e k connects the free hexagonal vertex in e k − to a trivalent vertex. Supposefurther that e j and e j +1 meet at opposite rays of the hexagonal vertex between them,1 ≤ j ≤ k − . Then the orange curves in the patterns all lift to 1-cycles which areessential in the surfaces Λ( P ) for the corresponding patterns P . These 1-cycles arereferred to as long edges or long I -cycles . (i,i+1) (i+1,i+2) (i,i+1) (i+1,i+2) ... (i,i+1) (i+1,i+2) (i+2,i+3) Figure 13.
Descriptions of 1-cycles in H (Λ( G ) , Z ) of the first type, gener-alizing γ ( e ) on the left of Figure 11. The lift of the orange curves generatethe first homology H (Λ( P ) , Z ) ∼ = Z for the corresponding patterns P .2. Three edges e , e , e of a graph G i connecting a hexagonal vertex with three trivalentvertices in G i defines a cycle γ ( e , e , e ) ∈ H (Λ( G ) , Z ). This is depicted on the rightin Figure 11. The 1-cycle γ ( e , e , e ) is drawn in the wavefront projection in Figure14. We refer to this type of 1-cycles γ ( e , e , e ) as a Y -cycle.
123 4 5 6
Figure 14.
The second type of 1-cycle γ ( e , e , e ) drawn in a slicing of thewavefront associated to the pattern on the left.We can also combine the above two constructions to associate a 1-cycle to any tree withleaves on trivalent vertices that passes directly through any hexagonal vertices, i.e. enteringand exiting along opposing edges, see Figure 99 for an example. For such a tree, we referto the pieces corresponding to edges as I -pieces, or edges, and the pieces that go througha hexagonal vertex as Y -pieces. In addition, we can decorate such 1-cycles with a number,indicating higher multiplicity . If we require the curves in the Legendrian surface to beconnected, then higher multiplicity in general requires these curves to be immersed. The topology of Λ( P ) is that of an annulus union disjoint 2-disks. Higher multiplicities will rarely feature in this manuscript, only in relation to Theorem 7.14. emark 2.11. Let Λ( G ) ⊆ ( J C, ξ st ) be a connected surface, and G ⊆ C a connected N -graph. The trivalent vertices of the N -graph G ⊆ C can be assumed to belong to G . Thisfollows once we impose certain equivalence relations on the set of N -graphs, which is donein Section 4. (cid:3) Combinatorial Homology.
Let G ⊆ C be an N -graph. We present a combinatorialmodel for the (chain-level) simplicial homology of Λ( G ). This can be achieved in general,but for this subsection we assume that G is a planar 3-graph, i.e. C = S and N = 3. Wewill think of G as bicolored — see Remark 2.3. This will ease notation, while containing theessential idea for higher N ∈ N and higher-genus C . Note that the results in this subsectionwill not be used in the rest of the manuscript, we have included them for completeness.The edges, faces and vertices of G lift to edges, faces and vertices of the Legendrian surfaceΛ := Λ( G ). Let us suppose that G and Λ are connected, and that the faces of G define apolyhedral decomposition ( F, E, V ) of the sphere. This decomposition lifts to a polyhedraldecomposition of Λ, as follows. Each face, edge and hexagonal vertex of G has three lifts toΛ; each trivalent vertex has two lifts. This yields χ (Λ) = 3 · − v, where v = | V ( G ) | is the number of trivalent vertices. For a point P ∈ S , we write P , P , P for the (up to) three pre-images in non-decreasing order of the z -coordinate. If P is on G ,we must choose a nearby point to define the ordering of z coordinates of sheets. If P is atrivalent vertex with label (12), in blue, then P = P while P = P for a label (23), in red.Lifts of edges and faces are labeled analogously. The chain complex C • associated to thispolyhedral decomposition of Λ computes the homology H ∗ (Λ; Z ). There is a simplified chaincomplex that computes H (Λ; Z ) , which we now explain.Lift each edge e = ( P, Q ) labeled ( i, i + 1) to a one-chain as follows (here i = 1 or 2). Inthe (any) orientation of the plane, if A is the sheet with lower z value in the region to theleft of P Q and B is the sheet with lower z value to the right of P Q then lift e to the chain P B Q B − P A Q A ; this only depends on e and not the ordering of P and Q . Write ˆ e for thislift of e . Extending by linearity, we get a map Z E → C . The embedded bicolored graph G is the union G = G B ∪ G R of embedded blue and a redgraphs intersecting at hexagonal vertices, where G B = ( F B , E B , V B ) . We define a complex A • as follows. A := Z F B ⊕ Z F R , A := Z E = Z E B ⊕ Z E R , and A is the image ∂ ˆ A , whereˆ A is the image of A in C under e (cid:55)→ ˆ e. A monochromatic face f ∈ Z F B ⊂ A has a lift to C as f − f , whereas f ∈ Z F R ⊂ A lifts to f − f . Summarizing, we have C (cid:47) (cid:47) C (cid:47) (cid:47) C A (cid:79) (cid:79) A (cid:79) (cid:79) (cid:47) (cid:47) A (cid:63)(cid:31) (cid:79) (cid:79) , where the map A → A sends e to ∂ ˆ e. The missing differential A → A is defined as follows.For a monochromatic face f ∈ Z F B or Z F R ⊂ A ,∂f = (cid:88) boundary edges e − (cid:88) interior edges e, which we extend by linearity. Proposition 2.12. A • is a chain complex and A • −→ C • is a chain map.Proof. Let f ∈ A be a blue face. A similar argument will work for red faces. We need tocheck that (cid:88) boundary edges ∂ ˆ e − (cid:88) interior edges ∂ ˆ e = 0 . his imposes a condition at all the interior and exterior vertices of f . In fact, the conditionis null at an interior vertex, since it must be monochromatic and hence trivalent, and ∂ ˆ e iszero over any trivalent vertex. Likewise for an exterior trivalent vertex, there is nothing tocheck. For an exterior hexagonal vertex, a local study is needed.Let h be a hexagonal vertex. Let e , e , e be three attached blue half-edges, with e = e (cid:48) , e = e (cid:48) , e = e (cid:48) the opposite red half-edges, respectively. Let h , h , h be the threepreimages of h . We can restrict the differential A | h → A | h to edges intersecting h andpoints over h , and in the chosen basis it takes the form(2.1) ∂ | h = − − −
11 0 − − − . The kernel is generated by e + e (cid:48) , e + e (cid:48) , e + e (cid:48) , e + e + e . The first three representlong two-colored edges passing straight through the hexagonal vertex, while the last is amonochromatic Y shape. The last generator could also have been taken to be e + e − e (cid:48) . This is the sum of two edges minus the edge in-between, precisely the configuration arising in ∂ f at a boundary hexagonal vertex. This concludes the calculation that ( A • , ∂ ) is a chaincomplex.To check that A • → C • is a chain map, we must show that for f ∈ Z F B ⊂ A , , we have ∂f − ∂f = (cid:88) e exterior ˆ e − (cid:88) e interior ˆ e. This is shown by direct calculation. (cid:3)
Let us now prove the following lemma before showing that A • is quasi-isomorphic to C • indegree one, and thus computes the first homology H (Λ; Z ). Lemma 2.13.
In the notation above, H ( A ) = 0 .Proof. This says that ∂ : A → A is injective. Suppose ∂f = 0. Let h be a hexagonal vertex,which must exist since both Γ and Λ are assumed connected. Label the edges adjacent to h by e , e , e , e = e (cid:48) , e = e (cid:48) , e = e (cid:48) as in the proof of Proposition 2.12. For 1 ≤ c ≤
6, let f c be the unique (opposite color) monochromatic face containing e a in its interior, and againwe notate f = f (cid:48) , etc. Now for i = 1 , , , write i, j, k for cyclically ordered elements of { , , } , i.e. j = i + 1 mod 3 , etc. Then e i is an exterior edge of f (cid:48) j and f (cid:48) k and by definitionan interior edge of f i . If we write f = (cid:80) i =1 a i f i + a (cid:48) i f (cid:48) i + · · · , then we must have a i + a j = a (cid:48) k for all i , and therefore (cid:80) a (cid:48) i = 2 (cid:80) a i . By the same token, (cid:80) a i = 2 (cid:80) a (cid:48) i , and therefore all a i and a (cid:48) i are zero.The faces f c with coefficients a c (cid:54) = 0 must therefore have no hexagonal vertices on theirboundary or interior. That said, the union U of such faces must have a boundary, andtherefore the coefficient of any face on the boundary of U must be zero. By iterating thisargument, all coefficients are zero. (cid:3) The 3-graphs associated to a 3-triangulation, and the 3-graph moves named candy twists and push-through , will be defined in Section 3, and Section 4 respectively. Now we establish thepoint of this subsection: roposition 2.14. Let Γ be a 3-graph for a 3-triangulation, or any graph related by candytwists or push-through moves. Then H ( A • ) ∼ = H ( C • ) ∼ = H (Λ; Z ) . Before the proof, a warning : H ( A • ) (cid:29) H ( C • ) in general. Here is an example of a weavewith topology of the twice-punctured plane.Despite b = 2, there is only one 1-cycle in H ( A • ) , represented by the tree with four leaves –the sum of edges (cid:98) e darkened in the picture. A choice for another generating 1-cycle is clear:it is a branch cut connecting the two trivalent vertices in the top (or bottom) – pictured asa dotted black cruve. This class can be represented in C • , but the chain connecting the twohexagonal vertices is not in A • . One could accommodate such chains with further notationalcomplexity, but we will not require them for our applications. Proof.
We need to prove the first equality only. Since A • → C • is a chain map, we need onlycompare the dimensions of their first homology groups. We prove this first for the 3-graphΓ T of a 3-triangulation T = ( F T , E T , V T ), then show that the result is invariant under candytwist and push-through moves.By definition, ∂ : A → A is surjective, so since by the lemma, ∂ : A → A is injective, weknow dim H ( A • ) = − χ ( A • ) . On the other hand, we know χ Λ = 6 − v = 6 − | F T | = 2 − h (Λ) , or h (Λ) = 3 | F T | − . We recall that each face of T has three blue vertices. It also has onehexagonal vertex which is a vertex of the blue and red graphs comprising Γ T . It similarly easyto compute that | F B | = | V T | + | E T | , | F R | = | V T | , | E B | = 2 | E T | + 3 | F T | , | E R | = | E T | , | V B | =3 | F T | , | V R | = 0 . Now | A | = | F B | + | F R | = 2 | V T | + | E T | , | A | = | E B | + | E R | = 3 | E T | + 3 | F T | ,and | A | = 2 | F T | is computed by noting that each hexagonal vertex contributes two possibledimensions to | A | via the rank-two matrix in Equation (2.1), and these dimensions arerealized as boundaries, while each trivalent vertex contributes nothing. We get h ( A • ) = − χ ( A • ) = − χ T + 3 | F T | = 3 | F T | − h (Λ) , as claimed.It remains to compute what happens after push-through or a candy-twist move. In fact,since the result only depends on the Euler characteristic of A • , we only need to show thatthis is invariant under candy twist and push-through. But these change the dimensions of( A , A , A ) by (2 , ,
4) and (1 , , A are in fact realized, and theresult follows. (cid:3) Combinatorial Constructions
In this section we introduce two combinatorial constructions for N -graphs, focusing primarilyon how to associate an N -graph to a given N -triangulation. The notion of an N -triangulationwas introduced in [FG06b, Section 1.15], and has since had an central role in higher Te-ichm¨uller theory [Gon17, GS18]. Legendrian weaves associated to an N -triangulation, viaour construction, place contact topology in the context of the recent developments in exactWKB analysis [GMN13, GMN14, Kuw20] and quiver Fukaya categories [BS15, Smi15].3.1. N -Triangulations. Let N ∈ N be a natural number, and consider the triangle t N := { ( x, y, z ) ∈ R : x + y + z = N x, y, z ≥ } . ubtriangulate this triangle t N with the lines( { x = s } ∪ { y = s } ∪ { z = s } ) ∩ t N , ≤ s ≤ N, which we refer to as an N -subdivision of the triangle t , following [FG06b, GMN14]. Thissubtriangulation has N triangles. Figure 15.
The Legendrian weaves Λ( G ( t )), Λ( G ( t )) and Λ( G ( t )) associ-ated to a 3-, 4- and 5-triangles.Now, let ( C, T ) be a triangulation T of a smooth closed surface C and subdivide each triangle t ∈ T according to the N -subdivision above. This yields a triangulation T N of the surface C . By definition, an N -triangulation on C is any triangulation isotopic to T N for sometriangulation ( C, T ).3.2.
Local Models.
The N -graph associated to an N -triangulation is obtained by gluinglocal models for the N -graph G ( t N ) associated to each triangle t N . We provide two equiv-alent definitions of this local N -graph, in terms of the following two constructions. Thereader content with using Figure 15 as a definition is invited to defer reading these technicaldescriptions. First Construction.
Consider the triangles in t N which point up, i.e. have a unique vertexwith highest z -value. For each of these (cid:0) N (cid:1) triangles, we insert a τ -trivalent vertex dual toit — that is, a trivalent vertex associated with the permutation (12) and such that the edgesof this piece of 2-graph intersects orthogonally with the edges of each triangle. By definition,the rest of the N -graph G ( t N ) is then uniquely determined by extending the edges from these τ vertices such that wherever three τ i -edges collide, we insert a hexagonal vertex with threeedges in τ i and three edges in τ i +1 . That is, the two rules to generate the N -graph for an N -triangulation are:(i) Insert exactly one (12)-trivalent vertex at the center of each upward pointing triangle,(ii) In the collision of three τ i -edges, a ( τ i , τ i +1 ) hexagonal vertex is inserted.We stress that the original triangles are not part of the N -graph.This construction of G ( t N ) can be considered as a dynamical description, in contrast with thestatic definition given by the second construction. Indeed, in this first construction one startsby placing the τ -vertices and lets the edges grow symmetrically from these trivalent vertices,such that each edge intersects the interior edges of the N -triangulation at the middle point.These edges must collide in the interior of the triangle, and these collisions are resolved viathe insertion of hexagonal vertices, creating τ i +1 -edges. This insertion of hexagonal verticesis iteratively performed when the τ i +1 -edges collide, creating τ i +2 -edges, and the processterminates when exactly three τ N − -edges are created at a unique hexagonal vertex. hus, given the triangle t N , we obtain a local model for an N -graph. The boundary conditionsfor this local model are such that the N -graphs associated to two N -triangles t N and t (cid:48) N ,which share an edge of the underlying t and t (cid:48) , match together. Remark 3.1.
This description, according to these two rules above, captures the propertiesof the spectral network associated to the WKB singular foliation for an SU (2) quadraticdifferential lifted via the unique N -dimensional irreducible representation of SU (2) — seeSections 2 and 4 in [GMN14]. The dynamical component, induced by the growing of theedges from vertices, corresponds with the time evolution of the differential equation definingthe WKB system. (cid:3) We now give a second equivalent construction of G ( t N ). Second Construction.
We describe a set of rays in the triangle t N , which will correspondto the edges of the N -graph. Let π = { ( x, y, z ) ∈ R : x + y + z = N } be the 2-planecontaining t N , as defined above, and let C be the set of centers of all triangles in t N . Nowconsider the honeycomb lattice H tiling the 2-plane π with regular hexagons such that thevertices of these hexagons in H are centered at the points in C . Equivalently, the hexagonsof this lattice honeycomb are the Voronoi cells associated to the vertices of the N -triangle t N . By construction, there are (cid:0) N − (cid:1) hexagons entirely contained in t N ⊆ H . In the inducedhoneycomb lattice H ∩ t N , add straight segments such that each internal regular hexagon issubdivided into six regular triangles, creating a set of rays H (cid:48) ⊆ t N . Finally, for each vertex v ∈ C which belong to an interior triangle t (cid:48) ⊆ t N pointing down with at least a vertex inthe boundary of t N , add straight segments from that vertex to each of the boundary points t (cid:48) ∩ t N parallel to the sides of the (non-subdivided) half hexagons containing such a vertex v . This creates a system of rays H ⊆ t N with H ∩ t N ⊆ H (cid:48) ⊆ H .Given the system of rays H ⊆ t N , the local N -graph G ( t N ) is defined as follows. Considerthe triangles in the N -triangle t N which are pointing up. Note that there are (cid:0) N (cid:1) of suchtriangles, starting with ( N −
1) of them in the bottom row, and decreasing by one up to aunique one triangle on the top row. By definition, there are (cid:0) N (cid:1) trivalent vertices for the N -graph G ( t N ); these are all τ -vertices and will be located at the center of the trianglespointing up, one trivalent vertex per triangle. By construction, we declare the remainingvertices of G ( t N ), as dictated by the system of rays H , to be hexagonal vertices.By definition, the edges of the G ( t N ) are to lie exactly above a ray in H , and it is allowedfor more than one edge to lie above a given ray, as long as the edges above a give ray areassociated to disjoint transpositions τ i ∈ S N . Remark 3.2.
In this convention, the graph G ( t N ) is not embedded, as multiple edges areallowed to overlap along a given ray. This configuration can be perturbed generically in orderto obtain an embedded N -graph. Nevertheless, this additional symmetry is in our favor whenstudying the transpositions associated to the boundary. (cid:3) Finally, it suffices to describe which edges lie above each ray, which in turn is uniquelyspecified by assigning a set of transpositions to the rays in H intersecting the boundary. Letus denote these 2 N − t N by r , r , . . . , r N − as ordered.Then the transpositions τ ( r i ) associated to each ray r i , i = 1 , . . . , N −
3, is given by thefollowing rules:- τ ( r i ) = τ ( r N − − i ), i.e. the association is symmetric with respect to the central ray r N − ,- For each ray r i , i ≤ N − τ ( r i ) := { τ j : j ≡ i mod 2, 1 ≤ j ≤ i } .This finalizes our second description of the N -graph G ( t N ) associated to an N -triangle t N . Remark 3.3.
Both these constructions provide a Legendrian front for the Legendrian liftof certain exact Lagrangian spectral curve for a local spectral network. In particular, this hows that the BPS graphs studied in [GLPY17], introduced as an interpolation betweenspectral networks and BPS quivers, are in fact the set of A singularities of the Legendrianfront Σ( G ( t N )) between the first two sheets. (cid:3) Note that the boundary of this local N -graph G ( t N ) can be compactly described as follows.Consider the permutation∆ N := N − (cid:89) i =1 (cid:89) j = i τ j = τ · ( τ τ ) · ( τ τ τ ) · . . . · ( τ N − τ N − · . . . · τ τ ) ∈ S N , which is the projection to the Coxeter group S N of the Garside element of the braid group B N in N -strands, i.e. a braid half-twist in N -strands. Then the edges of the N -graph associatedto t N along each of the three edges of t are precisely given by the ordered terms in ∆ N .That is, there exists an isotopy of the N -graph such that as one travels along an edge of t ,the edges of the N -graph that we encounter are first τ , then τ and τ , then τ , τ , and τ and iteratively until reaching τ for the ( N − τ ( r i ), i = 1 , . . . , N −
3, in the construction of G ( t N ) above.For context, these permutations along the boundary are particularly relevant for the studyof Legendrian surface weaves with boundary, whose Lagrangian projections yield interestingLagrangian fillings of their Legendrian boundary links. The braid description of these Legen-drian links is determined precisely by these permutations – see Section 7. We see again, conferRemark 2.8, that it is useful to think of Legendrian weaves as two-dimensional Legendrianbraids: their one-dimensional boundaries are positive braids.3.3. Global Model.
Given that the boundary conditions for the N -graphs in the localmodels for t N allow for gluing, we define the N -graph G ⊆ C associated to a global N -triangulation of C to be the N -graph obtained by concatenating the local models G ( t N )along each triangle t N in the N -triangulation. We study the flag moduli space invariants forthese N -graphs and their associated Legendrian weaves in Sections 6 and 8. Note that thegenus of these Legendrian weaves increases as N ∈ N , or the number of triangles, increases. Remark 3.4.
Trivalent vertices are dual to triangulations of surfaces. In particular, trian-gulations of surfaces with a large group of symmetries yield particularly interesting 2-graphs.From this perspective, Riemann surfaces with a conformal automorphism group of large ordergive rise to highly symmetric 2-graphs. For instance, Riemann surfaces associated to tilingsof the hyperbolic plane H with Schl¨afli symbol { n, } are highly symmetry, with { , } beingthe Klein quartic, { , } giving Bolza’s surface and { , } the M (3) surface. We expect theflag moduli space associated to the Legendrian surfaces of these 2-graphs, as defined in Sec-tion 5, to be algebraic spaces with correspondingly large symmetry. We begin an explorationof this kind with our Theorem 6.3 in Section 6. (cid:3) Bicubic graphs.
Here is a second construction of 3-graphs in a smooth surface C ,strictly disjoint from the class of 3-graphs arising from 3-triangulations.By definition, a graph is bicubic if it is both trivalent (cubic) and bipartite. Now consideran embedded bicubic graph G ⊆ C , and replace each vertex of G with a hexagonal vertex,doubling the edges as in Figure 16.The bipartite condition on the graph guarantees that these local models can be glued together,uniquely up to isotopy, yielding a 3-graph in C . Note that this 3-graph is entirely built fromhexagonal vertices, and no trivalent vertex is used. As a result, the topology of the Legendrianweave associated to such a 3-graphs is always that of a 3-component link of Legendrian 2-spheres. We will study a family of such 3-graphs in Section 6. igure 16. Example 3.5.
The bicubic graph G ⊆ S associated to the 1-skeleton of a 3-dimensionalcube, depicted in Figure 17, yields a 3-component Legendrian link Λ( G ) ⊆ ( J ( S ) , ξ st ) . Theflag moduli space M ( G ) will show that these three Legendrian spheres, even after satellitedto a Darboux ball ( R , ξ st ) are Legendrian knotted (and smoothly unknotted). (cid:3) Figure 17.
Remark 3.6.
Not every 3-graph which is exclusively formed by hexagonal vertices arisesfrom a bicubic graph, even up to candy-twist equivalence. In particular, two vertices mayhave just a single edge connecting them, with no vertices connected by three edges. Figure18 shows such an example.
Figure 18. n -gons. (cid:3) Example 3.7. ( An Explosion of Examples. ) Bicubic graphs can be readily generated asfollows. Let P (cid:48) be a polytope, not necessarily regular, and G (cid:48) its edge graph, i.e. G (cid:48) is theone skeleton of P (cid:48) . Suppose that P (cid:48) has v (cid:48) vertices, e (cid:48) edges and f (cid:48) faces. By definition, the xplosion of the polytope P (cid:48) is the polytope P formed by first truncating at the vertices andthen truncating the resulting polytope along the original edges of P (cid:48) . Then the 1-skeleton of P is cubic and has a unique bipartite coloring, up to an overall black-white swap, so thereforeis bicubic. Note that P has v = 4 e (cid:48) vertices, e = 6 e (cid:48) edges, and f = v (cid:48) + e (cid:48) + f (cid:48) faces.Even degenerate polytopes P (cid:48) give interesting examples. For instance, if P (cid:48) is the degeneratepolytope with two n -gon faces ( v (cid:48) = n, e (cid:48) = n, f (cid:48) = 2), then P is a 2 n -gon prism ( v = 4 n, e =6 n, f = 2 n + 2). The cube edge graph described in Example 3.5 is the bicubic graph whicharises when P (cid:48) has just two bigon faces. (cid:3) Diagrammatic Calculus For Legendrian Weaves
Let G ⊆ C be an N -graph. The geometric objects that we are interested in are the Legendrianweaves Λ( G ) ⊆ ( J C, ξ st ) and their invariants up to Legendrian isotopy. In this section weintroduce a series of combinatorial operations that can be performed to an N -graph G , andwe show how they affect the Legendrian isotopy type of Λ( G ). The geometric understandingof the Legendrian isotopy type through this diagrammatic calculus allows us to significantlysimplify computations of algebraic invariants associated to Λ( G ) in Section 5. Algebraiccomputations, using the results in this section, are detailed in Sections 6 and 7. Let us beginwith the combinatorial moves in G that preserve the Legendrian isotopy type of Λ( G ).4.1. Surface Reidemeister Moves.
Let Λ ⊆ ( J ( C ) , ξ st ) be a Legendrian surface, a Leg-endrian isotopy { Λ t } { t ∈ [0 , } will generically induce singularities of the Legendrian fibration J C −→ C × R . As a result, the front sets Σ(Λ t ) and their singularities will restructureas the parameter t ∈ [0 ,
1] ranges along a 1-parameter family. These modifications of theLegendrian fronts are referred to as perestroikas , or Reidemeister moves [Ad90, Chapter 3].
Remark 4.1.
The three classical 1-dimensional Reidemeister moves have been the mainmethod of study for smooth knots in geometric topology, since first introduced [Rei27, AB27].The corresponding seven moves for smooth surfaces are known as Roseman moves, after[Ros98, Theorem 1]. The corresponding Legendrian Reidemeister, and Legendrian Rosemanmoves, for Legendrian knots, and Legendrian surfaces, follow from the classification of (stable)wavefront singularities in dimensions dim(Λ) ≤ (cid:3) The combinatorial operations inducing surface Legendrian Reidemeister moves are the con-tent of the following theorem. In the moves, the local pieces of the N -graphs are actually 3-or 4-graphs. The color code follows our standard notation: blue and red are adjacent colors(corresponding to adjacent transpositions), red and yellow are adjacent colors, and blue andyellow are disjoint colors. Theorem 4.2.
Let G , G be one of the pairs of N -graphs depicted in Figures 19, 20, 21,22, 23, 24 and 25. Then the associated Legendrian surface Λ( G ) is Legendrian isotopicto Λ( G ) relative to their boundaries. That is, Moves I , II , III , IV , V , VI and VI’ are localsurface Legendrian Reidemeister moves. igure 19. (Move I) The first pair of local N -graphs G , on the left, and G on the right. We refer to this move as candy twist . Figure 20. (Move II) The second pair of local N -graphs G , on the left, and G on the right. We refer to this move as the push-through , since the trivalentvertex gets pushed through the hexagonal vertex. Figure 21. (Move III) The third pair of local N -graphs G , on the left, and G on the right. We refer to this move as the flop . Figure 22. (Move IV) The fourth pair of local N -graphs G , on the left,and G on the right. Note we must have N ≥
4. This moves implies the A generalized Zamolodzhikov relation depicted in Figure 105. Proof.
Let us start with Move I, the candy twist, as depicted in Figure 19. It illustratesthe method of proof for these surface Legendrian Reidemeister moves. There are essentiallythree equivalent viewpoints: exhibiting the Legendrian isotopy as N -graphs, visualizing thesurface wavefronts explicitly in ( J R , ξ st ), or studying these surface wavefronts as families of(possibly singular) Legendrian links. In the first perspective, we need to justify that all the N -graphs lift to embedded Legendrian surfaces. In the second, the challenge is visualizingthe actual front and ensuring that all the singularities lift to Legendrian embeddings. In igure 23. (Move V) The fifth pair of local N -graphs G , on the left, and G on the right, with N ≥
4. The blue and yellow colors are associated todisjoint transpositions.
Figure 24. (Move VI) The sixth pair of local N -graphs G , on the left, and G on the right, with N ≥ Figure 25. (Move VI’) Variation on the sixth pair of local N -graphs G , onthe left, and G on the right, with N ≥ not immediately lift to an embedded Legendrian surface, as the six-valentvertex is not a hexagonal vertex – the colors of the edges around it are not alternating, whichis the condition for the hexagonal vertices introduced in Section 2. Figure 26.
The 3-graph movie showing that the candy move - Move I - isa Legendrian isotopy. The geometric meaning of the central picture (not a3-graph) is explained in the text. evertheless, the center diagram in Figure 26 does in fact come from a Legendrian wavefrontwhose Legendrian lift is an embedded surface. Indeed, we have depicted such a front in thesecond front of Figure 27. Figure 27.
The homotopy of Legendrian wavefronts associated to Move I.The movie of wavefronts in Figure 27 geometrically constructs the homotopy of Legendrianfronts which lifts to the Legendrian isotopy corresponding to Move I. The three fronts in Fig-ure 27 lift to embedded
Legendrian surfaces, as the singularities are all Legendrian and thereare no vertical tangent planes. The singularities at the beginning of Figure 27 are segmentsof A -crossings, and two isolated A points. The singularities at the end of Figure 27 are justsegments of A -crossings. The singularity in the middle of the movie, not corresponding toan A segment, is not a stable front singularity, but it does lift to an embedded Legendriansurface, and thus the homotopy of fronts actually represents a Legendrian isotopy. Indeed,the tangent spaces at that singularity intersect transversely, and hence their lifts are disjoint.This concludes that Move I combinatorially represents a surface Legendrian Reidemeistermove. Remark 4.3.
For completeness, in Figure 28 we have drawn the homotopy of surface frontsfrom Figure 27 as a movie (of movies). It is thus a 2-homotopy of Legendrian links. Thesethree movies of links, one per each column, are obtained by slicing each of the respectivefronts in Figure 27 from left to right. This is the third viewpoint we mentioned above. (cid:3)
Let us now justify Move II, where a D − -singularity pushes-through an A -singularity. Theresulting front has a D − -singularity and two A -singularities. The clearest proof that thisis a Legendrian isotopy comes from carefully drawing and examining the right homotopy offronts. In this case, the required movie of fronts is depicted in Figure 29. These Legendrianfronts start with the front whose A -singularities yield the 3-graph G on the left of Move II,and end with the front whose A -singularities yield the 3-graph G on the right of Move II.These fronts describe a neighborhood R of a D − -singularity with a 2-plane Π ⊆ R whichstarts away from the D − -singularity. This 2-plane Π is drawn with a tilt in its slope. Thehomotopy of fronts consists of this 2-plane Π moving towards the D − -singularity and crossing through it. There exists a unique moment in this isotopy in which the D − -singularity is igure 28. The proof that Move I is a Legendrian isotopy by (transversely)slicing each of the Legendrian wavefronts in Figure 27.
Figure 29.
The homotopy of Legendrian fronts inducing Move II.contained in the 2-plane Π. The A -singularities right before that moment give rise to G for Move II, and right after this moment the A -singularities give rise to G for Move II.Since the 2-plane Π is not vertical, and the tangent 2-planes of the different branches atthe D − -singularity in all moments are distinct, each of the fronts in this homotopy lift toembedded Legendrian surfaces. Thus, the movie of fronts in Figure 29 shows that there existsa Legendrian isotopy with A -singularities as dictated by Move II, and Λ( G ) and Λ( G ) areLegendrian isotopic relative to their boundaries. This concludes Move II.For Move III, we can proceed analogously by drawing a homotopy of fronts which lifts toa Legendrian isotopy. Nevertheless, Move III can actually be deduced as a combination ofMoves I and II. We leave it as an exercise for the reader to visualize the spatial Legendrianfronts, and instead explain how to deduce Move III from the previous two moves. Startingwith one side of Move III, push both trivalent vertices through in the clockwise direction sing Move II. This creates additional hexagonal vertices and the two trivalent vertices dochange color. Perform Move II twice more, pushing-through these trivalent vertices again,and then cancel two pairs of hexagonal vertices with a candy twist (Move I) to obtain theright hand side of Move III.Let us now show that Move IV is a Legendrian isotopy. The corresponding spatial wavefrontsconsist of configurations of four 2-planes. The graph G on the left of Move IV is obtainedas the A -singularities, i.e. intersections, of the union of the four 2-planes π x = { ( x, y, z ) ∈ R : x + 0 . z = 0 } , π y = { ( x, y, z ) ∈ R : y + 0 . z = 0 } ,π z = { ( x, y, z ) ∈ R : z = 0 } , π = { ( x, y, z ) ∈ R : x + y + z = 1 } . These intersections and 2-planes are depicted, with the corresponding colors, in Figure 30.Now consider the 2-planes π t = { ( x, y, z ) ∈ R : x + y + z = t } , t ∈ [ − , π x ∪ π y ∪ π z ∪ π t , t ∈ [ − , Figure 30.
Front for the start of Move IV. The lines depict the intersectionsof the union of the four 2-planes π x ∪ π y ∪ π z ∪ π .This homotopy is not relative to the boundary, as the 2-planes π t , t ∈ [ − , π t , t ∈ [ − ,
1] through the triple intersection point π x ∩ π y ∩ π z . The A -singularity pattern ofthe resulting spatial wavefront is precisely as in the right graph G in Move IV, as required.Let us now address Move V, which depicts the local transition between two 4-graphs G , G in Figure 23. The corresponding spatial fronts consist of four 2-planes π , π , π , π ⊆ R ,where the only non-empty intersections are π ∩ π , corresponding to the blue segment in G (and G ), and π ∩ π , corresponding to the yellow segment in G , and G .The fact that the fronts giving G and G are homotopic as Legendrian fronts is proven inFigure 31. Each of the columns in the figure represents a spatial surface front, with thelinks in the columns corresponding to slices. The corresponding intersections, dictating the A -singularities, are marked with the same color as in Figure 23. The union of these slices inFigure 31 yield spatial fronts which lift to embedded Legendrian surfaces, and thus the movie igure 31. Each column represent slices of a spatial front. The A -singularities of the left column gives rise to G in Figure 23, and the A -singularities of the right column gives rise to G .of columns in Figure 31 exhibits a Legendrian isotopy from Λ( G ) to Λ( G ). Therefore, MoveV is a surface Legendrian Reidemeister move. Move VI in Figure 24 follows with the sameargument as for Move V, with a segment of A -singularities passing above, and disjointly, a D − -singularity — and likewise for Move VI’. This concludes the proof of Theorem 4.2. (cid:3) Remark 4.4.
The Legendrian Reidemeister moves in Theorem 4.2 provide a symplecticgeometric realization of A-type Soergel calculus. Moves I and V should be compared to[EW16, Figure 4.4]. Move II and Move VI are known as two-color associativity of type A × A , with Coxeter exponent m st = 2, and of type A , with Coxeter exponent m st = 3,and Move IV corresponds to the A relation [EW16, Figure 4.7]. It should be emphasizedthat the notation in Soergel calculus follows the notation for (rank three) parabolic subgroupof finite Coxeter groups, whereas we use the notation for Lie algebras whose irregular Weylorbits yield spatial wavefronts. See Appendix A for further details. (cid:3) Theorem 4.2 contains the Reidemeister moves that we use in the course of the article. Theyare all the possible (generic) Legendrian surface moves with only D − and A Legendriansingularities in the endpoints of the Legendrian isotopy. The complete set of surface Rei-demeister moves [Ad90, Section 3.3] also includes the moves associated to the A and D +4 -singularities, which will require the interaction of A -cusp edges A and A -swallowtails.Theorem 4.2 allows one to make local modifications to an N -graph G and obtain an N -graph G such that the Legendrian surfaces Λ( G ) ∼ = Λ( G ) ⊆ ( J C, ξ st ) are Legendrian isotopic.For the case C = S , we define in Subsection 4.7 an additional combinatorial move, which werefer to as a stabilization , going from an N -graph G to a ( N + 1)-graph G . This requires discussion on satellite constructions for Legendrian weaves, which is useful on its own, andalso needed for Subsection 4.5.4.2. Legendrian Satellite Weaves.
Let G ⊆ C be an N -graph. The Legendrian surfaceΛ( G ) defined by the weaving construction lies in the contact 5-manifold ( J C, ξ st ). Now,consider a contact 5-manifold ( Y, ξ ) and a Legendrian embedding ι : C −→ ( Y, ξ ). TheWeinstein Neighborhood Theorem [Wei71, Section 7] for Legendrian submanifolds gives acontactomorphism (cid:101) ι : ( J C, ξ st ) −→ ( O p ( ι ( C )) , ξ | O p ( ι ( C )) ) , where O p ( A ) is a sufficiently small neighborhood of A ⊆ Y , and such that the restrictionto the zero section C ⊆ J C is the initial Legendrian embedding ι . In particular, anyLegendrian Λ ⊆ ( J C, ξ st ) yields a Legendrian (cid:101) ι (Λ) ⊆ ( Y, ξ ). Thus, the contact 1-jet spacesserve as local contact manifolds, and a Legendrian embedding of C in an arbitrary ambientcontact 5-manifold allows one to embed a Legendrian weave there as well. In this context, theLegendrian surface (cid:101) ι (Λ) ⊆ ( Y, ξ ) is called the ι - satellite of Λ ⊆ ( J C, ξ st ) and the Legendriansurface ι ( C ) ⊆ ( Y, ξ ) is called the companion . This terminology parallels the theory ofsatellite knots, as introduced in [Sch53], and see also [NR13, EV18]. Notice that the smoothtopology of Λ and its satellite (cid:101) ι (Λ) is identical, only the ambient contact manifold (and thusthe Legendrian embedding type) are affected by this Legendrian satellite construction. Example 4.5.
Let ( Y, ξ ) = ( S , ξ st ) , C = S , and let ι = ι be the Legendrian embedding ofthe standard Legendrian unknot ι : S −→ S . Given any Legendrian Λ ⊆ ( J S , ξ st ) , wewill refer to (cid:101) ι (Λ) ⊆ ( S , ξ st ) as the standard satellite of Λ . Since ( S \ { pt } , ξ st ) ∼ = ( R , ξ st ) ,and the image (cid:101) ι (Λ) will avoid some point, this surface can be equivalently considered in (cid:101) ι (Λ) ⊆ ( R , ξ st ) ∼ = ( J ( R ) , ξ st ) . It can thereupon be described by its front projection to R = R × R . (cid:3) In case no Legendrian embedding ι is specified and C = S , the notation ι (Λ) will implicitlyrefer to the standard satellite (cid:101) ι (Λ) ⊆ ( R , ξ st ) as in Example 4.5. It is often the case thatthe Legendrians Λ( G ) ⊆ ( J C, ξ st ) that we introduce in this work do not have an a priori name nor they have been previously studied. Interestingly, for a certain variety of graphs G ⊆ S we will see how their standard Legendrian satellites are actually related to well-knownLagrangian surfaces, e.g. see Subsection 6.1.In addition, and in line with Markov’s Theorem for smooth 1-dimensional braids [Bir74,PS97], the satellite operation is also required for a meaningful stabilization operation. Finally,note also that even if Λ( G ) ⊆ ( J C, ξ st ) has no A -cusp edges, the spatial wavefronts for itsstandard satellite (cid:101) ι (Λ( G )) will always have A -cusp edges, as any front for the standardLegendrian unknot Λ ⊆ ( R , ξ st ) must have A -cusp edges. We now discuss A -cusp edgesand A -swallowtail singularites, which are required for such a stabilization operation andTheorems 4.10 and 4.21 below, regarding Legendrian surgeries and Legendrian mutations.4.3. Cusp Edges and Swallowtail Singularities.
Let G ⊆ C be an N -graph, the Legen-drian weave Λ( G ) ⊆ ( J C, ξ st ) associate to G is determined by its front π (Λ( G )) ⊆ C × R .By definition, these fronts only have D − , A and A singularities. The latter two are stable,i.e. a generic Legendrian isotopy Λ t ⊆ ( J C, ξ st ), t ∈ [0 , G ) = Λ , will haveeach of the A and A singularities of the front π (Λ ) persist for π (Λ t ), t ∈ (0 , D − is not: the fronts π (Λ t ), t ∈ (0 , ε ], will not have any D − -singularity for ε ∈ R + smallenough. igure 32. The Legendrian front of an A -swallowtail singularity (left). Theplanar diagrammatic depiction in our calculus (right).The generic (stable) singularities of fronts in 3-dimensional space are A , A , A , A A and A ,as shown in [Ad90, Section 3.2]. These singularities are depicted in Figure 33. The appearanceof A , A A and A singularities in a generic front forces us to extend our combinatorialdiagrammatics, as our Legendrian isotopies will (typically) be generic. In the figures for thissubsection, and only this subsection, we will draw edges around a hexagonal vertex withthe same color – this will simplify our diagrams, which are no longer N -graphs due to thepresence of A -cusp edges. Figure 33.
The generic Legendrian singularities of wavefronts in 3-space.The depicted A -singularity is known as the A -swallowtail, and the center A -singularity in the first row is referred to as the A -cusp edge. Note thatthe two D ± -singularities are not generic.We extend the diagrammatics with the following rule: orange segments will denote A -cuspedges of singularities, and orange dots will stand for A -swallowtail singularities. Figure 32depicts on its left a genuine spatial front for the A -swallowtail singularity. The singularitiesof this front consist of a segment of A -crossings, shown in blue, two A -cusp edges, in orange,and a unique A -swallowtail point. The planar diagram through which we represent this frontis shown on the right of Figure 32. It is simply a vertical view of the front (from above orbelow) with the A , A and A -singularities marked. Remark 4.6.
For the same reasons that we label A singularities with transpositions, inorder to indicate which two sheets are crossing, we should label A -cusp edges with the orresponding information. This is necessary information in order to recover the actual(homotopy type of the) Legendrian front, and thus the Legendrian itself. That said, in thisarticle, it should be clear from context where such A -cusp edges lie, so these labels will beomitted. (cid:3) The D − -singularities are the central pieces in the construction of our Legendrian weavesΛ( G ) ⊆ ( J C, ξ st ). It is important to emphasize that D − is not a generic singularity of a realspatial front, despite the fact that its complexification is a stable holomorphic Legendriansingularity. In particular, in our upcoming study of Legendrian surgeries, we will needgeneric Legendrian isotopies starting at Λ( G ), whose fronts will break the non-generic D − into generic singularities of real spatial wavefronts.The generic deformation of the D − -singularity is depicted in Figure 34 (left). It containsthree A -swallowtails arranged in a triangle and connected by A -cusp edges. Following ourconvention above, the associated planar diagram is shown in Figure 34 (right). Figure 34.
The spatial wavefront for a generic perturbation of the D − -singularity (left). The associated planar diagram for this stable spatial wave-front (right). Note that the A -edges around the hexagonal vertex all drawnwith the same color (blue), following the convention in this subsection.4.4. Legendrian Front Calculus with Cusp Singularities.
Let us continue our devel-opment of a diagrammatic front calculus for Legendrian surfaces, this time including A -cuspedges and A -swallowtails. Proposition 4.7 below is used to prove Proposition 4.9 and alsoTheorem 4.10, in the upcoming Subsection 4.5. Proposition 4.7.
Let G ⊆ C be an N -graph, N ∈ N . The four moves in Figure 35 areachieved by compactly supported Legendrian isotopies, relative to the boundary.Proof. Moves VII and VIII, on the creation and fusion of two A -swallowtails singulari-ties are immediate from the 3-dimensional First Reidemeister Move R1. Indeed, the left-to-right 1-dimensional Legendrian slices in Move VII correspond to a concatenation of R1and its inverse, i.e. an R1 is performed, corresponding to the appearance of the leftmost A -swallowtail, and then the same R1 is undone, corresponding to the appearance of therightmost A -swallowtail. This movie of 1-dimensional Legendrian slices can be isotoped toa movie with no R1 fronts, whose (big) front corresponds to the right of Move VII, withno swallowtails. For Move VIII, the R1 moves are performed in reverse order. That is, theleft-to-right 1-dimensional Legendrian slices correspond to the inverse of an R1 move (a pairof cusps being undone) and then the exact same R1 move. This homotopy of 1-dimensional igure 35. The five Legendrian front moves in Proposition 4.7. The movesare referred to as Move VII (upper left), Move VIII (upper right), Move IX(center left), Move X (center right) and Move XI (lower center).Legendrian fronts can be itself homotoped to a constant homotopy, which the local N -graphdepicted in the right of Move VIII.For Move IX, we proceed with our slicing techniques. The 1-dimensional vertical left-to-rightslices of the two fronts for Move IX are depicted in the left and right columns of Figure36. In the left column, the Reidemeister R1 move is performed for the upper piece of the1-dimensional Legendrian knot. In the right column, the R1 move is performed for the lowerpiece of the 1-dimensional Legendrian knot. The homotopy of Legendrian surface fronts isachieved by the center column in Figure 36, where both R1 are performed simultaneously.Since the homotopy of fronts preserves the boundary conditions, this lifts to a Legendrianisotopy of embedded Legendrian surfaces, thus proving that Move IX is a Legendrian Rei-demeister move. The fact that Move IX is a Legendrian Reidemeister move also followscarefully from visualizing the critical fronts associated to the generating family D +4 : F ( x, y, ξ , ξ , ξ ) = x y + y + ξ y + ξ y + ξ x, which leads to the above families in Figure 36.Move X consists of a sliding for a A -swallowtail along an A -crossing line, as depicted inFigure 37. The realistic surface fronts are depicted in the right of Figure 37, where the A -swallowtail singularity has been moved past the A -segment of singularities.The sliding lifts to a Legendrian isotopy, as the interaction between the A -swallowtail andthe A -line only sees a critical moment, where a A A singularitiy appears. At this criticalstage, the slopes are all distinct and non-vertical, thus the A -swallowtail is allowed to movepast with a homotopy of fronts. This concludes that Move X is a Legendrian Reidemeistermove. igure 36. The homotopy of surface fronts showing that Move IX is a Legen-drian Reidemeister move. The left-to-right slices for the left diagram in MoveIX are depicted in the left column, whereas the slices for the right diagram inMove IX are depicted in the right column.
Figure 37.
The front depiction of the non-trivial part in Move X. The A -swallowtail singularity slides across a orthogonal A -line, changing sheets asit slides through. igure 38. The homotopy of fronts for Move XI. The left front diagram ofMove XI is obtained as the union of the slices in the left column, whereas theright front in Move XI is the union of the slices in the right column.Finally, Move XI is proven in Figure 38. The middle singularity corresponds to the genericspatial front A A -singularity. In short, Move XI is obtained by performing a homotopywhich interpolates between a constant movie of Legendrian links, and a movie consisting ofdoing a Reidemeister R2 move and then undoing it, as in the left column of Figure 38. (cid:3) Remark 4.8.
It would appear that Reidemeister moves for Legendrian knots have beenmastered by the vast majority of contact topologists. This does not seem to be the case inhigher dimensions, including the Legendrian singularities appearing in surface fronts. Shouldthe reader be interested in that, [Ben86, Ad90] provides a starting presentation of the genericsingularities of surface fronts. Our present manuscript develops the diagrammatic calculusadding to that classification, which allows us to manipulate fronts in a versatile manner. Thecombination of the results of this article, along with [Ad90], should permit the reader to befluent in the manipulation of wavefronts for Legendrian surfaces in contact 5-manifolds. (cid:3)
Let us now address the move shown in Figure 39, which we prove in the following:
Proposition 4.9.
The combinatorial move depicted in Figure 39 is realized by a compactlysupported Legendrian isotopy of surfaces in a 5-dimensional Darboux ball ( J R , ξ st ) , relativeto the boundary.Proof. Let us start with the left front in Figure 39. Apply Move VII to create a canceling pairof A -swallowtails, as shown in the beginning of Figure 40. Now slide the A -swallowtail byperforming a Move X, and use the D +4 -singularity, i.e. Move IX to exchange the A -cusp edgewhere the A -swallowtail connects. This is depicted in the first and second steps of Figure40. The next two steps in Figure 40 consists of Legendrian isotopies where no singularitiesinteract with each other, it is a plain homotopy of fronts with the same singularities. Finally,the last step consists in joining the three existing A -swallowtails into a single D − -singularity,as depicted at the end of Figure 40. igure 39. (Move XII) This move allows us to exchange A -swallowtail sin-gularities with D − -singularities in the presence of a A -cusp edge. Figure 40.
The homotopy of fronts for Move XII. The initial A -swallowtailrequires two additional swallowtails to become a D − -singularity, and certainintermediate moves. The homotopy realizing this can be read in this picture. (cid:3) Legendrian Surgeries.
The theory of Legendrian surgeries was initiated in [Ad76,Ad79] in the study of critical points of the time function with respect to a Legendrianwavefront. Its modern description in terms of Lagrangian handle attachments is describedin [BST15, Theorem 4.2] and [DR16, Section 4]. A Legendrian surgery on Λ ⊆ ( Y, ξ ) isan operation which inputs an isotropic sphere within Λ, bounding ambiently, and outputs aLegendrian (cid:101) Λ ⊆ ( Y, ξ ). The Legendrians Λ and (cid:101)
Λ are not even homotopy equivalent, and thusLegendrian surgery is a useful method to create new
Legendrians by modifying the topologyof a given Legendrian Λ.In the context of Legendrian surfaces, there are different types of Legendrian surgeries [Ad90,Figure 48]. The following result characterizes the combinatorial operations that correspondto Legendrian 0-surgeries, 1-surgeries and Legendrian connected sums. heorem 4.10 (Legendrian Surgeries) . Let G ⊆ C , G ⊆ C be N -graphs and G ⊆ C an M -graph, for N, M ∈ N . The following statements hold:
1. ( ) The combinatorial move of adding an i -edge and two vertices along anexisting i -edge corresponds to a Legendrian -surgery. This move is shown in theupper right diagram in Figure 41.
2. ( ) The combinatorial move of removing an i -edge between two trivalent ver-tices corresponds to a Legendrian -surgery. This move is shown in the lower left ofFigure 41.
3. (
Connect Sum ) The kissing of two trivalent vertices v ∈ G and v ∈ G , where G ⊆ C , G ⊆ C are two disjoint graphs, corresponds to a connect sum ι (Λ( G )) ι (Λ( G )) ⊆ ( R , ξ st ) , for any satellite ι : Λ −→ ( R , ξ st ) . This is shown in the upper left of Figure 41.
4. (
Clifford Sum ) The combinatorial move of substituting a trivalent vertex by a trianglecorresponds to a connected sum of ι (Λ( G )) with a Clifford 2-torus T c ⊆ ( R , ξ st ) .This move is shown in the lower right of Figure 41.The 0-surgeries, 1-surgeries are local in any Λ( G ) ⊆ ( J C, ξ st ) . In contrast, the connectedsum in the third item requires to geometrically satellite the Legendrian weaves Λ( G ) ⊆ ( J C , ξ st ) and Λ( G ) ⊆ ( J C , ξ st ) via any Legendrian embedding ι : C ∪ C −→ ( R , ξ st ) . Figure 41.
The Legendrian Surgery Moves in Theorem 4.10Theorem 4.10 will be proven below. The Legendrian weaves in the statements involve only D − and A (and A -cusp edges for the connected sum, due to the satellite operation). Nev-ertheless the manipulation of their fronts in the proof of Theorem 4.10 requires the use offurther Legendrian front moves, involving A -swallowtails and A A -singularities and theirinteraction with the A , A and D − -germs, as developed in Subsection 4.4 above. Remark 4.11.
Should the reader be solely interested in the satellited Legendrian surface ι (Λ( G )) ⊆ ( R , ξ st ), the connected sum operation in Theorem 4.10.(3) is the strongest of the our statements (and the hardest to prove). Indeed, the satellite analogue of Items 1,2 and 4follow from Item 3. That said, Items 1,2 do not follow from Item 3 locally. (cid:3) In combination with Theorem 4.2, Theorem 4.10 yields the following two moves:
Corollary 4.12.
The two N -graph moves in Figure 42 corresponds to a Legendrian 1-surgery, i.e. upon performing (3’), or (3”), there exists an elementary index-2 exact La-grangian cobordism from the Legendrian weave on the left to the Legendrian weave on theright.In fact, in Move (3’) the Lagrangian 2-disk is attached along the 1-cycle represented by the(bi)chromatic horizontal edge between the two trivalent vertices. In Move (3”) the Lagrangian2-disk is attached along the 1-cycle represented by the (blue) tripod at the hexagonal vertexuniting the three trivalent vertices. Figure 42.
The two Legendrian Surgery Moves in Corollary 4.12, both rep-resenting Lagrangian 2-handle attachments.
Proof of Theorem 4.10.
We start by proving that adding an i -edge with two trivalent verticesto an existing i -edge effects a 1-surgery. The homotopy of spatial fronts is depicted in Figure43, according to the conventions in Subsection 4.3. The detailed description reads as follows.We first generically perturb the two D − -singularities in the first spatial front, which yieldsthe second front. Performing Move VIII and then Move I yields the third and fifth fronts,respectively, in Figure 43. Note that the homotopy from the third to the fourth front doesnot involve any change in the singularities of fronts, as the blue segment of A -singularitiesintersecting the orange A -cusp segment lies strictly below it in 3-space. The homotopy fromthe fifth to the sixth front emphasizes the yellow band where the (reverse) 1-surgery is to beperformed. The step from the sixth to the seventh fronts is precisely the reverse surgery: the A -cusp edges in the seventh front are surgered along the yellow band [Ad76, BST15], in thesixth front, to obtain the fifth front. The seventh front is homotopic to the eighth front byMove VII.Let us now show that removing an i -edge corresponds to a Lagrangian 2-handle attachment.The homotopy of fronts is depicted in Figure 44. Starting with the first front, genericallyperturbing yields the second front and two applications of Move VIII give the third front. Inthe fourth front we have shown the Legendrian 2-disk along which we perform the 2-surgery[Ad90, BST15], the result of which is the fifth front. Indeed, the 2-surgery opens up theinner circle of A -cusp edges and adds two horizontal (Legendrian) 2-disks. As a result, theeffect on its diagrammatic representation is removing the inner circle of A -cusps, as shownin the fifth front. The application of Move I gives the sixth front, which is readily homotopicto the seventh front. The eighth front is then obtained by performing a Move VII. igure 43. The diagrammatic homotopy of spatial fronts associated to theLegendrian 1-surgery move. It shows that the first front is a Legendrian 1-surgery on the eight front.
Figure 44.
The diagrammatic homotopy of spatial fronts associated to theLegendrian 2-surgery move. It shows that the first front is a Legendrian 2-surgery on the eight front.Now, we prove that joining two trivalent vertices in distinct graphs G ⊆ C , G ⊆ C is realized by a Legendrian surface connected sum, which is a 1-surgery whose attaching -sphere has its two points belonging to different boundary components. The require homo-topy of fronts is shown in Figure 45. In this case, we must satellite the Legendrian weavesΛ( G ) , Λ( G ) to a Darboux ball ( R , ξ st ). From the perspective of spatial fronts, we mustlocally add a A -curve and two A -cusp edges as depicted in the first front of Figure 45. TheLegendrian 1-surgery is performed from the first front to the second, along the Legendrianband given by the red dotted line. The homotopy from the second front to the third consistsof four applications of Move XI. Then, we use Move XII to obtain the fourth front. The fifthfront is achieved by applying Move VII, and the sixth front consists of two applications ofMove XI. Figure 45.
The diagrammatic homotopy of spatial fronts associated to theLegendrian connected sum.Finally, substituting a trivalent vertex by a triangle corresponds to a connected sum with thefour vertex graph G c ⊆ S in the left of Figure 46. It is an exercise to show that the spatialfront of the Legendrian weave ι (Λ( G c )) ⊆ ( R , ξ st ) is front equivalent to the front on the rightof Figure 46, which is known to be the Legendrian lift of the Clifford torus [DR11, CM19]. Figure 46.
The Clifford graph G c ⊆ S and a simplified spatial front for thesatellited Legendrian ι (Λ( G c )) ⊆ ( R , ξ st ). (cid:3) roof of Corollary 4.12. In Figure 42, Move (3’) follows by applying a sequence of MovesII to the leftmost trivalent vertex, pushing that vertex through all the hexagonal vertices –until it is connected to the rightmost trivalent vertex with a monochromatic edge – and thenusing Move (3) in Theorem 4.10. Move (3”) is more interesting, and its proof is shown inFigure 47.
Figure 47.
The Lagrangian 2-handle attachment in Move (3”) decomposedas a sequence of surface Reidemeister moves, from Theorem 4.2, and Move(2) in Theorem 4.10, in the guise of Corollary 4.12. (cid:3)
Theorem 4.10 provides a useful and efficient way to describe Legendrian surfaces in termsof N -graph combinatorics. Its statement is as strong as possible, in that the conclusion ison the Legendrian isotopy type of the associated Legendrian weaves. The computation ofalgebraic invariants then follows as a consequence of our geometric understanding.In particular, we have following. Corollary 4.13.
Let G ⊆ C be an N -graph and v ∈ G a trivalent vertex. The blow-upcombinatorial move on G , given by an insertion of a triangle at the vertex v , is a twisted -surgery on ι (Λ( G )) . (cid:3) The blow-up procedure was first studied in [TZ18, Section 5]. It is depicted in Figure 41(lower right). By definition, a twisted 0-surgery is a connected sum with a non-standardLegendrian torus in ( S , ξ st ). For now, we refer to [DR11, Section 4] for more details.The immediate consequence of Corollary 4.13 is that the Legendrian isotopy type of ι (Λ) isindependent of the choice of vertex v ∈ G . This question was initially asked in [TZ18] in thestudy of the dependence of the sheaf invariants in terms of v . Since the Legendrian isotopytype of ι (Λ) is independent of v , the algebraic invariants are also independent of v .Finally, note that the Legendrian 0-surgery in Theorem 4.10.(i) can be understood as aLegendrian connected sum with the 2-graph G ⊆ S shown in Figure 48 (Left). In fact, thestandard Legendrian satellite ι (Λ( G )) for this 4-vertex 2-graph is the standard Legendrian2-torus, a Legendrian front of which is shown in Figure 48 (Right). igure 48. A 2-graph G in the 2-sphere S (Left) and a Legendrian frontfor its Legendrian weave ι (Λ( G )) (Right). This is the standard Legendrian 2-torus T ⊆ ( R , ξ st ), given by Legendrian front spinning of the 1-dimensionalstandard Legendrian unknot Λ ⊆ ( S , ξ st ). Remark 4.14.
The Legendrian 0- and 1-surgeries in Theorem 4.10 physically correspondto partial puncture degenerations in the context of spectral networks [GMN13, GMN14].Indeed, the Legendrian weaves obtained as the Legendrian lift of the Lagrangian hyperk¨ahlerrotation of the spectral curve of a diagonalizable Higgs field are related by the Legendriansurgeries in Theorem 4.10. For instance, the process of a full puncture [1,1,1] degenerating toa simple [2,1] puncture in a punctured 3-sphere is precisely a Legendrian 0-surgery [GLPY17,Section 6]. (cid:3)
The Reidemeister moves and the stabilization operation in Subsections 4.1 and 4.7 preservethe Legendrian isotopy type of the (satellite) Legendrian weaves. The Legendrian surgeriesdiscussed in Theorem 4.10 generically change the topology of Λ( G ). The natural next step isto modify the Legendrian isotopy type of Λ( G ) without changing its topology, which we willdiscuss in Subsection 4.8. For now, we study an explicit example and present the stabilizationoperation.4.6. Example of a Closed Legendrian Weave.
Let us illustrate our spatial front calculusin an example. Consider the triangulation of C = S given by a tetrahedron, and the 3-graph G associated to this triangulation according to Section 3. This 3-graph is shown in Figure49 (upper left). The 3-graph G is depicted in the plane as an unfolded triangulation, thusthe triangles should be identified according to the faces of the tetrahedron: the outer threevertices of the dashed triangle are identified, and the dashed lines are glued accordingly. Inparticular, the 3-graph G has twelve trivalent vertices and four hexagonal vertices. The ques-tion is to describe the Legendrian isotopy type of this Legendrian surface ι (Λ( G )) ⊆ ( R , ξ st ).In addition, we would like to compute Legendrian invariants, such as the augmentation va-riety of 3-dimensional Lagrangian fillings in ( D , ω st ). In this context, understanding theLegendrian isotopy type readily implies the computation of this Legendrian invariant.We will exploit Theorem 4.2 and Theorem 4.10 to understand this Legendrian weave, andnote that the closed surface ι (Λ( G )) := (cid:101) ι (Λ( G )) has genus 4 . First, we describe the sequenceof Legendrian moves and surgeries in Figure 49. In Diagram (1) on the upper left, first notethat there are three blue triangles each having one vertex in the central triangle, one eachin two outer triangles, and passing through one glued edge. There is another blue trianglewith one vertex on each of the outer triangles. By Theorem 4.10, we conclude that Diagram(1) corresponds geometrically to a connected sum of the weave from Diagram (2) with fourcopies of the Clifford 2-torus T c . The 3-graph of Diagram (2) is still complicated, so we useTheorem 4.2 to simplify. First apply Move III, flopping the four vertices in the upper right ofthe 3-graph. This brings us to Diagram (3). Now do a Move I to undo the newly appearingcandy twists. igure 49. Simplification of a 3-graph with Theorem 4.2 and Theorem 4.10.This brings us to Diagram (4). So we have proven that the standard satellite ι (Λ( G )) isLegendrian isotopic to ι (Λ( G (cid:48) )) i =1 T c , where G (cid:48) is the 3-graph in Diagram (4) of Figure 49.It now suffices to understand the Legendrian ι (Λ( G (cid:48) )) ⊆ ( R , ξ st ). Assertion : Let G (cid:48) ⊆ S be the 3-graph in Figure 50 (upper left). The Legendrian 2-sphere ι (Λ( G (cid:48) )) is Legedrian isotopic to the standard Legendrian unknot Λ ⊆ ( R , ξ st ). Proof of the assertion : By Theorem 4.10, we can undo the two bigons in Diagram (5)of Figure 50, and understand them as two connect sums with the standard Legendrian 2-torus T , defined as any Lagrangian 1-handle attachment to the standard Legendrian unknotΛ ⊆ ( R , ξ st ).By applying Move I in Theorem 4.2 to the 3-graph in Diagram (6), we arrive at the 3-graph G (cid:48)(cid:48) in Diagram (7) of Figure 50, simplifies to the three concentric circles of alternating colors inDiagram (8). The Legendrian weave ι (Λ( G (cid:48)(cid:48) )) ⊆ ( R , ξ st ) is readily seen to be the standard3-component unlink Λ ∪ Λ ∪ Λ ⊆ ( R , ξ st ). Hence ι (Λ( G (cid:48)(cid:48) )) is obtained by performingLagrangian 1-handle attachments to Λ ∪ Λ ∪ Λ ⊆ ( R , ξ st ), and thus ι (Λ( G (cid:48)(cid:48) )) must be thestandard Legendrian unknot. (cid:3) The conclusion of the above discussion is that the Legendrian isotopy type of the Legendriansurface ι (Λ( G )) ⊆ ( R , ξ st ) associated to 3-triangulation of the tetrahedron, i.e. Diagram (1)of Figure 49, is that of the connected sum of four copies of the Clifford 2-torus T . Hence, wenow have a complete geometric understanding of ι (Λ( G )). In particular, this readily implies[Siv11, DR11] that the C -moduli of objects of the category of microlocal rank-one sheaves in R supported in ι (Λ( G )) is isomorphic to ( C \ { , } ) . (cid:3) N -Graph Stabilization. The Reidemeister moves introduced in Theorem 4.2 consti-tute combinatorial operations on a given N -graph G which yield the same Legendrian isotopy igure 50. Diagrammatic proof that the standard satellite of the Legendrian2-sphere associated to the 3-graph in Diagram (5) is the standard Legendrianunknot two-sphere Λ ⊆ ( R , ξ st ).type for the associated Legendrian weave Λ( G ), as a Legendrian in ( J C, ξ st ). In particular,the resulting graph is still an N -graph.In this section we discuss a different type of combinatorial move, where the number of sheets N ∈ N is increased. This operation, which we call stabilization , inputs an N -graph G ⊆ C andoutputs an ( N + 1)-graph s ( G ) ⊆ C . The main property of stabilization, proven in Theorem4.17 below, is that it preserves the Legendrian isotopy type of the standard Legendriansatellite ι (Λ( G )) ⊆ ( R , ξ st ), and as a result it is a non-characteristic operation. Remark 4.15.
The relative homology class of the surface Λ( G ) ⊆ J C has order N , and thusno combinatorial operation that modifies the number N ∈ N of sheets for a Legendrian weavewill ever yield a Legendrian isotopic surface in the 1-jet space J C . Therefore, preserving theLegendrian isotopy type for the (standard) satellite is the optimal statement for a stabilizationoperation. (cid:3) Let us describe the Legendrian weave stabilization. Given an N -graph G , the first step is tointroduce a ladybug trivalent graph B in ( N, N + 1) as depicted in blue in the left of Figure51 in such a way that G is completely contained in one face of B , i.e. G is inside one of thewings of the ladybug B . The second step is the introduction of descending halos centered atan ( N + 1)-graph G , which consists of a nested set of N − A -crossings indexed bythe permutations ( N − , N ) , ( N − , N − , · · · (23) , (12) reading outward. This is depictedin the right of Figure 51.The concatenation of these two operations leads to the following: Definition 4.16.
Let G ⊆ C be an N -graph. The stabilization of G is the ( N + 1)-graph s ( G ) ⊆ C obtained from G by placing a ladybug B around G , labeled with the transposition( N, N + 1), and a sequence of descending halos centered at the ( N + 1)-graph G ∪ B . (cid:3) The construction is independent of the choice of such face. igure 51. Ladybug graph B around G (left) and halos centered at G (right).Figure 52 depicts the stabilization for the cases N = 2 ,
3. The ladybug graph B is shown inblue. Figure 52.
Stabilization of a 2-graph (left) and of a 3-graph (right).The stabilization in Definition 4.16 is the Legendrian surface generalization of the TypeII Markov move for smooth N -strand braids [Mar35, Bir74]. The main property of graphstabilization is the following geometric result: Theorem 4.17.
Let G ⊆ S be an N -graph. Then the standard satellites ι (Λ( G )) and ι (Λ( s ( G ))) are Legendrian isotopic in ( S , ξ st ) .Proof. Let us provide a detailed proof for the case N = 2, where the stabilization is a 3-graph. The argument for higher N ≥ ι (Λ( s ( G ))), which yields the diagram on the left of Figure 53. Indeed, the standard satelliteclosure of a 3-graph introduces three circles of A -crossings, drawn in dark grey, and threecircular cusp edges, drawn in orange. Perform a Legendrian isotopy which exchanges the(12)-circle of A -crossings with the adjacent (34)-circle of A -crossings; this gives the diagramin the right of Figure 53. This move is possible thanks to the cusp sliding shown in the firsttwo columns of Figure 54. For a general N -graph, a front for the standard satellite closure of the Legendrian weave contains N additional sheets, ( N + 1) , ( N + 2) , . . . , N . The bottom N sheets 1 , . . . , N are woven according to G , andthe top horizontal N sheets are parallel. The bottom and top sheets are then connected by circles worth of A -crossings, according to the half-twist ∆ ∈ Br + N , and N circles worth of A -cusp edges. igure 53. Exchange of (12) and (34) circles of A -crossings. Figure 54.
The left three diagrams depict slices in the dotted segments forFigure 53. The rightmost diagram depicts a slice for the dotted segment inthe right of Figure 55.Then use the innermost cusp circle and perform a Move XI, also denoted R as it consists oftwo Reidemeister I moves, to remove two of the A -crossings as in the left of Figure 55, thiscorresponds in the slice to the third column of Figure 54. Iterate with an R in the samecusp edge with the (34)-circle of crossings and the ladybug piece B , arriving at rightmostdiagram in Figure 55.Finally, eliminate the two half-moons in the cusp edge and isotope the cusp edge above thegraph G (1 , , which is possible thanks to the configuration shown at the rightmost columnof Figure 54. The resulting diagram is that on the left of Figure 56, which is Legendrianisotopic to the diagram on its right by applying two butterfly moves and an inverse R . (cid:3) Reidemeister moves in Subsections 4.1 and the Stabilization in Theorem 4.17 complete theset of combinatorial moves that is available to us when manipulating an N -graph, should theLegendrian isotopy type of the associated (satellite) Legendrian weave be preserved.4.8. Legendrian Mutations.
We now discuss the N -graph combinatorics of Legendrianmutations, a new geometric operation that we define in this manuscript. This operationinputs a Legendrian surface Λ ⊆ ( R , ξ st ) and an isotropic 1-cycle γ ⊆ Λ, and outputs aLegendrian surface µ γ (Λ) ⊆ ( R , ξ st ). The Legendrian surface µ γ (Λ) ⊆ ( R , ξ st ) will be igure 55. Performing an R -move with (34) and the ladybug. Figure 56.
From N = 2 to N = 3 (left) and N = 3 to N = 4 (right).ambiently (relatively) smoothly isotopic to Λ, and oftentimes not Legendrian isotopic to Λ.The choice of notation aims at emphasizing its relation to the wall-crossing phenomenon[GMN10, KS10, KS14], Lagrangian mutation [Pol91, Aur07, Aur09] and [FOOO09, Chapter10], and quiver mutations [FZ02, Via14]. Definition 4.18.
Let G ⊆ C be an N -graph and e ∈ G and i -edge between two trivalentvertices. The mutation of G along e is the N -graph µ e ( G ) obtained by performing theexchange depicted in Figure 58 (left), also shown in Figure 4 (3). (cid:3) By Theorem 4.21 below, the Legendrian weaves Λ( G ) and Λ( µ e ( G )) will be mutation-equivalent, according to the upcoming 4.19 – this motivates Definition 4.18 from the perspec-tive of contact topology. Note that the operation in Definition 4.18 is the simplest possiblemutation, corresponding to the combinatorics associated to a Whitehead move, i.e. an edgeflip in the context of triangulations dual to 2-graphs. Indeed, consider the two unique non-degenerate triangulations T , T of the square, the dual 2-graphs G , G differ precisely by amutation along their unique internal edge.Correspondingly, the standard satellites of their associated Legendrian weaves are two Leg-endrian cylinders with coinciding Legendrian boundary, smoothly isotopic relative to theirboundary but which are not Legendrian isotopic relative to their boundary.In general, given a 1-cycle γ ∈ Λ( G ) which is expressed combinatorially in G , it is pos-sible to describe the mutation of G along such 1-cycle γ . The mutated graph µ γ ( G ) caneither be defined in an ad hoc way, or rather be understood as a graph which is equivalentvia Reidemeister moves, as in Subsection 4.1, to the mutated graph µ e ( γ )( G (cid:48) ). Here G (cid:48) s Reidemeister equivalent to G and e ( γ ) is an i -edge between trivalent vertices such that[ e ( γ )] = [ γ ] ∈ H (Λ( G ) , Z ) under the canonical identification H (Λ( G ) , Z ) ∼ = H (Λ( G (cid:48) ) , Z )given by a Legendrian isotopy. Here is the definition: Definition 4.19 (Legendrian Mutation) . Two Legendrian surfaces Λ , Λ ⊆ ( R , ξ st ) aremutation-equivalent if and only if there exists a compactly supported Legendrian isotopy { (cid:101) Λ t } t ∈ [0 , relative to the boundary ∂ Λ , with (cid:101) Λ = Λ , and a Darboux ball ( B, ξ st ) such that(i) The two restrictions (cid:101) Λ | ( R \ B ) = Λ | ( R \ B ) coincide away from this Darboux ball,(ii) There exists a global front projection π : R −→ R such that each of the spatialfronts π | B ( (cid:101) Λ ) and π | B (Λ ) respectively coincide with each the of two fronts in Figure57. (cid:3) Figure 57.
Legendrian mutation in a local spatial wavefront.The two fronts depicted in Figure 57 coincide at their boundaries and lift to Legendriancylinders. These Legendrian cylinders are not
Legendrian isotopic relative to their boundary.Indeed, compactifying the upper sheet of the fronts with an A -cusp edge and a flat 2-disk,and the lower sheet with a different A -cusp edge and a flat 2-disk, yields the standardLegendrian unknot Λ ⊆ ( R , ξ st ) for the left front in Figure 57, and a loose Legendrian2-sphere s (Λ ) for the right front in Figure 57. The Legendrians Λ , s (Λ ) ⊆ ( R , ξ st ) are notLegendrian isotopic [EES05a, EES05b].A strong motivation for the study of the above mutations is the production of Legendriansurfaces which are not Legendrian isotopic, even though they belong to the same formalLegendrian isotopy class [Gro86, EM02]. In order to distinguish Legendrian isotopy classeswe will be using flag moduli spaces , which synthesize Legendrian invariants coming from thestudy of microlocal sheaves in terms of algebraic geometry.
Remark 4.20.
The conic Legendrian singularity for the front in Figure 57 (left) is not ageneric singularity. It is explained in detail in [DR11, CM19], and its generic perturbationcontains four A -swallowtail singularities. (cid:3) Theorem 4.21 (Legendrian Mutations) . Let G , G be one of the pairs of N -graphs depictedin Figure 58. Then the associated Legendrian surface Λ( G ) is a Legendrian mutation of Λ( G ) relative to their boundaries.Proof. Let us start by showing that the exchange move in Figure 58 (left) corresponds toa Legendrian mutation, as in Definition 4.19. By [CMP19, Theorem 6.3], the Lagrangianprojections Π(Λ ) , Π(Λ ) ⊆ R of the Legendrian lifts of the fronts π (Λ ) , π (Λ ) ⊆ R inFigure 57 correspond to the two Polterovich surgeries associated to the normal crossing oftwo Lagrangian planes R × { } , { } × R ⊆ ( R , ω st ). The Lagrangian projection of the igure 58. The Legendrian Mutation Moves in Theorem 4.21Legendrian lifts for each two 2-graphs in the exchange move in Figure 58 (left) are exactLagrangian fillings L , L of the Hopf link Λ Hopf ⊆ ( S , ξ st ) ∼ = ∂ ( R , ω st ). It thus suffices toshow that L , L ⊆ ( R , ξ st ) are the positive and negative Polterovich surgeries of the twoLagrangian planes R × { } , { } × R ⊆ ( R , ω st ) at their intersection points. Indeed, Figure59 (center) depicts the 2-graph for the singular Legendrian whose Lagrangian projections isthe Lagrangian union ( R × { } ) ∪ ( { } × R ). Figure 59.
The 2-graphs associated to a Legendrian Mutation. The middle2-graph yields a spatial front which lifts to a singular
Legendrian surface, con-sisting of the union of two 2-planes intersecting at a point. This is illustratedin Figure 60.The 2-graph in Figure 59 (center) describes a topological surface which is the union of 2-planes intersecting at a point, both for the Lagrangian surfaces in ( R , ω st ) and the Legendriansurfaces in ( R , ξ st ). Figure 60 helps illustrate these two 2-planes. Topologically, the front inFigure 59 (center) is the cone over the annular projection of the (2 , . The first 2-plane, shown in yellow, is obtained by taking the upper sheet in two ofthe four regions, opposite to each other, and the lower sheet in the remaining two. Similarly,the second 2-plane, depicted in orange, is obtained by taking the lower sheet in the tworegions where the yellow 2-plane is the upper sheet, and the upper sheets in the remainingtwo regions. The fronts of these two 2-planes intersect along A -crossing edges and a moresingular point (drawn in green). The A -crossing edges lift to embedded pieces, as the slopesof the branches are different, but the green point in the middle lifts to an intersection pointof these two 2-planes.Finally, the Lagrangian projections of the Legendrian lifts of Figure 59 (left) and Figure59 (right) are realized as Polterovich surgeries of the corresponding Lagrangian projectionin Figure 59 (center). Since the Legendrian lifts of Polterovich surgeries are Legendrianmutations [CMP19, Theorem 6.3], this concludes the first part of Theorem 4.21. This is consistent with the fact that the Hopf link is the boundary of two transversely intersecting planesin the 4-ball D . For the max-tb Legendrian
Hopf link, these two planes should be taken to be Lagrangian. igure 60. The 2-graph (left) associated to the singular moment in theLegendrian mutation transition and its spatial front (right). The front canbe divided into four regions, each of which has a lower and upper sheet. Byalternatingly choosing upper and lower sheets, we explicitly obtain the frontsfor two distinct Legendrian 2-planes, depicted in orange and yellow.Let us now show that the exchange move in Figure 58 (right) also corresponds to a uniqueLegendrian mutation. This is proven directly through the homotopy of fronts in Figure 61.Indeed, the first step in Figure 61, starting from the upper left, consists of applying Move II,pushing a trivalent vertex through a hexagonal vertex. The second and third steps are alsoa direct application of a Move II, pushing the remaining two trivalent vertices through thenewly created two hexagonal vertices. The fourth move, starting at the left of the secondrow, is a mutation of 2-graphs. This yields the 3-graph at the center of the second row, thearrow being labeled by the letter µ . Finally, we apply a Move III, flopping the four verticesnearest to the center, in order to achieve the 3-graph at the right of Figure 58 (right). Thisshows that the exchange move in Figure 58 (right) is a Legendrian mutation. Figure 61.
The Legendrian mutation for 3-graphs as a sequence of Legen-drian isotopies and 2-graph mutation. (cid:3)
For our applications to Lagrangian fillings, it is important to understand how 1-cycle rep-resentatives of classes in H (Λ( G ) , Z ) change under the mutations depicted in Figure 58.Following Subsection 2.4, we focus on 1-cycles represented by monochromatic edges – ormore generally long edges – and by Y -cycles. Figure 62 explicits shows how to transport ertain I -cycles along the mutation. In addition, mutation along a long edge is dictated bythe following: Figure 62.
The 2-graph mutation with the additional information of the1-cycles, before and after the 2-graph mutation (Left). The Y -cycle and anincident 1-cycle transforming before and after a mutation along the Y -cycle(Right). Corollary 4.22.
Let [ γ ] ∈ H (Λ( G ) , Z ) be represented by a long edge in an N -graph G ,as shown in the first row of Figure 63. Then the Legendrian mutation µ γ (Λ( G )) is theLegendrian weave associated to the graph µ γ ( G ) as depicted in the second row of Figure 63. Figure 63.
The two cases, left and right, of a Legendrian mutation along a1-cycle γ represented by a long-edge.Theorem 4.21 and Corollary 4.22 describe mutations along Y -cycles and I -cycles, eithermonochromatic or long edges. In general, we might be interested in mutating along a cycle γ which is a tree, both with Y -pieces and I -piece, as introduced in Section 2.4. Thus, we nowdevelop local rules for Legendrian mutations that will allow us to mutation along any suchcycle γ . These rules also imply Corollary 4.22.4.9. Diagrammatic Rules for N -graph Mutations. Let γ be a 1-cycle in an N -graph,given by a tree with Y -pieces and I -pieces. In this subsection we gather the necessary rulesfor performing a general mutation along γ and also diagrammatically carrying a 1-cycle afterthe mutation at γ . The rules are local , either near a hexagonal vertex or a trivalent vertex,and there are three cases that we need to draw: Legendrian mutation being performed at a Y -piece, at a I -piece, and mutation near a trivalent vertex.First, we draw the rules for the effect of mutating at a cycle which contains Y -pieces:(i) Figure 64 shows how the Y -cycle at which we mutate transforms, this cycle is depictedin green. Note that the resulting cycle locally contains only one Y -piece.(ii) Figure 65 explains how to transform the other Y -cycle, in ochre (a darker yellow),under mutation at the green Y -cycle in Figure 64. iii) Figure 66 then depicts the transformation of edge I -cycles through a hexagonal vertexunder mutation at the green Y -cycle in Figure 64.(iv) Finally, Figure 68 provides the last information needed for carrying any cycle uponmutating at the green Y -cycle in Figure 64. These are the three ways in which a1-cycle must be continued if the 1-cycle is coming from the extremes of one of thesides.Second, the rules for mutating at a long edge of an I -piece of a 1-cycle:(v) Figure 68 shows how to transform an I -piece upon mutation at the green I -piece.(vi) Figure 69 then depicts the transformation of a Y -piece of a cycle, in ochre, uponmutation at the green I -piece in Figure 68.Finally, the local rules for mutating near a trivalent vertex are shown in Figure 70. Theserules are derived by performing Legendrian Reidemeister moves, especially Move II, untilthe given cycle at which we want to mutate becomes a monochromatic (short) edge. Thena monochromatic edge mutation is performed, as in Theorem 4.21, and Legendrian Rei-demeister moves are performed back to the starting configuration. The two non-cancelingapplications of a push-through move, before and after a monochromatic edge mutation, areresponsible for the tripling behavior seen in the diagrams. = Figure 64.
Case Mutation at Y -cycle: Internal Mutation along Y -piece in green. Figure 65.
Case Mutation at Y -cycle in Figure 64: Effect for ochre Y -cycleof Internal Mutation along Y -cycle in green in Figure 64. igure 66. Case Mutation at Y -cycle in Figure 64: Effect for ochre I -cycleof Internal Mutation along Y -cycle in green. igure 67. Case Mutation at Y -cycle in Figure 64: Effect for side I -cycles ofInternal Mutation along Y -cycle in green. === Figure 68.
Case Mutation at I -cycle in green (upper Left). In second andthird row: effect of this mutation for ochre I -cycle of Internal Mutation along I -cycle in green. igure 69. Case Mutation at horizontal I -cycle as in Figure 68: Effect forochre Y -cycle of Internal Mutation along I -cycle in green in Figure 68 (left). === Figure 70.
Case Mutation near trivalent vertex for green cycle (first row).Second and third rows: Effect for ochre Y -cycle of Internal Mutation at greencycle near trivalent vertex.4.10. Sufficiency For Stabilized Legendrians.
Finally, we conclude this section by in-troducing the following combinatorial idea, motivated by the topology of Legendrian surfacesin 5-dimensional contact manifolds.
Definition 4.23. An N -graph G ⊆ C is said to have a bridge if there exists two disjoint2-disks D , D ⊆ C such that the complement G \ ( G ∩ D ∪ G ∩ D ) consists of ( N − τ , τ , . . . , τ N − consecutive with respect to a transverse orientedcurve in C \ ( D ∪ D ). (cid:3) For the N = 2 case, where G is a trivalent graph, a bridge for G according to Definition4.23 coincides with the standard graph-theoretic notion of a bridge [BM08, Die17]. A general N -graph G ⊆ C with a bridge is depicted in Figure 71 (left), and an example of a 4-graphwith a bridge is shown in Figure 71 (right).The geometric motivation for this definition is based on the theory of loose Legendriansurfaces, also known as stabilized Legendrians [Mur12]. This class of loose Legendrians areknown to satisfy an h -principle and has proven to be very useful in the study of Weinsteinmanifolds [CE12, CM19]. The reader is referred to [CE12, Mur12] for further details. For thepresent manuscript, we will assume known its definition and state the following property: Figure 71.
Structure of an N -graph with a bridge (left) and instance of a4-graph with a bridge (right). Proposition 4.24.
Let G ⊆ C be an N -graph with a bridge. Then ι (Λ( G )) is a looseLegendrian surface.Proof. The proof is a simple argument in the theory of spatial fronts. Indeed, consider the1-dimensional front slice along the dashed orange line in Figure 71. The braid shown alongthis slice is depicted in Figure 72 (left). Its closure as a satellite of the standard Legendrianunknot is shown in Figure 72 (center). This Legendrian link is isotopic, via a sequence ofReidemeister II moves, to the Legendrian link given by the front in Figure 72 (right). Theloose chart is exhibited in yellow in this figure. Note that this chart has arbitrarily largethickness due to the dilation freedom in ( R , ξ st ) and the fact that our front is global. Thisproves that ι (Λ( G )) is a loose Legendrian if G has a bridge. (cid:3) Figure 72.
The front for the Legendrian link obtained in a 3-dimensionalslice of a bridge (left). The front for the corresponding satellite closure (center)and a homotopic front exhibiting a loose chart (right).Proposition 4.24 immediately has the following consequence.
Corollary 4.25.
Let G ⊆ C be an N -graph with a bridge. Then ι (Λ( G )) ⊆ ( S , ξ st ) admitsno exact Lagrangian filling L ⊆ ( D , ω st ) . (cid:3) Corollary 4.25 should be contrasted with the fact that many of the Legendrian surfaces ι (Λ( G )) ⊆ ( S , ξ st ) admit exact Lagrangian fillings. For instance, it follows from Theorem4.10 that any 2-graph G obtained from the unique two-vertex 2-graph by adding bigons , i.e.a 1-surgery, yields a Legendrian surface ι (Λ( G ))) which admits exact Lagrangian fillings. Onthe other hand, simple Example 4.26 (Exact Lagrangian Cobordisms To a Loose Legendrian) . Consider the Leg-endrian Clifford 2-torus T c ⊆ ( S , ξ st ) associated, via the standard satellite, to the 2-graphin Figure 73 (Left). By applying our combinatorial Legendrian surgery from Theorem 4.10,Figure 41.(3), we obtain an exact Lagrangian cobordism from T c to the Legendrian 2-sphereΛ l associated Figure 73 (Right). By Proposition 4.24, the Legendrian Λ l is a loose Legen-drian surface. This proves that the Legendrian Clifford 2-torus T c ⊆ ( S , ξ st ) is a subloose egendrian surface, and we will show in Section 6 that T c is not a loose Legendrian. Inparticular, this also proves that T c ⊆ ( S , ξ st ) admits no 3-dimensional exact Lagrangianfillings L ⊆ ( D , λ st ) in the standard symplectic 6-disk. The points in the non-empty flagmoduli associated to T c will in fact be geometrically represented by non -exact Lagrangianfillings. (cid:3) Figure 73.
An exact Lagrangian cobordism from a non-loose Legendrian2-torus to a loose Legendrian 2-sphere.5.
Flag Moduli Spaces
In this section we introduce one of the central algebraic invariants in this article, the flag moduli space M ( G ) of an N -graph G and its associated Legendrian weave. We will provethat these flag moduli spaces are moduli spaces of constructible sheaves associated to aLegendrian weave, but we first present their explicit and self-contained definition.5.1. Preliminaries on the Flag Variety.
Let N ∈ N be a natural number and R acommutative ground ring, which will oftentimes be a field. We denote by GL N the generallinear group, a scheme whose value over R is GL( N, R ), and likewise for PGL N , the projectivegeneral linear group. By definition, a (full or complete) flag is an element F • ∈ { F ⊂ F ⊂ F ⊂ · · · ⊂ F N − ⊂ F N : dim F i = i, ≤ i ≤ N } , i.e. a sequence of nested linear subspaces F i ⊆ R N = R ⊕ ( N ) . . . ⊕ R , 0 ≤ i ≤ N . Let B ⊆ GL N be the Borel subgroup of upper triangular matrices preserving the standard coordinate flag.Since GL N acts transitively on the set of bases, the space that parametrizes such full flags isthe homogeneous space B = GL N /B . This is an algebraic variety, known as the flag variety.The relative position of two flags ( F • , G • ) ∈ B × B is encoded algebraically by the Bruhatdecomposition GL N = (cid:71) w ∈ S N BwB, where the symmetric group S N = W (GL N ) is identified with the Weyl group. That is,the orbits of the diagonal action of GL N on a pair of flags are indexed by the symmetricgroup S N . Computationally, F • and G • are in relative position w ∈ S N if and only ifdim( F i ∩ G j ) = rk( w ( i, j )). Here w ( i, j ) is the principal submatrix of the permutationmatrix of w with lower right corner in the ( i, j )-entry, and the permutation matrix has 1in the entries ( w ( i ) , N − w ( i ) + 1), and zeroes elsewhere. By definition, F • and G • are in “Vexillary” is the appropriate adjectival form of “flag”. Hence, it should technically be named the vexillary moduli space. The word is possibly too obscure, and we thus favor flag moduli space, as in flag variety. This is a maximal Zariski closed and connected solvable algebraic subgroup. Since B is a minimalparabolic subgroup of GL N it preserves the most geometric linear structure in R N , which is precisely a flag F • . ransverse position (or totally transverse or completely transverse ) if their relative position isid ∈ S N , which is the generic relative position between two points in the flag variety B . Inparticular, an elementary transposition τ i ∈ S N determines a relative position between twoflags F • and G • in which only their i th vector spaces differ, and no others.We will require a slight generalization of the above when the surface C is not simply con-nected: compatible local systems of flags, rather than flags of subspaces of a fixed vectorspace. This will not be required for our applications in Sections 6, 7 and 8, so the readeris welcome skip this paragraph. Let E −→ X be a local system on a topological space X .By a local system of flags, we mean a complete filtration (flag) E • of E by local systems E k such that the monodromy preserves the filtration. In this sense, the flag itself makes globalsense. Let U ⊂ X be a subspace and let F • be a flag of sub-local systems on U , so that F k ⊆ E for all 0 ≤ k ≤ N . We say that F • is compatible with E • if the monodromiesare: specifically, for γ ∈ π ( U, u ) and v ∈ F k , i k ( γ · v ) = i k, ∗ ( γ ) · i k ( v ) , where the symbol · denotes (ambiguously) the action of any group on a vector space. Note that by monodromyinvariance, we may speak of the relative position of two compatible sub-local systems of flags F • and G • on subspaces U and U (cid:48) of X .With these algebraic preliminaries, we turn to describing the flag moduli space associated toan N -graph.5.2. Description of the Flag Moduli Space of an N -graph. Let G be an N -graph ona connected surface C , thought of as the union of the embedded graphs G i . By a face of G we mean the closure of a connected component of the complement C \ G .We first give a general description of the flag moduli space for C not necessarily simplyconnected. We will not use this in our applications, so the reader is welcome to skip to thesimpler Definition 5.2, which is equivalent when C is simply connected.Let Σ( G ) ⊂ C × R be the wavefront of the Legendrian weave, woven according to G ⊆ C .Call a region a connected component of the complement ( C × R ) \ Σ( G ) . Definition 5.1.
Let C be a connected surface and let G ⊆ C be an N -graph. The framedflag moduli space (cid:102) M ( C, G ) associated to G is comprised of the following data.i) A rank- N local system E −→ C , equivalently a vector space V and a representationof the based fundamental group π ( C ) on V .ii) For each face F of the N -graph G , a compatible local system of flags F • ( F ).iii) For each pair of adjacent faces F , F , sharing an i -edge e , their two associated com-patible local systems of flags F • ( F ) , F • ( F ) are in relative position τ i ∈ S N , andalong the common edge e we have chosen isomorphisms F j ( F ) ∼ = F j ( F ) , ≤ j ≤ N, j (cid:54) = i, and no other information, as F i ( F ) (cid:29) F i ( F ).iv) By gluing, these isomorphisms define local systems in each region, since the j th stepof a flag of local systems F j compatible with E defines a local system on the regionbetween the j th and ( j + 1)st sheets — and these are not separated by a τ i crossingof sheets when j (cid:54) = i . We require that such local systems in regions, each of whichare sub-local systems of E via upward generization morphisms, are compatible with E . This condition is not local in the N -graph, G . he group PGL N acts on the space (cid:102) M ( C, G ) diagonally, i.e. as isomorphisms of E and onall flags of local systems at once. By definition, the flag moduli space of the N -graph G isthe quotient stack M ( C, G ) := (cid:102) M ( C, G ) / PGL N . We simply write M ( G ) when C is understood. (cid:3) Definition 5.2.
Let C be a connected, simply connected surface and let G ⊆ C be an N -graph. The framed flag moduli space (cid:102) M ( C, G ) associated to G is comprised of tuples of flags,specifically:i) There is a flag F • ( F ) assigned to each face F of the N -graph G .ii) For each pair of adjacent faces F , F ⊆ C \ G , sharing an i -edge, their two associatedflags F • ( F ) , F • ( F ) are in relative position τ i ∈ S N , i.e. they must satisfy F j ( F ) = F j ( F ) , ≤ j ≤ N, j (cid:54) = i, and F i ( F ) (cid:54) = F i ( F ) . The group GL N acts on the space (cid:102) M ( C, G ) diagonally, i.e. on all flags at once. By definition,the flag moduli space of the N -graph G is the quotient stack M ( C, G ) := (cid:102) M ( C, G ) / PGL N . We simply write M ( G ) when C is understood. (cid:3) We will equivalently exchange between the linear and projective perspective for a full flag.In the projective setting, flags F • (or local systems of flags) are understood as a sequenceof nested projective planes P ( F ) • , given by the projectivization of the linear spaces of thelinear flag F • . For a ground field R , the moduli space M ( C, G ; R ) is representable by anArtin stack of finite type [LO08, LO09], and is typically an algebraic variety (unless G is sosymmetric that an admissible configuration of flags might be fixed by PGL N ).In Subsection 5.3 we explain why the moduli space M ( C, G ; R ) is an invariant of the Leg-endrian isotopy type of the associated Legendrian weave Λ( G ) ⊆ ( J C, ξ st ). The algebraicquestions we are interested in this article are about the different properties and computationsof the moduli M ( C, G ; R ) — for instance the cardinality of |M ( C, G ; F q ) | over a finite fieldor how M ( C, G ; R ) changes upon performing the combinatorial moves in Section 4, includingLegendrian mutations and surgeries. To ease notation, we will denote flags F • by F .5.3. Sheaf Description of Flag Moduli and Invariance.
Let C be a smooth surface, R a commutative ring, and Sh( C × R ) the category of constructible sheaves, i.e. the R -linear dg-derived category of complexes of sheaves of R -modules on C × R with constructiblecohomology sheaves. For algebraic preliminaries on (derived) dg-categories we refer thereader to [Kel94, Tab05, Toe07, LO10], and for simplicity we will choose R a field. In thissection, we use the identification J ( C ) ∼ = T ∞ , − ( C × R ) of the first jet bundle of C withdownward covectors of C × R — see [NRS +
15, Section 2.1]. Now given an N -graph G ⊆ C ,the Legendrian Λ( G ) ⊂ J ( C ) ∼ = T ∞ , − ( C × R ) ⊂ T ∞ ( C × R ) can be used to define thesubcategory Sh Λ( G ) ( C × R ) ⊂ Sh( C × R ) whose objects are constructible sheaves whosesingular support at contact infinity is contained in Λ( G ) ⊂ T ∞ ( C × R ) — see [TZ18, Section4].We write C ( C, G ) := Sh G ) ( C × R ) ⊂ Sh Λ( G ) ( C × R ) for the subategory of microlocalrank-one sheaves which are zero in a neighborhood of C × {−∞} , or C ( G ) for short. Thishas a simple description, which we now explain. The dg-category Sh Λ( G ) ( C × R ) is itself asubcategory of sheaves constructible with respect to the stratification defined by the frontprojection Σ( G ), and thus has a combinatorial description. By [KS90, Theorem 8.1.11], it isequivalent to the dg-category of functors from the poset of strata to k -mod (chain complexes)— see also [Nad09, Section 2.3] and [STZ17, Section 3.3]. The subcategory cut out by C ( G ) s the one whose objects are isomorphic to ones with the following properties: the chaincomplex assigned to a neighborhood of C × {−∞} is zero; the complexes in each region of( C × R ) \ Σ( G ) are rank-one local systems (or just vector spaces if C is simply connected);the morphisms assigned to all downward restriction maps are isomorphisms; and the upwardmorphisms from small open sets intersecting Σ( G ) to the regions above them which do notare codimension-one inclusions.The combinatorial model for this description leads to the flag moduli space M ( G ) of isomor-phism classes of objects in C ( G ) . Indeed, the flag moduli space M ( G ) associated to an N -graph G ⊆ C , as introduced in Definition 5.2, relates to the category C ( G ) := Sh G ) ( C × R ) according to the following result, which itself generalizes [TZ18, Section 4.3] to N -graphs with N ≥ Theorem 5.3.
The flag moduli space M ( C, G ; R ) is isomorphic to the moduli space of objectsin C ( G ) := Sh G ) ( C × R ) , the subcategory of microlocal rank- objects in Sh Λ( G ) ( C × R ) supported away from C × {−∞} . Proof.
We first assume that C is simply connected. The argument parallels that of [STZ17,Sections 6.2 and 6.3], with the additions required by the strictly two-dimensional behavior.The moduli space of objects is defined locally, meaning that it is the fiber product overits restriction-to-boundary maps of the moduli spaces (cid:102) M in neighborhoods of C . We canassume that these neighborhoods of C are chosen small enough so that they are contractibleand contain no more than one “feature” of the given N -graph G . That is, for some suchneighborhood U ⊆ C , either U ∩ G is empty or contains part of an edge, a single trivalentvertex, or a single hexagonal vertex. We then have a local study for each of these cases.In the case where U is empty or contains part of an edge, the front of the Legendrian weaveover U is either N parallel sheets or N sheets with a single crossing labeled τ i , and can beidentified with σ × R , where σ is a front of a one-dimensional Legendrian knot being either N parallel lines or N lines with a single crossing. Then, since the R factor is contractible,we can identify the moduli space (cid:102) M over U using the one-dimensional study in [STZ17,Sections 6.2 and 6.3], concluding that it is either the flag variety or pairs of τ i -transverseflags, respectively.The moduli Sh G ) ( C × R ) is local with respect to G ⊆ C and the topology of the surface C , i.e. it is globally described as fibered products for the local pieces of G ⊆ C . It thereforeremains to show that Sh G ) ( C × R ) coincides with M ( C, G ) for the local graphs G tri ⊆ D and G hex ⊆ D , respectively given by a trivalent vertex and a hexagonal vertex, as introducedin Section 2. We do these in turn.The trivalent vertex case was studied in [TZ18, Section 4] for 2-graphs, and we will make theneeded adjustments to N -graphs. The computation for the local N -graph G tri consists of ananalysis of the moduli of constructible sheaves supported at the D − -wavefront singularity, asdirectly carried out in [TZ18]. The boundary conditions for an object in Sh G tri ) ( D × R )consist of a triple of flags ( F , F , F ) such that F i ∈ S τ k ( F j ) for i (cid:54) = j , 1 ≤ i, j ≤
3, if theedges of G tri are labeled by τ k . This can be seen by combining the result for a neighborhoodof a single crossing edge above, taking the fiber product over the spaces of flags in the emptyneighborhoods in-between. Then [TZ18, Section 4.1] implies that these are all the requiredconditions (and strata) and thus Sh G tri ) ( D × R ) coincides with M ( D , G tri ). Note thatthe analysis in [TZ18, Subsection 4.1.2] restricts to the case where the local model is a 2-graph G tri ⊆ D , it is readily seen that this model suffices for the analysis of the local model N -graph G tri ⊆ D . igure 74. Constructible sheaf in R microlocally supported on the A -swallowtail singularity (left). The sheaf convolution given by the Guillermou-Kashiawa-Schapira [GKS12] quantization upon performing a Reidemeister R1move (right).Alternatively, it is possible to directly conclude the analysis of the D − -singularity by perform-ing a generic perturbation of the D − -wavefront, as depicted in Figure 34, and studying thecategory of constructible sheaves supported at a A -swallowtail singularity. Indeed, Figure74 (left) shows the conditions for a constructible sheaf microlocally supported along the frontof an A -swallowtail singularity, which consists of a choice of injective map f : C −→ C ,where C ∼ = R k and C ∼ = R k +1 , for some k ∈ N . The crucial fact is that the (stalk ofthe) sheaf in the remaining 3-dimensional open strata I is uniquely determined to be thecone of the map ( f, − f ) : C −→ C ⊕ C . This is a consequence of the Guillermou-Kashiwara-Schapira quantization [GKS12, Theorem 3.7] of Legendrian isotopies: since the A -swallowtail is the big wavefront [Ad90] of the first Reidemeister move for 1-dimensionalLegendrian fronts, it follows that the sheaves in the strata I are uniquely determined by f : C −→ C by the sheaf kernel associated to the first Reidemeister move. It is readilyseen [STZ17] that the result of the convolution with such a kernel yields the sheaf transfor-mation in Figure 74 (right). By the non-characteristic property of the category of microlocalsheaves [GKS12], the sheaves microlocally supported on the wavefront of the D − -singularityis equivalent to that for a generic perturbation of such D − -singularity. The generic pertur-bation consists of three A -swallowtails and the conditions for the constructible sheaves onthese stratification follow from the above analysis. In conclusion, we obtain an isomorphismSh G tri ) ( D × R ) ∼ = M ( D , G tri ).Let us now address the hexagonal vertex G hex . Since the Legendrian weave Σ( G hex ) is thebig wavefront of the third Reidemeister move for 1-dimensional Legendrian fronts, it sufficesto understand the kernel of its quantization. Figure 75 shows the local transformation forconstructible sheaves near the third Reidemeister move [STZ17, Section 4.4.3]. Figure 75.
The explicit flag exchange given by the Guillermou-Kashiawa-Schapira [GKS12] quantization upon performing a Reidemeister R3 move. n Figure 75, the C i , 1 ≤ i ≤ E , E are complexes of vectors spaces, which we canactually assume to be vector spaces [STZ17, Section 3.3]. If C ∼ = R k , for some k ∈ N ,the microlocal rank 1 condition implies that E , E , C , C ∼ = R k +1 , C , C ∼ = R k +2 and C ∼ = R k +3 . The four flags at one of the sides of the hexagonal vertex are F (1)1 = C −→ C −→ C −→ C , F (1)2 = C −→ E −→ C −→ C , F (1)3 = C −→ E −→ C −→ C , F (1)4 = C −→ C −→ C −→ C , and the four flags at the other side of the hexagonal vertex are F (2)1 = C −→ C −→ C −→ C , F (2)2 = C −→ C −→ E −→ C , F (2)3 = C −→ C −→ E −→ C , F (2)4 = C −→ C −→ C −→ C . The three crossings in Figure 75 (left) imply, from left to right, that F (1)1 ∈ S τ k +1 ( F (1)2 ) , F (1)2 ∈ S τ k +2 ( F (1)3 ) , F (1)3 ∈ S τ k +1 ( F (1)4 ) . Similarly, the three crossings in Figure 75 (right) imply, from left to right, that F (2)1 ∈ S τ k +2 ( F (2)2 ) , F (2)2 ∈ S τ k +1 ( F (2)3 ) , F (2)3 ∈ S τ k +2 ( F (2)4 ) . These are precisely the conditions for the flag moduli space M ( D , G hex ) in Definition 5.2,and hence Sh G hex ) ( D × R ) ∼ = M ( D , G hex ).This concludes the argument for the case where C is simply connected. We now turn tothe case where C is not simply connected. There are no further local conditions. The onlyadditional concerns regard compatibilities of local systems.Let Σ( G ) ⊂ C × R be the wavefront of the Legendrian weave, and recall that we call a region a connected component of the complement ( C × R ) \ Σ( G ) . A constructible sheaf in C ( G )restricts to a local system on each region, since there is no singular support away from thewavefront. There are two distinguished regions R top and R bot containing neighborhoods of C × {∞} and C × {−∞} , respectively. A constructible sheaf in C ( G ) restricts to 0 in R bot (bydefinition) and to a local system on R top ∼ C that we assign to be the data E from Definition5.1(i). Now, as explained in Definition 5.1(iv), the data of a point in (cid:102) M ( G ) defines a localsystem in each region. Commutativity of sheaf restriction maps requires that a section whichis parallel transported around a region and then included into E arrives at the same place asa section which is included first and then parallel transported around R top , and this is therequirement of Definition 5.1(iv). (cid:3) Local Flag Moduli Computations.
Let us prove the following useful lemmas on theflag moduli, which can be implicitly used when performing computations on M ( C, G ; R ). Inthis section, and subsequent computations, we will consider a ground field R = k , with k = C and finite fields k = F q as the main fields of interest.We start with the study of the flag moduli space at a trivalent vertex, as depicted in the leftof Figure 8, and characterize that local flag moduli space. Lemma 5.4.
Consider the neighborhood O p ( N ) of a τ i -trivalent vertex in an N -graph. Thenthe local moduli of flags M ( O p , G ; k ) is set-theoretically a point, and the PGL N -action on (cid:102) M ( O p , G ; k ) has stabilizer ( k ∗ ) N − × k ( N ) − .Proof. Let F , F , F be the three flags in O p ( N ). The GL N -action is transitive on thespace of flags, and thus F can be mapped to the standard flag S , defined by S j = { x j +1 = x j +2 = . . . = x N − = x N = 0 } , where k = Spec k [ x , . . . , x N ] . The GL N -action allows us to also map the two flags F and F , respectively, to S and S ,defined by S j = S j = S j , ≤ j ≤ N, j (cid:54) = i, j = { x i = x i +2 = . . . = x N − = x N = 0 } , S j = { x i − x i +1 = x i +2 = . . . = x N − = x N = 0 } . This implies that the quotient of the moduli (cid:102) M ( O p , G ; R ) by the gauge group PGL N isset-theoretically a point. In order to recover its structure as a quotient stack, it suffices toidentify the stabilizer of the triple of flags S , S , S . For that, notice that the stabilizer of S is the projectivization of the Borel subgroup of upper triangular matrices, isomorphic to( k ∗ ) N − × k ( N ). The condition of fixing the flag S transversely cuts out a k -coordinate inthe interior of the upper triangle, since it sets the ( i, i + 1) entry equals to zero. This cutsthe stabilizer down to ( k ∗ ) N − × k ( N ) − , and finally stabilizing S imposes the equality ofthe two diagonal entries ( i, i ) and ( i + 1 , i + 1), thus transversely cutting down a k ∗ . Theresulting stabilizer is ( k ∗ ) N − × k ( N ) − , as claimed. (cid:3) In its simplest instance of N = 2, this is the statement that three distinct points in theprojective line P ( k ) can be sent to { , , ∞} with trivial stabilizer. A lesson from Lemma5.4 is that for any N , near at least one trivalent vertex of an N -graph, we are allowed to usethe gauge group PGL N and fix the flags around that vertex. The (proof of the) lemma alsoprovides the (geometric) degrees of freedom left after this choice. Example 5.5.
Consider the -graph G associated to the triangulation of C = S with twotriangles. Then the flag moduli space M ( S , G ; C ) consists of a point {∗} . In fact, this point {∗} of the flag moduli space geometrically corresponds to the conjecturally unique Lagrangian3-disk filling of the standard Legendrian unknot Λ ⊆ ( S , ξ st ) . (cid:3) Lemma 5.4 is a statement about a particular triple of flags. It ought to be noted that a generic triple of flags is part of a moduli space of dimension (cid:0) N − (cid:1) , with birational coordinates givenby generalized triple ratios – see Section 7 and [FG06b, Section 9]. The flags appearing inthe context of our N -graphs are in general a combination of non-generic flags, arising fromthe local vertices, with a flag being modified at exactly one degree when crossing an edge.Let us now address our second local model at a vertex, that of a hexagonal vertex, as depictedin the right of Figure 8. Lemma 5.6.
Consider the neighborhood O p ( N ) of a hexagonal vertex, with edges τ i , τ i +1 and consecutively ordered flags F j , j ∈ Z / Z . Then any pair of opposite flags F k , F k +3 determines the others.Proof. By symmetry, it suffices to show that the flags F , F determine F and F . Weassume that F and F are separated by a τ i +1 edge — a similar argument will work if it isof type τ i . By the prescribed transversality, we have F j = F j and F j = F j for j (cid:54) = i + 1.Now since F i (cid:54) = F i , and F i +21 = F i +24 , there exists a unique linear subspace V ⊆ F i +21 which contains F i , F i . So we must have F i +15 = F i +16 = V , uniquely determining the flags F and F . (cid:3) A direct application of Lemma 5.6 is the invariance of the moduli of flags under the N -graphReidemeister Move I from our Theorem 4.2 above: Corollary 5.7.
The flag moduli space M ( C, G ; R ) is invariant under the candy twist. (cid:3) The candy twist – Move I – is the move depicted in Figure 19 above, and the proof ofCorollary 5.7 follows immediately from Lemma 5.6, since the interior faces of the local modelare uniquely determined by two opposing boundary flags, and they in turn determine theremaining ones. Corollary 5.7 also follows from Theorem 1.1 and the Legendrian invarianceproven in [GKS12, Theorem 3.7]. The invariance of the moduli of flags under the other movesin Theorem 4.2 can be proven similarly by direct means. emma 5.6 discusses the flags in a neighborhood of a hexagonal vertex and allows for acomputation of the local flag moduli space M ( O p ( N ); R ) at a hexagonal vertex, since itreduces it to the study of a quadruple of flags. Example 5.8.
Let us illustrate this point by computing M ( O p (3); C ) , which we claim isisomorphic to a point stabilized by the subgroup ( C ∗ ) ⊆ PGL(3 , C ) . Indeed, the incidenceproblem at a hexagonal vertex is given by six flags F = ( p , l ) , F = ( p , l ) , F = ( p , l ) , F = ( p , l ) , F = ( p , l ) , F = ( p , l ) where p i and l i , for ≤ i ≤ , are points and lines in P ( C ) and the notation ( p i , l i ) stands forthe projectivized flag p i ∈ l i . Since the three points p , p , p are pairwise distinct, PGL(3 , C ) acts on them transitively, and their stabilizer is the (projectivization) of a maximal torus in GL(3 , C ) , which is isomorphic to ( C ∗ ) . Lemma 5.6 provides a more direct route: it sufficesto observe that the PGL(3 , C ) -stabilizer of the two completely transverse flags F , F is theset of diagonal matrices in PGL(3 , C ) , i.e. ( C ∗ ) . (cid:3) It is an exercise to extend the argument for Lemma 5.4 above in this context and show that:
Lemma 5.9.
Consider the neighborhood O p ( N ) of a ( τ i , τ i +1 ) -hexagonal vertex in an N -graph. Then the local moduli of flags M ( O p , G ; k ) is set-theoretically a point, and the PGL N -action on (cid:102) M ( O p , G ; k ) has stabilizer ( k ∗ ) × (cid:16) ( k ∗ ) N − × k ( N ) − (cid:17) . (cid:3) Having computed the local models at trivalent and hexagonal vertices, in Lemmas 5.4 and5.9, we now address the local flag moduli space around a τ i -edge connecting two trivalentvertices for 1 ≤ i ≤ N −
1. Thanks to our discussion in Subsection 2.4 on the homology ofthe associated Legendrian weaves, we know that this is the flag moduli space associated toa Legendrian cylinder. In contrast to Lemmas 5.4 and 5.9 above, we will now discover thatthe local flag moduli space around a monochromatic edge is (set-theoretically) non-trivial.
Lemma 5.10 (Flag Cross-ratio) . Let G be an N -graph, and e ∈ G a monochromatic edgebetween two trivalent vertices. The local flag moduli space M ( O p ( e ) , G ; k ) in a neighborhood O p ( e ) is isomorphic to k ∗ with stabilizer ( k ∗ ) N − × k ( N ) − , under the PGL N -action. Lemma 5.10 appears in the study of cluster coordinates for 2-graphs in the works [FG06b,TZ18], yet a treatment of it here, in the context of N -graphs, seems in order. The interestingpart in Lemma 5.10 is the existence of a non-trivial flag moduli space around the edge e ∈ G .The stabilizer only appears due to the dependence on N . Note also that, by using Lemma5.6, the statement in Lemma 5.10 can readily be generalized for a long edge e , i.e. an I -cyclebetween two trivalent vertices, as described in Section 2. Proof.
For an edge e ∈ G between two trivalent vertices, it suffices to discuss the case of amonochromatic edge, since the push-through move preserves the flag moduli. In this case,let v , v be the two endpoints of e . By Lemma 5.4, the local flag moduli space around v can be fixed to be a point with stabilizer ( k ∗ ) N − × k ( N ) − . In this normalization, theflag moduli space around v is determined in two of the sectors, and thus it is uniquelydescribed by the remaining choice of flag. This is tantamount to the choice of a fourth pointin P ( k ) \ { , , ∞} , which yields a modulus of k ∗ . (cid:3) In general, the existence of a non-trivial 1-cycle γ ∈ H (Λ( G ) , Z ) provides the flag modulispace with a k ∗ factor, which can be geometrically interpreted as being a contribution ofthe microlocal monodromy of the associated local system induced in the Legendrian surfaceΛ( G ), as we explain in Section 7. The following example illustrates this point in the case ofa Y -cycle in G . xample 5.11. Let us compute the local flag moduli space in an N = 3 neighborhood ofa Y -cycle, as depicted in Figure 11 (Right). The configurations of points for this incidenceproblem are given by the following conditions: (a) Three distinct points p , p , p , and three points q i ∈ l i = (cid:104) p i , p i +1 (cid:105) , where the index ≤ i ≤ is understood modulo , (b) The triples { p i , p i +1 , q i } , ≤ i ≤ , are triples of distinct points.The action of PGL allows us to set p = [1 : 0 : 0] , p = [0 : 1 : 0] and p = [0 : 0 : 1] witha ( k ∗ ) Cartan stabilizer, and this stabilizer can then be used to fix q = [1 : 1 : 0] ∈ l and q = [0 : 1 : 1] . The remaining choice of q yields the k ∗ contribution to the flag moduli spacesince it is a choice of a point q ∈ l distinct from p , p . (cid:3) This concludes our local computations of flag moduli spaces M ( C, G ). We now study thebehavior of the invariant M ( C, G ) under Legendrian surgery, and Sections 6 and 8 willdevelop global computation of flag moduli spaces. Given an N -graph G ⊆ C , we easenotation by writing M ( G ) for M ( C, G ).5.5.
Flag Moduli under Legendrian Surgeries.
Let
G, G (cid:48) be N -graphs such that G (cid:48) isobtained by Legendrian surgery on G , as described in Theorem 4.10. The following resultrelates the flag moduli spaces M ( G ) and M ( G (cid:48) ) before and after Legendrian surgery. Theorem 5.12.
Let k be a field and G an N -graph. For any τ i -edges of G , the flag modulispace M ( G ) satisfies the following local relations: Figure 76.
The change of the flag moduli spaces M ( G ) under combinatorialchanges in a piece of an N -graph G . Proof.
The relations can be verified with our description of the flag moduli space in Subsection5.2. We can also argue directly thanks to the geometry developed in Section 4. Indeed,the moduli of objects of the category of constructible sheaves microlocally supported at aLegendrian connected sum Λ is a direct product of the moduli of objects microlocallysupported at Λ and those microlocally supported at Λ . By Theorem 4.10, the right andleft graphs G r , G l for the Relations ( i ) and ( ii ) geometrically correspond to Legendrianconnected sums with the standard Legendrian 2-torus T , and the Legendrian Clifford 2-torus T c , respectively. The flag moduli for the former is k ∗ , and for the latter it is k \ { , } ,which concludes ( i ) , ( ii ). Finally, the relation ( iii ) follows from Proposition 4.24, as there donot exist constructible sheaves microlocally supported at a loose Legendrian. (cid:3) Note that, by construction, there exists a 3-dimensional exact Lagrangian cobordism L ( G, G (cid:48) )from Λ( G ) to Λ( G (cid:48) ), in the symplectization of ( J C, ξ st ). Thus, from the standard results in loer theory [EES05b, EGH00], we expect a map from M ( G ) × H ( L ( G, G (cid:48) ) , k ) to M ( G (cid:48) ).Theorem 5.12 gives a strong indication of what these maps should be, i.e. for ( i ) , ( ii ), M ( G (cid:48) )is a k ∗ - or a ( k \ { , } )-bundle over M ( G ), with the map being a section for this bundleprojection.5.6. Non-characterstic Property of Stabilization.
We conclude Section 5 with an in-teresting and direct computation of flag moduli spaces. First, note that the proof of Theorem4.17, showing that the standard satellites of Λ( G ) and Λ( s ( G )) are Legendrian isotopic, andTheorem 5.3 imply the isomorphism M (Λ( G )) ∼ = M (Λ( s ( G ))) , where s ( G ) is the stabilization we introduced in Subsection 4.7. We will nevertheless providea self-contained sheaf-theoretical proof of that equivalence, which we now illustrate in thecase N = 2. Proof of flag moduli space equivalence N = 2 . In that case, the moduli of objects in the cat-egory M (Λ( s ( G ))) parametrizes flags in P up to PGL(3 , C ) equivalence abiding the con-straints imposed by the 3-graph on the left of Figure 77. We assume that the 2-graph G = G , , before stabilizing, contains at least a vertex. Figure 77.
The flag configuration for N = 2 stabilization.The graph G (1 , imposes constraints on the points lying in a line l ⊆ P , the ladybugchanges this line to distinct lines l , l , also different from l , and the descending (12)-haloprovides the freedom of a point p ∈ l . The fact that G (1 , is contained in a wing of theladybug implies that l ∩ l ∩ l is a point, which for now we denote ∞ . Let us show thatthis moduli space coincides with the moduli space of points in l imposed by G (1 , . Forthat, note that the stabilizer of three non-collinear points p , p , p ∈ P is isomorphic to C ∗ × C ∗ ; indeed, it is isomorphic to the space of invertible diagonal matrices in PGL(3 , C ).Geometrically, each of the C ∗ allows us to move any point in one of the three possible linesspanned by two of the three points { p , p , p } around that line, on the complement of thesetwo spanning points.Hence we can start by using the PGL(3 , C ) and fix the points 1 , p, ∞ ∈ P in the configurationshown in the right of Figure 77, which determine the lines l , l . From the C ∗ × C ∗ we canuse the first C ∗ in order to send the third point in l imposed by G (12) to 0 ∈ l , and thesecond C ∗ to choose a point in the line l = (cid:104) , p (cid:105) , which in turn determines a line l ⊆ P bytaking its span with ∞ ∈ l ∩ l . This fixes the configuration of lines l , l , l and the points To our knowledge, these maps have yet to be studied in the context of microlocal sheaf theory. Theexpectation that they exist comes from the fact that the flag moduli space M ( G ) should correspond to anaugmentation variety for Λ( G ), and these maps are known to exist between augmentation varieties. , , ∞ , p ⊆ P with { , , ∞} ⊆ l , and that is precisely the three points being fixed by thePGL(2 , C ) symmetry acting in G (12) . (cid:3) This argument is self-contained, yet hopefully illustrates how in general the geometric con-clusion from Theorem 4.17, and the invariance of the flag moduli space M under Legendrianisotopy, are stronger and neater tools than the strict algebraic invariance of the flag modulispace. Let us now move forward with the following Sections 6, 7 and 8, which display severalapplications of the techniques developed in Sections 2, 3, 4 and 5, and in particular proveTheorems 1.5, 1.6, 1.8 and 1.9 stated in the introduction.6. Applications and Vexillary Computations
In this section we study applications of our diagrammatic calculus for Legendrian weavesΛ( G ) associated to an N -graph G , and their flag moduli spaces M ( G ). In particular, we willprove Theorem 1.5 and Theorem 1.6.6.1. First Pair of Computations.
Let us start with two simple examples of Legendrianweaves and their flag moduli: the Legendrian Clifford torus and the double t ∪ t ⊆ S ofthe 4-triangle t in the 2-sphere S .6.1.1. The Legendrian Clifford Torus.
Let us consider the 2-graph G = ( ∂ ∆ ) (1) ⊆ S inFigure 78, which has already featured in the proof of Theorem 4.10. The flag moduli space M ( G ) is readily seen to be the pair of pants P \{ , , ∞} . Indeed, there are four contractibleconnected components in S \ G , which implies that (cid:102) M ( G ) = { ( p , p , p , p ) ∈ ( P ) : p i (cid:54) = p j , i (cid:54) = j } where P ∼ = GL(2 , C ) /B is the flag variety of lines in C . Since PGL(2 , C ) acts 3-transitivelyon P , we can assume that ( p , p , p ) = (0 , , ∞ ), and the quotient (cid:102) M ( G ) /P GL (3 , C ) isgiven by M ( G ) = { λ ∈ P : λ (cid:54) = 0 , , ∞} . This flag moduli space is shown in Figure 78 (left), which is uniquely determined by thechoice of λ ∈ P \ { , , ∞} . Figure 78.
The tetrahedral 2-graph G as a planar projection of the 1-skeleton (∆ ) (1) of the tetrahedron ∂ ∆ (left). A front projection for theLegendrian 2-torus ι (Λ( G )) (right).Let us illustrate the Legendrian geometry in this case. The Euler characteristic of the Leg-endrian weave Λ( G ) is χ (Λ( G )) = 2 · χ ( S ) − G ) is a closed 2-torus. Adifferent front for Λ( G ) is depicted in Figure 78 (right), where the cone singularity [CM19,Section 2] is used, in line with the description in [DR11, Section 3]. The flag moduli space M ( G ) for the 2-graph G is read in this front as the moduli space of constructible sheaves in R microlocally supported with rank-1 in the front Figure 78 (right). This latter moduli isgiven with the data of a 1-dimensional vector space C in the bounded region in the interior f the front and a linear monodromy map λ : C −→ C . The monodromy must be an iso-morphism, and thus λ ∈ GL(1 , C ) ∼ = C ∗ , and also satisfy the additional constraint imposedby the cone singularity. By generically perturbing this singularity, it is readily seen that thecondition is that the monodromy λ does not have 1 has an eigenvalue, which in this casereduces to λ ∈ C \ { , } ∼ = P \ { , , ∞} . This is precisely the flag moduli space M ( G ). (cid:3) Remark 6.1 ([Nad17b, TZ18]) . This particular wavefront allows for a direct Legendriananalysis of the Landau-Ginzburg model ( C , z z z ), as follows. The regular fiber F ⊆ C ofthe superpotential is isomorphic to F ∼ = ( C ∗ ) , and its Lagrangian skeleton is thus an exact2-torus T ⊆ F , i.e. the vanishing cycle for the (non-isolated) singularity W . Its LegendrianliftΛ := { ( z , z , z ) ∈ C : | z | = | z | = | z | = 1 / , arg( z ) + arg( z ) + arg( z ) = 0 } ⊆ ( S , ξ st ) , has vanishing (singular) thimble the conic Lagrangian L = { ( z , z , z ) ∈ C : W ( z , z , z ) ∈ R + , | z | = | z | = | z |} . By performing a real blow-up at the origin, we introduce a real 2-sphere S at the origin anda projection map π : Λ −→ S from our Legendrian 2-torus onto this exceptional 2-sphere S . In coordinates, the map π ( z , z , z ) = ( (cid:60) ( z ) , (cid:60) ( z ) , (cid:60) ( z )) is just given by taking thereal parts of the complex coordinates and realizes the Legendrian surface Λ ⊆ ( S , ξ st ) asthe Legendrian weave ι (Λ( G )) associated to the four-vertex 2-graph G ⊆ S , given by the1-skeleton of the tetrahedron. Thus, the mirror of the Landau-Ginzburg model ( C , z z z )is the Legendrian 2-torus in ( J S , ξ st ) which satellites to the Clifford 2-torus T c ⊆ ( S , ξ st ).This leads to the description of the A-model Landau-Ginzburg model ( C , z , z , z ), given bythe category µ Sh L ( C ) of wrapped sheaves, as the bounded dg-category of finitely-generatedtorsion complexes on the flag moduli space M ( T c ) ∼ = P \ { , , ∞} . (cid:3) The Double of the 4-Triangle.
Let us consider the 4-graph G ( t ) associated to a 4-triangle t , as depicted in Figure 79 (left), and described in Section 3. Let G = G ( t ) ∪ ∂ G ( t ) ⊆ S be the 4-graph obtained by gluing two copies of this 4-graph along their bound-aries, i.e. G is the 4-graph associated to the 4-triangulation of S with two underlying t -triangles. The 4-graph G is depicted in Figure 79 (right), where the circle at the boundaryis identified to a unique point, which is a hexagonal vertex. Figure 79.
The local 4-graph G ( t ) associated to a 4-triangle t (left). Theglobal 4-graph G ( τ ) given by the 4-triangulation τ of the 2-sphere S withtwo triangles. or the computation of the flag moduli space M ( G ), we employ our geometric techniques inSection 4. Theorems 4.10 allows us to remove the initial three (blue) τ -bigons, by consideringa direct sum with three copies of the standard Legendrian 2-torus T st , see Section 4. Byapplying Move I in Theorem 4.2 three times, we obtain the 3-graph in Figure 80 (left).Further removing three of the bigons, we reach the 3-graph G in Figure 80 (right). Theframed flag moduli space (cid:102) M ( G ) for the 3-graph G is given by the choice of two flags F = ( p , l , π ) , F = ( p , l , π ) ∈ GL /B in projective 3-space, and a choice of threepoints p , p , p ∈ P k such that- ( l , π ) and ( l , π ) are completely transverse, i.e. l (cid:54)∈ π and l (cid:54)∈ π , and p (cid:54) = p ,- p ∈ l , p (cid:54) = p ,- p ∈ l , p (cid:54) = p ,- p ∈ π ∩ π , p (cid:54) = p .In particular, M ( G ) ∼ = (cid:102) M ( G ) / PGL , and the flag moduli space M ( G ) is described bythe data above. By Theorem 4.10, and the fact that each bigon contributes to k ∗ oncethe Legendrian weave is connected, we deduce that our original flag moduli space must beisomorphic to M ( G ) ∼ = M ( G ) × ( k ∗ ) . Figure 80.
The 4-graph G ( τ ) in Figure 79 after three index 1 anti-surgeries- accounted by the connected sums with T st - and simplified with Move I(right). The 4-graph obtained by three additional index anti-surgeries (right).This simplification, from the original 4-graph G to G , allows for a direct description aboveof the flag moduli space M ( G ), from which further information can be readily extracted. Forinstance, the F q -rational count for M ( G )( F q ) is immediately: |M ( G )( F q ) | = ( q − ( q − q − q )( q − q )( q − q ) · q · ( q − q − q − q − ( q + 1) q, as | PGL(4 , F q ) | = ( q − q − q )( q − q )( q − q )( q − − , the rightmost multiplicativefactor is the count for the two flags F , F , and the q factors stands for the final choice of( p , p , p ). (cid:3) We conclude this initial gallery of computations with the following:
Example 6.2 (Concentric Circles) . Let τ = ( τ i , τ i , . . . , τ i n ) be an ordered collection of n simple transpositions τ i j ∈ S N − , 1 ≤ j ≤ n , n ∈ N . Consider the N -graph G ( τ ) ⊆ S described by n concentric circles C i ⊆ S , 1 ≤ i ≤ n , with center on the North Pole, andstrictly increasing radius. This N -graph is depicted in Figure 81 (left). igure 81. The N -graph G ( τ ) associated to the sequence of transposi-tions τ = ( τ i , τ i , . . . , τ i n ) (left). The 4-graph G ( τ ) associated to τ =( τ , τ , τ , τ , τ , τ ) (right).The Legendrian weave Λ( G ( τ )) ⊆ ( J S , ξ st ) is a radial version of the N -stranded positivebraid closure of β = σ i σ i · . . . · σ i n . Smoothly, it is a link of N two-spheres S . Themoduli space of rank-one sheaves in R supported along the positive braid β is the openBott-Samelson variety O ( β ) [STZ17, Tri19, CG20]. By Section 5.2, since C = S is simplyconnected, there is no further monodromy information and M ( G ( τ )) = O ( β ). In particular,the links with different n have a different number of points over F q and cannot be Legendrianiosotopic. We note further that [STZ17, Theorem 6.34] relates this number to the HOMFLY-PT polynomial of the (topological) knot in R defined by the braid β. (cid:3) Symmetry groups for Legendrian weaves.
Let G be an arbitrary finite group andΛ ⊆ ( S , ξ st ) a Legendrian surface, with underlying smooth surface S (Λ). Let L (Λ) be thespace of embedded Legendrian surfaces in ( S , ξ st ) Legendrian isotopic to the Legendriansurface Λ, with base point Λ. In addition, let L (Λ) be the monoid of 3-dimensional exactLagrangian concordances in the symplectization ( S × R ( t ) , e t λ st ), up to Hamiltonian isotopy,based on the Legendrian surface Λ ⊆ ( S , ξ st ). Let ϕ t : S (Λ) −→ ( S , ξ st ) be a S -family ofLegendrian embeddings, t ∈ S . Then the graph mapgr : π ( L (Λ)) −→ L (Λ) , [ ϕ t ] (cid:55)−→ ( ϕ t ( S (Λ)) , t ) , allows us to relate loops of Legendrian surfaces with Lagrangian concordances.These spaces L (Λ) , L (Λ) are challenging to study. Already in the 1-dimensional case ofLegendrian links Λ ⊆ ( S , ξ st ), it was only established recently that there exist Legendrianlinks such that the fundamental groups π ( L (Λ)) can admit (infinite order) non-Abeliansubgroups [CG20, Corollary 1.6], and L (Λ) actually contains elements of infinite order [CG20,Corollary 1.7]. To our knowledge, the only previous result about the fundamental group π ( L (Λ)) or the monoid L (Λ) for Λ ⊆ ( R , ξ st ) a Legendrian surface was proven in [SS16],where Legendrian surfaces Λ Z n , n ∈ N , were built such that π ( L (Λ Z n )) admits the finitecyclic group Z n as a subgroup. Legendrian weaves and their flag moduli space are well-suitedto address these questions. We present the following result for Legendrian surfaces in ( S , ξ st ): Theorem 6.3.
Let G be an arbitrary finite group. Then there exists a Legendrian surface Λ G ⊆ ( S , ξ st ) such that (i) G is a subfactor of the fundamental group π ( L (Λ G )) , (ii) G is a subfactor of the 3-dimensional Lagrangian concordance monoid L (Λ G ) .In fact, the latter is the image of the former via the graph map gr : π ( L (Λ)) −→ L (Λ) . roof. We begin by describing a construction of 2-graphs. Let (
C, T ) be a closed smoothsurface, T a triangulation with e ( T ) edges, and G ( T ) the trivalent 2-graph dual to the tri-angulation T . Consider the 2-graph G (cid:48) obtained by adding a bigon at each edge of G ( T ),using Move I in Figure 4. By Theorem 4.10, the Legendrian ι (Λ( G (cid:48) )) is obtained by per-forming a connected sum of ι (Λ( G ( T ))) with e ( T ) copies of the standard Legendrian torus T ⊆ ( S , ξ st ). Then [DR11, Proposition 4.6], or Theorem 5.12, implies that the com-plex flag moduli space M ( G (cid:48) ) is isomorphic to the product M ( G ( T )) × ( C ∗ ) e ( T ) , and thus H ∗ ( M ( G (cid:48) ) , Q ) ∼ = H ∗ ( M ( G ( T ) , Q ) ⊗ H ∗ (( C ∗ ) e ( T ) , Q ) by the K¨unneth formula.Now let C be a Hurwitz surface [Hur92, LT99] with symmetry Hurwitz group G ( C ). Thetopological surface underlying the Riemann surface C admits a triangulation T ( G ( C )) withsymmetry group G ( C ). In particular, the dual graph G ( G ( C )) also has symmetry group G ( C ). Let us now consider the 2-graph G (cid:48) , associated to G ( G ( C )) as above, where theedge bigons are added such that G ( C ) is still a subgroup of the symmetry group of G (cid:48) .The flag moduli space M ( G (cid:48) ) is a Legendrian isotopy invariant of the Legendrian surfaceΛ( G (cid:48) ) ⊆ ( S , ξ st ), and thus G ( C ) acting faithfully in H ∗ ( M ( G (cid:48) )) implies that G ( C ) is asubfactor of π ( L (Λ( G (cid:48) ))) and L (Λ G ). Since G ( C ) acts faithfully on the set of edges of thetriangulation T , G ( C ) acts faithfully on 1 ⊗ H ∗ (( C ∗ ) e ( T ) , Q ) ⊆ H ∗ ( M ( G (cid:48) ) , Q ) piece of thecohomology of the flag moduli space M ( G (cid:48) ). This concludes the statement for Hurwitz groups G ( C ). Now, let G be an arbitrary finite group. Then G is a subgroup of the alternating group A n for large enough n ∈ N . By [Con84, Section 3], see also [LT99], A n is a Hurwitz group G ( C ) for n ≥ G injects into such a Hurwitz group G ( C ). The argumentabove thus implies that G is a subfactor for π ( L (Λ( G (cid:48) ))) and L (Λ( G (cid:48) )). Hence, the choiceΛ G = Λ( G (cid:48) ) completes the proof of Theorem 6.3. (cid:3) We do not know whether or not a result analogous to Theorem 6.3 holds for 1-dimensionalLegendrian knots Λ ⊆ ( S , ξ st ). That could be a good question in low-dimensional contacttopology. Any answer – positive or negative – would be of interest.There is a complement to Theorem 6.3 for certain groups G of infinite order, including non-Abelian groups such as PSL(2 , Z ), by using results of the first author. Indeed, the Legendrianweave associated to the 4-graph G ( τ ) with the eighteen concentric circles τ = ( τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ , τ )represents a 3-component Legendrian link Λ( G ( τ )) of 2-spheres. The geometric Br -braidaction constructed in [CG20], modulo its center Z (Br ), acts faithfully on the flag modulispace M (Λ( G ( τ ))). This flag moduli space is described in Example 6.2. Then [CG20, Theo-rem 1.1] shows that the modular group PSL(2 , Z ) acts faithfully on the cluster charts for thespace obtained by forgetting the monodromies in the Grothendieck resolution M (Λ( G ( τ ))).Hence, PSL(2 , Z ) is a subfactor of π ( L (Λ( G ( τ )))) and L (Λ( G ( τ ))) for these Legendrianweaves Λ( G ( τ )).6.3. Flag Moduli and Bipartite Graphs.
In Section 3, we introduced the construction ofa 3-graph G ⊆ C associated to an embedded eponymous bipartite graph G . This subsectionexplains how to compute flag moduli spaces for such 3-graphs.We will employ a useful notation, local to this subsection. If a, b ∈ V are distinct vectors in a3-dimensional vector space V , we denote by ab the unique 2-plane spanned by a, b . Similarly,given two 2-planes A, B, ⊆ V , the intersection A ∩ B will be denoted by AB .At a hexagonal vertex, traveling between opposite faces requires crossing three edges ofalternating colors, and thus opposite faces are assigned completely transverse flags A =( a, A ) = aA and B = ( b, B ) = bB . Note that a single such pair A , B determines theremaining four regions, by Lemma 5.6: if crossing red, blue, red from A to B , the flags Note that A n , for n ≤ A m , for a greater m ≥ n , and thus all cases A n are covered. n succession are ( a, A ) , ( AB, A ) , ( AB, B ) , ( b, B ). If crossing blue, red, blue, the flags are( a, A ) , ( a, ab ) , ( b, ab ) , ( b, B ). This is depicted as follows:( a, A )( b, B ) ( a, ab )( AB, B )( AB, A ) ( b, ab ) • • a bABabA B Now consider an edge of the bicubic graph G . In the associated 3-graph, this edge generatestwo hexagonal vertices which are connected by two adjacent edges of different colors. Thislocal configuration is said to be a hexagonal edge. Let us denote the two flags on oppositeregions along the axis connecting the hexagonal vertices by A = aA and C = cC . Let B bethe flag in the interior region of the hexagonal edge, transverse to both A and C . There aretwo further conditions on the flag B : AB ⊂ C, c ⊂ ab. The Weyl group W ( A ) ∼ = S is the symmetric group on three elements, and thus there aresix possible relative positions for the two flags A , C ∈ GL /B . Here we consider the case ofa finite field k = F q . In a hexagonal edge, the relative position of the two outer flags A , C isrestricted: Lemma 6.4.
The two outer flags A , C in a hexagonal edge must coincide or be completelytransverse. In addition, with A , C fixed, number the of choices of flag B in the interior of thehexagonal edge is q , in the case A = C , and q − , in the case A (cid:54) = C .Proof. Let us analyze their possible relative positions, labeled according to the elements W ( A ) = { , , , , , } :- Type 0: C = A . Then the conditions are automatic, and B is simply transverse to A = C . There are q such choices.- Type 1: c = a, C (cid:54) = A. The second condition is then automatic, but C ⊃ a = c and C ⊃ AB means C = A. This is a contradiction.- Type 2: c (cid:54) = a, C = A. The first condition is then automatic, but c ⊂ C = A and c ⊂ ab means c = a . This is a contradiction.- Type 12: a (cid:54) = c, C (cid:54) = A but a ⊂ C . Then a ⊂ C and AB ⊂ C means C = A This is acontradiction.- Type 21: a (cid:54) = c, C (cid:54) = A but c ⊂ A. Then c ⊂ A and c ⊂ ab means c = a. This is acontradiction.- Type 121: In this case, the flag B is determined by either equivalent choice: a line b in ac not equal to a or c (then B is the plane bAC ) or a plane B containing AC notequal to A or C (then b is acB ). The number of such choices is q − B has either q or q − A and C to be either equal or completely transverse. The other configurationshave no solutions. (cid:3) We now apply Lemma 6.4 and the discussion above to prove Theorem 1.5 in the introduction. .4. Non-isotopic Links of Legendrian Spheres.
Let n ∈ N and consider the bipartiteLadder Graph L n ⊆ S depicted in Figure 82 (bottom). The number n ∈ N denotes half thenumber of square faces, and the right and left sides of the bipartite graph are identified in S . In particular, S \ L n has 2 n + 2 connected components, 2 n squares and two 2-disks, atthe north and south poles of S . We consider its associated 3-graph L n ⊆ S , as described inSection 3, which is shown in Figure 82 (bottom). The Legendrian weave Λ( L n ) ⊆ ( J ( S ) , ξ st )consists of a 3-component link of Legendrian 2-spheres, independent of n ∈ N . Figure 82.
The bipartite Ladder Graph L n , where the right and left sidesare identified after n rungs (bottom). The 3-graph L n associated to L n (top).Note that the Legendrian link Λ( L n ) ⊆ ( J ( S ) , ξ st ) is smoothly isotopic to the surface unlink,as the codimension of this smooth embedding is three. We now show that the Legendrianisotopy type of the Legendrian link Λ( L n ) ⊆ ( J ( S ) , ξ st ) is different for each n ∈ N . Thiswill be achieved by counting the number of points of their flag moduli spaces M ( L n ) over afinite field. The precise statement reads: Theorem 6.5 (Theorem 1.5) . Let L n ⊆ S be the (2 n ) -runged ladder graph and F q a finitefield, q a prime power. Then the flag moduli space M ( L n ) has orbifold point count |M ( L n )( F q ) | = q n − − q n − + q n − + q − q − . Hence, the Legendrian surface links Λ( L n ) and Λ( L m ) are Legendrian isotopic iff n = m .Proof. Let us consider the two flags A , C ∈
GL(3 , C ) /B located in the strata correspondingto the neighborhoods of the north and south poles. We have shown these flags in Figure 83.The flags in the vertical regions will be denoted B i , 0 ≤ i ≤ n −
1, with the cyclic condition B = B n .By Lemma 6.4, the existence of the flags B i in the vertical hexagonal edges, 0 ≤ i ≤ n − A , C must either be trivial, i.e. A = C ,or completely transverse, i.e. the projective lines A (cid:54) = C are distinct, and a (cid:54)∈ C and c (cid:54)∈ A .The F q -count is divided into these two cases.First, let us consider the case where A and C are completely transverse, i.e. they belongto the Bruhat GL(3 , C )-orbit labeled by w = (12)(23)(12) ∈ W ( A ). We claim that afterchoosing the flag B = ( b, B ), the remaining flags B i , 1 ≤ i ≤ n − igure 83. The flag configuration at a point of the flag moduli space M ( L n )where A = ( a, A ), C = ( c, C ) are the inner and outer flags. Observe that thechoice of A , C partially fills the flags in the horizontal eye-shaped regions. Figure 84.
Flag configuration at a point of the flag moduli space M ( L n )in the case A = ( a, A ) is completely transverse to C = ( c, C ). For theseconfigurations, the choice of flag ( b, B ) uniquely determines the point in theflag moduli.Let us prove this. Since A and C are completely transverse, they determine the flags( AC, A ),( a, ac ) in the horizontal eye-shaped spaces in the upper row, and the flags (
AC, C ),( c, ac )in the corresponding horizontal spaces along the bottom. The additional choice of B = ( b, B )determines the flags ( AC, B ) , ( b, ac ) in the left and right regions adjacent to that of B . Notethat B (cid:54) = ac and b ∈ B ∩ ac . Similarly, b (cid:54) = AC and the two points AC, b ∈ P span the line B . The flag B must have b ∈ P as its point, and its line must contain AC, b ∈ P . Hence theflag B = ( b, B ) is uniquely determined, and coincides with B . By an analogous reasoning, B determines the flag ( AC, B ) on the adjacent region at its right, and hence the line in B must be B . Since the point in B must be the intersection B ∩ ac , we conclude B = ( b, B )and thus B = B = B . Iteratively applying these two steps, we show that B i = B for all1 ≤ i ≤ n −
1. The cyclic condition B = B n is automatically verified in this case. Inconclusion, in this completely transverse case, the choices are the three flags A , B , C , beingpairwise completely transverse. This configuration is depicted in Figure 85 (left).The counts over a finite field are | PGL(3 , F q ) | = ( q − q − q )( q − q ) q − , | P ( F q ) | = | P ( F q ) ∗ | = ( q − q −
1) = q + q + 1 , and a projective line P ( F q ) has q + 1 points. Also, note that there are | P ( F q ) | = q + 1choices of lines through a point. Now, the choice of the flag A = ( a, A ) gives a count of igure 85. The projective flags A , C and B = ( b, B ) in the case A , C arecompletely transverse (left). The configuration of projective flags in the case A = C , where admissible flags ( b , B ) , ( b , B ) , ( b , B ) , ( b , B ) , ( b , B ) aredepicted (right). | P ( F q ) | · | P ( F q ) | . The choice of the completely transverse flag C = ( c, C ) gives q , as wemust have a (cid:54)∈ C , and c ∈ C but c (cid:54) = A ∩ C . The line B in the third transverse flag B = ( b, B )must contain the point A ∩ C , and its point b = B ∩ ac is uniquely determined by the choiceof such B . Since B must be distinct from A and C , we get q − B . Thisyields a total count of ((1 + q + q )(1 + q )) · ( q ) · ( q − q + q )( q − q )( q − q ) = 1 q − , for the case where the flags A , C are completely transverse. Thus, A , B , C can be fixed,mutually completely transverse, and a factor of ( q − − remains.Second, let us consider the case where A = C . In this case, the flags B i , 1 ≤ i ≤ n , will notall be equal. We proceed with the same systematic analysis as before. The initial choice is B = ( b , B ), and this determines the flags ( b , ab ) , ( AB , B ) in the left and right adjacentregions of B . In turn, this determines the line in B to be B ⊆ P . The point in B remainsundetermined at this stage, and this is a choice of b ∈ B , with a count of q , since b ∈ B and b (cid:54) = A ∩ B . This is depicted in Figure 84. The choice of the point b ∈ B readilydetermines the point in the flag B , whose line is undetermined. There are exactly q choicesfor a line B ⊆ P in B , as it must contain b and be different from B . This is an iterativeprocess, where the count of choices that determine the flag B i , 2 ≤ i ≤ n is exactly q , eitherbecause of the choice of a point or a line. The flag configuration is depicted in Figure 86. Figure 86.
Flag configuration at a point of the flag moduli space M ( L n ) inthe case A = C . For these configurations, the sequence of flags ( b i , B i ) arepart of the choice that determine the points in the flag moduli. t this stage of the case A = C , we need to impose the cyclic condition B = B n given bythe ladder graph. This is not automatic, and it will actually reduce the naive count of q n forthe choices of B i , 0 ≤ i ≤ n −
1. Let us use the PGL(3 , F q ) symmetry to fix the flags A = C and B . We will now use affine coordinates, so the flag A will be understood as a line a ⊆ F q and a plane A ⊆ F q . Thus, we assume that the line a ⊆ F q in A is spanned by andthe plane A ⊆ C is the kernel of the covector (0 , , B is given by the pair , (1 , , F × q ) , and we will divide our count for fixed A , B by the isotropy factor of ( q − .Let us parametrize the remaining degrees of freedom for flags B i , 1 ≤ i ≤ B n − by thechoice of coordinates x i ∈ F q and a i ∈ F q , respectively used for each line b i and plane B i ,1 ≤ i ≤ n . By labeling lines and planes by their normalized vectors and covectors, we obtainthe description: B : b = B = (1 , , B : b = x B = (1 , , B : b = x B = (1 , a , − a x ) B : b = − a x x + x B = (1 , a , − a x ) B : b = − a x x + x B = (1 , a + a , − a x − a ( x + x )) B : b = − a x − ( a + a ) x x + x + x B = (1 , a + a , − a x − a ( x + x )) B : b = − a x − ( a + a ) x x + x + x B = (1 , a + a + a , − a x − a ( x + x ) − a ( x + x + x ))... ... B k : b k = − (cid:80) ki =2 (cid:16)(cid:80) i − j =1 a j (cid:17) x i (cid:80) kj =1 x j , B k = , k (cid:88) j =1 a j , − k (cid:88) i =1 a i i (cid:88) j =1 x j Since the dot product B k · b k = 0 for all 1 ≤ k ≤ n , the 2-planes B k contain the points b k , asrequired. Define the new variables α i = i (cid:88) j =1 a j , y i = x i +1 , X = n (cid:88) j =1 x j , nd the vectors α = ( α , α , . . . , α n − ), y = ( y , . . . , y n − ). This is an allowed change ofvariables, as it is a triangular and invertible transformation. The equation B = B n givesfour equalities. Two of the equalities are α n = 0, X = 0. The third equation reads α · y = 0 , i.e. n − (cid:88) i =1 α i y i = 0 . The fourth equation, imposed by the vanishing of the third coordinate of B n is dependenton the first three equations, as b n ∈ B n . We are now in position to count solutions of thissystem over F q :(i) Suppose that the vector α ∈ ( F q ) n − is non-vanishing. There are ( q n − −
1) suchpossibilities for α. Then the equation α · y = 0 imposes exactly one linear relationamong the y i variables, 1 ≤ i ≤ n −
1. This yields a choice of q n − possibilities forthe vector y . The contribution in this case is thus ( q n − − q n − .(ii) Suppose that instead α = 0 is the zero vector. Then the equation α · y = 0 is vacuous.The choice of an arbitrary vector y ∈ F n − q completes the count with a factor of q n − .In conclusion, the case A = C yields a total count of( q n − − q n − + q n − ( q − . Finally, adding together the two cases for the relative position of the two flags A , C , we obtaina finite field count of |M ( L n )( F q ) | = 1( q −
1) + ( q n − − q n − + q n − ( q − = q n − − q n − + q n − + q − q − . (cid:3) Note also that the proof of Theorem 6.5 shows that the moduli space of n -gons M n [MGOT12,OST13] admits an embedding into our flag moduli space M ( L n )( C ). In the next section, wewill consider N -graphs G ⊆ D with non-empty boundary ∂G (cid:54) = ∅ , which feature prominentlyin our study of Lagrangian fillings through N -graphs G .7. Microlocal Monodromies and Lagrangian Fillings
This section explains how to use N -graphs G in order to study 2-dimensional exact La-grangian cobordisms between 1-dimensional Legendrian links in ( S , ξ st ) – in particular, thestudy of their exact Lagrangian fillings. Briefly, the Legendrian mutations we developed inSection 4 will be used to construct Lagrangian fillings, and we use microlocal monodromies –and the connection to cluster algebras – to distinguish them. The proof of Theorem 1.8, usingthese two steps to build infinitely many distinct Lagrangian fillings for a class of Legendrianknots, is also given here.7.1. Exact Lagrangian Cobordisms.
This manuscript has heretofore focused on the studyof Legendrian surfaces in an ambient 5-dimensional contact manifold. In fact, the theory of N -graphs and Legendrian weaves that we have developed is also useful for studying exactLagrangian fillings of 1-dimensional Legendrian links Λ ⊆ ( S , ξ st ) and, more generally, exactLagrangian cobordisms between such Legendrian links. This is also the context in whichapplications to both Spectral Networks and Soergel Calculus should arise.There are two advantages to studying exact Lagrangian fillings L of ∂L ⊆ ( S , ξ st ) fromthe perspective of N -graphs. First, the manipulation of their Hamiltonian isotopy class ⊆ ( D , ω st ) becomes combinatorial, as do operations such as Polterovich surgery (seeTheorem 4.10). Second, the computation of cluster coordinates for the augmentation varietyAug(Λ) associated to the Legendrian link ∂L = Λ ⊆ ( S , ξ st ) is accessible. Remark 7.1.
The cluster structures in the coordinate rings of Aug(Λ) have proven to bean effective method for proving new results for Legendrian knots in the 3-sphere [STWZ19,CG20]. We do not know how to prove these cited results using Floer-theoretic methods (suchas the Legendrian DGA [Che02, Etn05]), nor is there currently a Floer-theoretic description for the cluster coordinates induced by an exact Lagrangian filling L ⊆ ( D , ω st ). (cid:3) In this section we present the context in which Legendrian weaves Λ( G ) provide exact La-grangian cobordisms. This is a viewpoint that we will use extensively in the reminder of thearticle, including Section 8 and Appendix A.7.1.1. The geometric setup.
Let ( R , ξ st ) have coordinates ( x, y, z, s, t ) ∈ R , contact 1-form α st = e s ( dz − y dx ) − dt , and let π : ( R , ξ st ) −→ ( R , λ st ) be the projection π ( x, y, z, s, t ) =( x, y, z, s ). Consider the contact 3-planes ( R l , ξ st ) := { t = 1 , s = l } ⊆ R and choose twoLegendrians Λ ⊆ ( R , ξ st ) and Λ = ( R , ξ st ). Suppose that Λ ⊆ ( R , ξ st ) is a Legendriansurface with isotropic boundaries ∂ Λ = Λ (cid:116) Λ , and Λ = Λ ∩ ( R , ξ st ), Λ = Λ ∩ ( R , ξ st ).The crucial geometric fact is that the projection π (Λ) ⊆ ( R , λ st ) is an immersed exactLagrangian, whose immersion points are in bijection with the Reeb chords of Λ ⊆ ( R , α st ).In particular, if the Legendrian surface Λ ⊆ ( R , ξ st ) has no Reeb chords, then the Lagrangianimage π (Λ) ⊆ ( R , λ st ) is an embedded exact Lagrangian with boundary Λ (cid:116) Λ . It is readilyverified that π (Λ) is an exact Lagrangian cobordism from Λ to Λ (and not viceversa). Theparticular case of Λ = ∅ yields exact Lagrangian fillings of Λ .In line with the constructions in this article, the Legendrians Λ , Λ ⊆ ( R , ξ st ) that we studyarise from positive braids – see [CG20, Section 2] – and thus can be described as satellites ofthe standard Legendrian unknot Λ st ⊆ ( R , ξ st ). The description in the paragraph above isthen modified as follows. Consider ( J ( S × [1 , , ξ st ), two Legendrian linksΛ ⊆ ( J ( S × { } )) , Λ ⊆ ( J ( S × { } )) , and a Legendrian surface Λ ⊆ ( J ( S × [1 , , ξ st ) such thatΛ ∩ ( J ( S × { } )) = Λ , Λ ∩ ( J ( S × { } )) = Λ . Now, suppose that the surface Λ has no Reeb chords, then the Lagrangian projection π (Λ) ⊆ ( J S × R , λ st ) in the symplectization of ( J S , ξ st ) provides an exact Lagrangiancobordisms from Λ to Λ . The case in which Λ = ∅ can be compactified to ( J D , ξ st ) inthe ( J ( S × { } ) , ξ st ) end, which symplectically corresponds to adding a standard symplectic4-disk ( D , ξ st ) in the concave end of the symplectization, i.e. as an exact symplectic fillingof ( S , ξ st ). Diagrammatically, this implies that we can describe exact Lagrangian fillings ofa positive Legendrian braid Λ = Λ( β ) ⊆ ( S , ξ st ) in ( D , ω st ) by drawing N -graphs in D whose free edges meet the boundary according to a positive braid word β . Here Λ( β ) denotesthe standard satellite of the Legendrian in ( J S , ξ st ) whose front in S × R is given by thepositive braid (word) β .In short, exact Lagrangian fillings between Legendrian links can be studied via the spatialwavefronts of their Legendrian lifts to the contactization, and the techniques we have devel-oped for Legendrian surfaces can be applied. In particular, we can use our diagrammatic N -graph calculus to study and distinguish exact Lagrangian cobordisms. As far as we know, this remains an open question even if the exact Lagrangian filling is given by apinching sequence [EHK16, Pan17b, Pan17a]. .1.2. Free N -Graphs. Let G β be the set of N -graphs on a 2-disk D with boundary braidword β . As stated above, in order to construct embedded exact Lagrangian fillings L ⊆ ( D , ω st ) for Λ( β ) ⊆ ( S , ξ st ) as N -graphs G ⊆ D in G β , we must have that the Legendrianweave Λ( G ) ⊆ ( R , ξ st ) has no Reeb chords. Let us introduce the following: Definition 7.2. An N -graph G ⊆ D is said to be free if its associated Legendrian frontΣ( G ) can be woven with no Reeb chords. (cid:3) In this section many of the N -graphs G ⊆ D can be checked to be free by direct inspection. Example 7.3.
Let G ⊆ D be a 2-graph such that ( D \ G ) / ( ∂ D ∩ ( D \ G )) is simply-connected. Then G is free if and only if G has no faces contained in the interior of D . Figure87 shows four examples of 2-graphs. Figure 87.
Two free 2-graphs ( i ) and ( ii ), shown on the Left. Two 2-graphs,( iii ) and ( iv ), whose woven front must have a Reeb chord (Right). Each ofthe fronts associated to the non-free two 2-graph can be woven with exactlyone Reeb chord, as indicated. In both cases, the green lines depict the twosheets of a woven front and the orange segments indicate the distance betweenthese sheets. On the left, these length of the distance grows as we approachthe boundary, whereas for the 2-graph ( iii ) there must be a maximum for thisdistance, forcing a Reeb chord.The two 2-graphs ( i ) , ( ii ) on Figure 87 (Left) are free. For that, consider a smooth 1-dimensional foliation of D \ G whose leaves are open intervals and such that the closure ofeach leave intersects ∂ D . The radial-like yellow foliations depicted in Figure 87 (Left) suffice.Then choose a woven front for such 2-graphs such that the differences between the heightsof the two sheets of the front strictly increase along each of the leaves of this foliation, being0 at G and having positive value at ∂ D . These woven fronts do not have Reeb chords, asthe functions giving the differences of heights between the sheets do not have critical points.In contrast, such foliations do not exist for the two 2-graphs ( iii ) , ( iv ) on Figure 87 (Right),as D \ G contains a region whose closure is contained in the interior of D . It can be shownthat any front woven with respect to ( iii ) or ( iv ) must have a Reeb chord and there exists awoven front with a minimal number of Reeb chords, one per each interior face of G . (cid:3) From the perspective of Lagrangian fillings, the 2-graph ( i ) in Figure 87 is an embedded(exact) Lagrangian filling for the 2-component standard unlink, which is the union of twodisjoint Lagrangian disks D ∪ D . The 2-graph ( ii ) yields the embedded Lagrangian fillingfor the standard unknot, which is the standard flat Lagrangian disk D ⊆ D . This standsin contrast with the immersed Lagrangian fillings represented by ( iii ) and ( iv ). The 2-graph( iii ) is an immersed exact Lagrangian annulus with boundary the 2-component standardunlink, and ( iv ) is an immersed exact Lagrangian once-punctured 2-torus filling the standardLegendrian unknot. In general, the following criterion is useful: Lemma 7.4.
Let G ⊆ D be a free N -graph. Then the N -graph µ ( G ) ⊆ D , obtained from G by performing a Legendrian mutation at any I -cycle or Y -tree of G , is also free. roof. Consider the 2-graph mutation at a monochromatic i -edge of an N -graph G . Let O p ( e ) be a neighborhood of a monochromatic edge e in a free N -graph. The 2-graph mutationalong the 1-cycle γ e can then be performed by the exchange in Figure 88, which builds onFigure 58 (Left). Since both 2-graphs G and µ e ( G ) in the exchange coincide in a neighborhoodof the boundary, we can force that the front woven with respect to µ e ( G ) coincides identically– not just up to homotopy of Legendrian fronts – with the given front Σ( G ) woven with respectto G . Let us choose a 1-dimensional foliation in D with respect to G , as in Example 7.3,such that the difference between the heights of any pair of sheets in the woven front strictlyincrease (or decreases) as we move along the sheets of the foliations away from G . (Thisfoliation exists because G is free.) We have depicted such a foliation for G in Figure 88. Figure 88.
Mutation for an N -graph G along a monochromatic i -edge e . Themutated graph µ e ( G ) admits a woven front Σ( µ e ( G )) which coincides with anyfront Σ( G ) woven with respect to G near the boundary of the neighborhood O p ( e ). The yellow foliation near the boundary fixes the difference betweenthe i th and ( i + 1)th sheets in both fronts Σ( G ) and Σ( µ e ( G )). This foliationis extended to the interior in two different ways, yellow or red, depending onthe graph being G or µ e ( G ).In order to guarantee that µ e ( G ) is free, we construct a front Σ( µ e ( G )) woven with respectto µ e ( G ) as follows: this new front is identical to that of G near the boundary of the neigh-borhood of the monochromatic edge, and the j -th sheets for Σ( µ e ( G )) coincide with thoseof Σ( G ) except for the sheets corresponding to j = i, i + 1. The i th and ( i + 1)th sheetsof Σ( µ e ( G )) are woven according to µ e ( G ) such that the difference in heights between the i th and the ( i + 1)th sheets increases (or decreases) strictly along the 1-dimensional red fo-liation as we move away from µ e ( G ) as shown in Figure 88 (Right). Since the red foliationis drawn to coincide with the yellow foliation at the boundary of the neighborhood O p ( e ),this is consistent with the sheets coinciding in that neighborhood. Given that the leaves ofthe 1-dimensional red foliation are intervals with a free end, it is possibly to build such afront, meeting the condition that the difference of heights between i th and ( i + 1)th strictlyincreases (or decreases). In addition, we can draw the front Σ( µ e ( G )) such that the slopes ofeach sheet are arbitrarily close to the slopes of Σ( G ). This guarantees that µ e ( G ) is free asrequired.For a general N -graph mutation along a I - or Y -cycle, it suffices to observe that Subsections4.8 and 4.9 show that such mutations are given by a composition of Legendrian Reidemeistermoves, as presented in Subsection 4.1, and mutations along monochromatic edges. Legen-drian Reidemeister moves are local, relative to the boundary, and can be performed withoutever introducing Reeb chords. Thus an N -graph mutation µ ( G ) of a free G is free if thestatement holds for 2-graph mutations, which we have already proven above. (cid:3) Lemma 7.4 allows us to perform Legendrian mutations to the N -graph and obtain potentiallynew embedded exact Lagrangian fillings. Examples of this are now illustrated. We will mplicitly apply Lemma 7.4 in Subsection 7.3, in order to realize cluster mutations as N -graph mutations of embedded exact Lagrangian fillings.7.1.3. Explicit Examples of Lagrangian Fillings.
For the case of free 2-graphs on a disk D ,this immediately yields that the max-tb Legendrian (2 , n )-torus positive link Λ(2 , n ) has atleast a Catalan C n number worth of exact Lagrangian fillings [EHK16, Pan17b, STZ17, TZ18].This is because C n counts binary trees, which are equivalent to free 2-graphs. These exactLagrangian fillings are distinguished, up to Hamiltonian isotopy, through the use of clustercoordinates – see Subsection 7.2.1. Now, the ability to increase N ∈ N greatly expands the class of Legendrian links for which their Lagrangian fillings can be studied with N -graphcalculus, including all Legendrian positive braids Λ( β ), β ∈ Br + N for any N ∈ N . Example 1 : Recently, the first examples of Legendrian links with infinitely many exact La-grangian fillings were described in the article [CG20]. We exhibit them here in terms of 3-graphs. For any ( p, q ) ∈ N , the max-tb Legendrian ( p, q )-torus positive link Λ( p, q ) ⊆ ( S , ξ st )is the satellite of the braid ∆( σ σ · . . . · σ p − ) q ∆ along the standard Legendrian unknot. Letus now illustrate how to diagrammatically visualize these infinitely many Lagrangian for theLegendrian link Λ(3 , Remark 7.5.
Similar p -graphs can be drawn for Λ( p, q ) for all ( p, q ) ∈ N and they produceinfinitely many Lagrangian fillings if p ≥ , q ≥ p, q ) = (4 , , (4 , p, q ) ⊆ ( S , ξ st ), p ≥ , q ≥ ,
6) [CG20, Corollary 1.5]. (cid:3)
Consider the braid word β = ( σ σ ) = ∆( σ σ ) ∆ in the 1-jet space ( J S , ξ st ). This braid β can be depicted as a set of points in the circle S labeled with two colors, corresponding to σ , σ . Figure 89 shows this braid β in two circles, the inner circle S × { } and outer circle S × { } in the annulus S × [1 , , β is the (3 , , G ⊆ S × [1 ,
2] depicted in Figure 89 describes a Legendrian surface Λ( G ) ⊆ ( J S × [1 , , ξ st ) with boundary Λ(3 , (cid:116) Λ(3 , G ) can be assumed to have no Reeb chords, and thus π (Λ( G )) is an exact Lagrangian cobordism from Λ(3 ,
6) to itself. Since the graph G hasno trivalent vertices, Λ( G ) has the topology of Λ(3 , × [1 ,
2] and it is in fact an exactLagrangian concordance. The remarkable property of the 3-graph G , and its Lagrangianprojection π (Λ( G )), is stated in the following: Theorem 7.6 ([CG20]) . The -graph exact Lagrangian concordance in Figure 89 has infiniteorder. In particular, for any fixed exact Lagrangian filling of Λ(3 , , iterated concatenationof this -graph yields infinitely many Lagrangian fillings of the Legendrian link Λ(3 , ⊆ ( S , ξ st ) . (cid:3) In fact, it is possible to describe the entire faithful modular PSL(2 , Z )-representation in[CG20] with the diagrammatics of 3-graphs. Similarly, the diagrammatics of 4-graphs giveexplicit spatial wavefronts for the M , -worth of the (Legendrian lift of the) Lagrangian fill-ings for the Legendrian link Λ(4 , ⊆ ( S , ξ st ). The non-triviality, and infinite order, of thisLagrangian concordance is detected by studying its action on the cluster structure of thecoordinate ring of the moduli space of isomorphism classes of simple objects in Sh Λ(3 , ( R ). This is particularly relevant for the study of exact Lagrangian fillings, as it is expected that any Λ( β )with β ∈ Br +2 has only finitely many exact Lagrangian fillings, and we will show in Theorem 7.14 that this is not the case already for N = 3. igure 89. Legendrian weave whose Lagrangian projection defines an infi-nite order element in the fundamental group of the space of Legendrian linksisotopic to Λ(3 , , n, m ), n ≥ , m ≥
6, are obtained byconcatenating this 3-graph.
Example 2 : Let us address the following question. Given a positive braid β , and the Legen-drian link Λ = Λ(∆ β ∆), how do we diagrammatically produce an N -graph which representsan embedded exact Lagrangian filling for Λ ⊆ ( S , ξ st ) ?Let us begin with a simple example, with β = ∆ = ( σ σ ) the full-twist, which is smoothlythe (3 , τ i -edges along the boundary ∂ D of a (planar)2-disk D according to the braid word β and complete these edges to an N -graph G inside D . The only rule is that the Legendrian weave Λ( G ) should not have Reeb chords, or elseit would yield an immersed Lagrangian filling, and thus we require G to be free.Consider the free 3-graph G in Figure 90 (upper Left). This represents an embedded exactLagrangian filling L of the max-tb Legendrian (3 , ,
3) = Λ(∆ β ∆) = Λ(∆ ).We can now apply the Legendrian mutation moves in Theorem 4.21 in order to produceanother Lagrangian filling L which is not Hamiltonian isotopic to the exact Lagrangianfilling L . (Note that L and L are smoothly isotopic relative to their boundaries, and L will be also embedded thanks to Lemma 7.4.) In Figure 90 we perform a Lagrangian disksurgery on L along a Lagrangian 2-disk which bounds the 1-cycle in H ( L , Z ) graphicallygiven by the Y -cycle in surrounded by the dashed green curve.At this stage we can manipulate L with Theorem 4.2, in this case Figure 90 (upper right)to 90 (bottom left) shows how to apply Move II to push-through a hexagonal vertex througha trivalent vertex (as indicated by the green arrow). This is an interesting move because itmakes a new 1-cycle for L readily visible, as represented by the blue monochromatic edge in90 (bottom left) surrounded by a dashed green curve. We can perform Lagrangian surgeryat this monochromatic edge, as in Theorem 4.21, to obtain another exact Lagrangian filling L , also embedded by Lemma 7.4. It is immediate that L and L are not not Hamiltonianisotopic to L , as the cluster coordinates associated to these 3-graphs, as explained in Sub-section 7.2.1, show that L and L are not Hamiltonian isotopic. In conclusion, the 3-graphsin Figure 90 represent three distinct embedded exact Lagrangian fillings for Λ(3 , igure 90. Four 3-graphs representing exact embedded Lagrangian fillingsfor the maximal-tb (3 , , Example 3 : Let us illustrate what a generic β ∈ Br +3 .The pictures in the case of β ∈ Br + N , N ≥ N − β = ( σ σ σ ) σ σ σ σ σ ( σ σ σ ) , which has no particular significance to us. To obtain exact Lagrangian fillings, we drawblue and red edges around a circle S ⊆ R , according to σ or σ , and construct 3-graphswith no Reeb chords and these boundary constraints. Figure 91 shows four free 3-graphs G i , i ∈ [1 , π ( ι (Λ( G i ))) ⊆ ( D , ω st ) are embedded exactLagrangian fillings which are distinct up to Hamiltonian isotopy for i (cid:54) = j , i, j ∈ [1 , Remark 7.7.
From our experience drawing 3-graphs, the pictures in Figure 91 accuratelyrepresent the generic appearance of exact Lagrangian fillings described by free 3-graphs. Wepresently do not know any example of a Lagrangian filling for a positive braid which does not arise as an N -graph, for some N ∈ N . (cid:3) Remark 7.8.
There exists a technique for producing many such free N -graphs G , filling β -boundary conditions at a circle and thus representing embedded exact Lagrangian fillings.This is ongoing work by the first author, which in particular proves that any Legendrianlink Λ( β ) arising from a positive braid β ∈ Br + N admits an embedded Lagrangian fillingwhose Legendrian lift is a Legendrian weave. In precise terms, it can be proven that for eachtriangulation of a | β | -gon, one can assign a free N -graph which represented an embeddedLagrangian filling of β , where | β | is the length of the positive braid β . (cid:3) Microlocal monodromies and cluster structures.
In this section, we demonstratehow notions of cluster theory are borne out with N -graphs. This is an important ingredientin showing that microlocal monodromies can be used to distinguish exact Lagrangian fillings,as we do in Section 7.3 and as has been mentioned previously.To orient the discussion, we recall that the cluster structures on the Fock-Goncharov modulispaces of framed local systems described in [FG06b] were given a sheaf-theoretic descriptionin [STW16, STWZ19]. In these works, the spectral surface associated to a bipartite graph, asdefined in [Gon17, Section 2.2], is described symplectically as an exact Lagrangian filling ofthe zigzag Legendrian curve. In the case of bipartite graphs associated to an N -triangulation, igure 91. Four exact embedded Lagrangian fillings for the braid β in Ex-ample 3. Their satellites in ( R , ω st ) are smoothly isotopic relative to theirboundaries, but not Hamiltonian isotopic.)as in [Gon17, Section 1], the zigzag curves isotope to concentric circles around the vertices ofthe triangulation, and the singular support of such a configuration translates to the data ofa local system with a monodromy-invariant flag at each vertex. Sheaf quantization [GKS12]then implies that local systems on the exact filling embed as a cluster chart of objects, thechart being provided by the bipartite graph (and its dual quiver), and the cluster coordi-nates given by microlocal monodromies. The intersection form in H ( L, Z ), or its negative,corresponds to the skew-symmetric bilinear form in cluster theory. For us, the crucial pointis that we can represent all these Lagrangian fillings by N -graphs, as in the diagrammaticsof Subsection 7.1, and the cluster coordinates can be read directly from the N -graph, as wewill now explain. Remark 7.9.
In [TZ18], the case of Legendrian surfaces defined by trivalent 2-graphs wasstudied, giving a sheaf-theoretic description of the constructions in [DGG16]. In this setting,the microlocal monodromy functor µmon induces, at the level of moduli of objects, a mor-phism from the sheaf moduli space to the cluster chart defined by the triangulation dual tothe 2-graph. The image is a (holomorphic) Lagrangian in a (holomorphic) symplectic leaf, s in [DGG16], in a manner compatible with quantization of algebra of functions. Fur-thermore, in that work, the potential describing the local exact structure of the Lagrangianwas interpreted as a generator of BPS states or disk invariants, following the analysis ofAganagic-Vafa [AV00, AV12]. Here we generalize some of the constructions to N -graphs, N ≥ (cid:3) In this article, the Legendrians surfaces are described by N -graphs, a more complex construc-tion, but we will now explain how the basic features should persist. That is, the microlocalmonodromy functor allows us to read cluster coordinates for the moduli spaces of isomor-phism classes of simple objects in Sh Λ ( R ), equivalently augmentation varieties, directly from N -graphs with boundary Λ. Examples of these constructions are provided below.7.2.1. Microlocal monodromies as cluster coordinates.
By definition, microlocal monodromyis a functor µmon : Sh Λ → Loc (Λ)from the category Sh Λ of sheaves microsupported on the Legendrian surface Λ, as defined inSubsection 5.3, to the category of local systems on Λ [STZ17]. This functor carries microlocalrank-one sheaves F ∈ Sh , i.e. simple sheaves, to rank one local systems on the surface Λ.Since it is locally defined, the monodromy of the local system µmon ( F ) around a loop γ ∈ H (Λ) can be evaluated by restricting the constructible sheaf F to an annular tubularneighborhood of γ . Below, these annuli are depicted as thin purple loops. In short, thecalculation for Legendrian weaves can be done using the microlocal monodromy functor µmon as it is used for knots, as described in [STZ17].The main point in these computations is that the stalk µmon ( F ) | λ at a point λ ∈ Λ isthe cone of the restriction map corresponding to λ , and for flags this is the inclusion ofsubspaces, whence cones become cokernels. The transversality of adjacent flags ensures thatthese cokernels propogate as a local system. Let us now perform these calculations for 1-cycle γ ∈ H (Λ , Z ), starting at the I -cycle represented by a monochromatic edge.Let us consider a monochromatic edge with label τ i , as depicted in Figure 92. ab c de Figure 92.
Neighborhood of a monochromatic edge e with the data de-termining a constructible sheaf F . As we show, the microlocal monodromy µmon ( F ) along the 1-cycle γ ( e ) is given by the cross-ratio (cid:104) a, b, c, d (cid:105) .Near such a monochromatic edge, a sheaf object in a simply connected face is specified bythe data of a quadruple of flags. Each of these flags has the same subspaces F j in eachregion for j (cid:54) = i , and for j = i we additionally require the data in each region of a line l in the two-dimensional space V := F i +1 / F i − . This is the data of four lines a, b, c, d ⊆ V ,as specified in Figure 92. Restricted to the purple oval shown, we have a cylindrical braidof type β = σ i , where σ i is the lift of the transposition τ i from the Coxeter group S N tothe braid group Br N . Given the prescribed transversality imposed by the flag moduli of an N -graph, we further know that the cyclic chain of inequalities a (cid:54) = b (cid:54) = c (cid:54) = d (cid:54) = a holds. Wethus have the chain of isomorphisms of cokernels a ∼ = V /b ∼ = c ∼ = V /d ∼ = a, In work in progress with Linhui Shen, the second author will develop the relation to cluster theory moresystematically, and prove Lagrangianicity of the moduli space. hich computes the microlocal monodromy. In this case, the isomorphism that we obtain isthe cross ratio (cid:104) a, b, c, d (cid:105) = a ∧ bb ∧ c · c ∧ dd ∧ a of the four lines a, b, c, d, and it is equal to the cluster coordinate associated to 1-cycle γ asprescribed in [FG06b, Section 9].Let us now consider the cluster coordinate associated to a Y -cycle, which is a new type of1-cycle, as it only appears for N ≥
3. Figure 93 depicts a Y -cycle, drawn as a purple circle,along with the data determining a constructible sheaf in a neighborhood of this 1-cycle. • •• ( a, A )( c, C ) ( b, B ) ( a , a b )( b , a b ) ( c , b c )( b , b c ) ( c , a c ) ( a , a c ) Figure 93.
Neighborhood of a Y -cycle with the data determining a con-structible sheaf F . As we compute, the microlocal monodromy µmon ( F )along the associated 1-cycle γ is given by the triple ratio of the three trans-verse flags.Following the notation in Section 6, we denote by ab the unique plane containing the two lines a and b , while AB denotes the intersection of the planes A and B . The braid associated tothe Y -cycle γ , as drawn by the purple circle in Figure 93, is given by β = ( σ i σ i +1 σ i ) , where σ i corresponds to the crossing coming from a τ i -edge. By considering the three-dimensionalvector space V := F i +2 / F i − , a given flag is specified by a line and a plane in V . Since theword σ i σ i +1 σ i represents the half-twist ∆ for flags on V , and τ i τ i +1 τ i is the Coxeter elementin S , the complete data specifying a constructible sheaf near the Y -cycle is given by threetransverse flags ( a, A ) , ( b, B ) , ( c, C ) in V . In this notation, the line is written in lower caseand the covector defining the the plane in upper case, thus ( a, A ) determines a flag. Now,the microlocal monodromy functor µmon along γ is computed as the composition of theisomorphisms a ∼ = V /B ∼ = c ∼ = V /A ∼ = b ∼ = V /C ∼ = a. Let v a ∈ a, v b ∈ b, v c ∈ c, v d ∈ d be non-zero vectors defining the corresponding one-dimensional lines. Then the parallel transport from a to c in this basis is given by thequotient B ( a ) /B ( c ), where B ( a ) is the pairing between the vector v a and the covector B .Iterating these isomorphisms, we conclude that the microlocal monodromy along the Y -cycleis given by (cid:104) ( a, A ) , ( b, B ) , ( c, C ) (cid:105) := B ( a ) C ( b ) A ( c ) B ( c ) C ( a ) A ( b ) . This expression is precisely the triple product of transverse flags as defined in [FG06b], andthus we have shown that the microlocal monodromy along a Y -cycle determines a clustercoordinate. .2.2. Legendrian Mutations are cluster transformations.
The coordinate transformationsupon Legendrian mutations can also be computed, as we will demonstrate in an example.The conclusion is that Legendrian mutations induce cluster transformations. The case of amonochromatic edge follows from the analysis in [TZ18, STWZ19], and we now study themutation at a Y -cycle. To do so, consider the local geometry shown in Figure 94. We wantto compute how the cluster coordinate associated to the unique monochromatic (blue) edge– as in Subsection 7.2.1 – changes as we perform a Legendrian mutation along the Y -cyclespecified by the unique hexagonal vertex. • •• • ( a, A )( c, C ) ( b, B ) ( a , a b )( b , a b ) ( c , b c )( b , b c ) ( c , a c ) ( a , a c ) ( b, B (cid:48) ) Figure 94.
The geometric setup before performing a Legendrian mutationat the Y -cycle, where the cluster coordinate associated to the monochromaticedge is given by the cross-ratio (cid:104) B, bc, ab, B (cid:48) (cid:105) .The monochromatic blue edge has monodromy equal to the cross ratio z := (cid:104) B, bc, ab, B (cid:48) (cid:105) ofthe four planes in the projective line of planes containing b . (This can be computed directlyor by intersecting the four lines with any transverse line – see Subsection 7.2.1.) Now, afterLegendrian mutation at the Y -cycle, the resulting 3-graph is shown Figure 95. ••• •• ( a , a b )( b , a b ) ( A B , A )( A B , B ) ( c , b c )( b , b c ) ( B C , B )( B C , C ) ( c , a c ) ( a , a c ) ( A C , C ) ( A C , A ) ( b, B (cid:48) )( a, A )( c, C ) ( b, B ) Figure 95.
The result of applying a Legendrian mutation to Figure 94 alongthe Y -cycle, along with the data of a constructible sheaf.The 1-cycle determined by the blue monochromatic edge in Figure 94 becomes a (bichromaticedge) 1-cycle contained in the 3-graph shown in Figure 96, which is itself a piece of Figure95: • ( b, B (cid:48) )( b, B ) ( AB,B ) ( AB, A )( AC, A ) ( a, A ) ( a,ab ) ( b, ab ) Figure 96.
Local geometry near the 1-cycle after mutation.By applying Move II, we can push the red trivalent vertex in Figure 96 through the hexavalentvertex. This allows us to represent the 1-cycle as a monochromatic edge again, as shown inFigure 97: • • ( b, B (cid:48) )( b, B )( b, bAC ) ( b, ab ) Figure 97.
The constructible sheaf near the 1-cycle after Legendrian muta-tion and Move II. The new coordinate is thus the cross-ratio (cid:104)
B, bAC, ab, B (cid:48) (cid:105) .The required conclusion, stating that the new cross-ratio z (cid:48) = (cid:104) B, bAC, ab, B (cid:48) (cid:105) is obtainedby a cluster transformation, follows from this:
Lemma 7.10.
Let x = (cid:104) ( a, A ) , ( b, B ) , ( c, C ) (cid:105) be the triple ratio of flags and z = (cid:104) B, bc, ab, B (cid:48) (cid:105) the cross-ratio of lines. Denote by z (cid:48) = (cid:104) B, bAC, ab, B (cid:48) (cid:105) the new microlocal monodromy. Then z (cid:48) = z (1 + x ) . Proof.
By PGL invariance, we may assume that a = , A = (0 , , , b = , B = (1 , , , c = − , C = (1 , x, x ) . Since the cross-ratio z is prescribed, we find that B (cid:48) = ( z, , AC = x − . This implies that bAC = (1 , x, z (cid:48) = z (1 + x ). (cid:3) Note that z (cid:48) = z (1 + x ) in Lemma 7.10 is the transformation expected for a cluster-Xtransformation. This concludes that a Legendrian mutation at the Y -cycle induces a cluster transformation for the microlocal monodromy coordinate at the monochromatic blue edge inFigure 94. The computation is analogous if we choose a different blue monochromatic edgeto be added near the Y -cycle. In particular, if we had chosen instead the blue edge attaching The rule for a cluster-X transformation upon mutating at loop k is that the monodromy z i transformsto 1 /z k if i = k and otherwise z (cid:48) i = z i (1 + z − sgn (cid:15) ik k ) − (cid:15) ik , where (cid:15) i,k is the skew-symmetric cluster form. Weget agreement on the nose if we make this form the negative of the intersection pairing. t the lower-right of the Y -cycle and pointing upward, and again called its monodromy z ,then we would have A (cid:48) = (0 , z,
1) and would obtain z (cid:48) = (cid:104) ab, aBC, A, A (cid:48) (cid:105) = z (cid:18) x (cid:19) − , in agreement with the cluster transformation. Example 7.11. Flip of a N -triangulation . Let ( C, τ ) be a punctured surface C , τ an idealtriangulation and τ (cid:48) an ideal triangulation obtained from τ by a flip. Denote by t N , resp. t (cid:48) N ,the N -triangulation refinement of τ , resp. τ (cid:48) . It is an exercise [Gon17, Prosition 1.1] to showthat the Legendrian weave Λ( G ( t (cid:48) N )) differs from Λ( G ( t N )) by a sequence of (cid:0) N +13 (cid:1) G ( t (cid:48) N )) can be obtained from Λ( G ( t N )) by performing (cid:0) N +13 (cid:1) Legendrianmutations along 1-cycles represented by monochromatic edges.For instance, [Gon17, Figure 9] translates into four monochromatic edge mutations for a flipin a N = 3 triangulation, as we have depicted in Figure 98. We can see how to perform thecorresponding moves for 3-triangulations with 3-graphs. Indeed, referring to the notationin Figure 104, perform a monochromatic edge mutation at z and w , then perform MoveIII, a flop of the two trivalent and two hexagonal vertices in the center, and proceed with amutation at the remaining two monochromatic edges. In conclusion, the constructions of thispaper can therefore be used to give a geometric understanding of the intermediate quiversarising when flipping N -triangulations. Figure 98.
Flip in a 3-triangulation realized as four monochromatic edgemutations. In general, the Legendrian weaves associated to two N -triangulations which differ by a flip of the underlying (1-)triangulation dif-fer by a sequence of (cid:0) N +13 (cid:1) such 2-graph mutations. Note the Move III flopisotopy in-between the two mutation pairs. We remark that the case x = − c ∈ ab, andthus not in the domain of the birational cluster map. .3. N-graph Realization of Quiver Mutations.
In this subsection we explain how touse N -graphs in order to construct infinitely many Lagrangian fillings for certain Legendrianlinks in the standard contact 3-sphere ( S , ξ st ). These Lagrangian fillings are distinguishedby the microlocal monodromies/cluster coordinates in Subsection 7.2.Let β ∈ Br + N be a positive braid, with a fixed braid word w ( β ). Consider an N -graph G ⊆ D on the 2-disk such that the labels of the edges of G near ∂ D , read cyclically, form the word w ( β ). Following Subsection 7.1, the Lagrangian projection L ( G ) = π ( ι (Λ( G ))) ⊆ ( R , ω st )of ι (Λ( G )) ⊆ ( R , ξ st ) is an exact Lagrangian filling of the Legendrian link Λ( β ) ⊆ ( S , ξ st )associated to the positive braid β . All the N -graphs G ⊆ D which feature in this subsectionwill be free, and thus the Lagrangian projections are embedded, equivalently Λ( G ) has noReeb chords.Now consider a free N -graph G ⊆ D , b ( G ) := rk( H (Λ( G ) , Z )) and a basis B = { [ γ ] , . . . , [ γ b ( G ) ] } for H (Λ( G ) , Z ) ∼ = Z b ( G ) , equivalently a basis for the first homology group of its Lagrangianprojection L ( G ). For a choice of basis B , we denote by Q ( B ) the intersection quiver of the1-cycles γ i , i ∈ [1 , b ( G )]. The vertices v i of the quiver Q ( B ) are in bijection with elements ofthe homology basis B , and the number of arrows between two distinct vertices v i , v j is givenby the geometric intersection number | γ i ∩ γ j | . The direction of each arrow is given by thesign of each geometric intersection, and there are no loops, i.e. no edges from a vertex v i toitself. The quiver obtained by mutation of a quiver Q at the vertex v i will be denoted µ i ( Q ).We will study the realization of quiver mutations, algebraic in nature, as Legendrian mu-tations of free N -graphs, which are geometric. Suppose that there exists a subset B µ ofclasses [ γ i ], i ∈ [1 , k ], for some k ≤ b ( G ), such that γ i ∈ B µ is represented by a 3-graphcycle with no multiplicity. That is, each 1-cycle γ i is represented by either a Y -cycle, a tree,a monochromatic edge I -cycle or a long edge. Let { x , . . . , x k } be the cluster coordinatesassociated to { γ , . . . , γ k } via microlocal monodromies, as in Subsection 7.2.1. Remark 7.12.
In general, this set of cluster coordinates { x , . . . , x k } is only a partial subsetof the entire cluster seed { x , . . . , x b ( G ) } for H ( L ( G ) , Z ). The ability to work with a subsetis an advantage that allows for our methods to be applied in more generality. From theviewpoint of cluster algebras, the vertices of Q ( B ) which are not in Q ( B µ ) are to be consideredas frozen vertices, and the variables { x k +1 , . . . , x b ( G ) } as frozen coordinates. (cid:3) By Subsection 4.8, and Lemma 7.4, we can perform a Legendrian mutation at γ i ∈ B µ andobtain a free N -graph µ i ( G ). The intersection quiver Q ( µ i ( B )) associated to the mutatedbasis µ i ( B ) is the mutated quiver µ i ( Q ( B )). The 1-cycle in the mutated graph µ i ( G ) cor-responding to γ j ∈ B , under mutation at γ i , is denoted by µ i ( γ j ). By Subsection 7.2.2, thecluster coordinate associated to µ i ( γ j ) is given by the j -th coordinate in the cluster trans-formation of { x , . . . , x b ( G ) } at x i . Therefore, the exact Lagrangian filling represented bythe free N -graph µ i ( G ) has intersection quiver µ i ( Q ( B )) and cluster coordinates obtained bymutation of the cluster seed { x , . . . , x b ( G ) } for L ( G ) at x i . In conclusion, if the 1-cycles arerepresented by trees, performing one quiver (or cluster seed) mutation as a Legendrian mu-tation is possible, and the microlocal monodromies after the Legendrian mutation accuratelyreflect cluster mutation. Remark 7.13.
The challenging aspect of the geometric side is that iterating this procedureis not necessarily possible, or at least readily accessible. This aspect is not reflected in he algebra of quiver mutations (or cluster coordinate mutations) since, by definition, twoopposite edges between vertices are canceled . (cid:3) The technology of 3-graphs and their mutations, as developed in Subsection 4.8, allows us toiterate Legendrian mutations in an abundance of cases, including arbitrarily high genus. Wewill illustrate explicit cases in which an infinite sequence of quiver mutations can be realizedas an infinite sequence of N -graph mutations. These cases can be inserted in (infinitely many)other examples, and the first consequence is the production of new families of Legendrianlinks with infinitely many exact Lagrangian fillings: Theorem 7.14.
Let Λ s,t = Λ( β s,t ) ⊆ ( S , ξ st ) be the Legendrian link given by the standardsatellite of the positive braid β s,t = ( σ σ )( σ σ ) s σ σ ( σ σ ) t ( σ σ )( σ t +12 σ σ s +22 ) , s, t ∈ N , s, t ≥ . Then Λ s,t ⊆ ( S , ξ st ) admits infinitely many embedded exact Lagrangian fillings in ( D , λ st ) realized as -graphs G s,t ⊆ D and their Legendrian mutations.Proof. The argument is uniform for all s, t ∈ N and all the difficulties, and their solutions,are already present for the simplest case. Let us thus assume s = t = 1 for now. First,we need to construct a free 3-graph G = G , which represents a Lagrangian filling for theLegendrian link Λ( β ) associated to β = β , . This 3-graph is shown in Figure 99:
12 3 4 56
Figure 99.
The 3-graph G and the initial Quiver Q .The exact Lagrangian L ( G ) associated to G is a genus-4 surface with two boundary com-ponents, and thus b ( G ) = 9. Let us consider the subset B µ = { γ , γ , γ , γ , γ } given bythe following 1-cycles: γ is represented by the yellow 1-cycle in Figure 99, which is a treeof Y -pieces, and γ , γ , γ , γ are represented by monochromatic edges, in purple in Figure99. In addition, we consider the 1-cycle γ represented by the monochromatic edge, in green. In previous attempts to geometrically iterate Lagrangian mutations, such as [STW16, Section 2], thisobstruction manifests itself as embedded curves becoming immersed upon performing Dehn twists, a problemwhich presently has no known solution. We thank Dylan Thurston for useful discussions on quivers and their mutations. In particular, forproviding the infinite sequence of mutations that we use in this proof. he intersection quiver Q = Q ( B µ ∪ { γ } ) is given by the quiver drawn in Figure 99. Thequiver Q is of infinite mutation type, as it is associated to hyperbolic Coxeter diagram[Law17, Table 1]. In fact, we claim that the sequence of quiver mutations µ s n , where s n = n ≡
12 if n ≡
23 if n ≡
34 if n ≡
45 if n ≡ , is an infinite sequence of quiver mutations. Indeed, each time we apply the sequence ofmutations µ µ µ µ µ , the number of arrows from the vertex v to v increases by two, andthe number of arrows from v i to v , for i ∈ [2 ,
5] increases by one. In particular, at the k thiteration there are 2 k + 1 arrows from v to v and k arrows from v i to v , i ∈ [2 , Y -tree which represents the 1-cycle γ . Theresulting 3-graph, which is free by Lemma 7.4, is depicted in Figure 100:
12 3 4 56
Figure 100.
Mutated 3-graph µ ( G ) at the 1-cycle γ (yellow) correspondingto vertex 1 in Q , and its associated intersection quiver µ ( Q ).Now, upon this Legendrian mutation at γ the 1-cycles γ , γ , γ , γ are still represented bymonochromatic edges. These new 1-cycles µ ( γ ) , µ ( γ ) , µ ( γ ) , µ ( γ ) are circled in purplein Figure 100. The figure also displays the mutated quiver µ i ( Q ( B µ ∪ { γ } )) and the cycle Precisely, the quiver Q corresponds to the rank 6 paracompact hyperbolic Coxeter group L = [3 [1 , , , , ]. in green. Similarly, upon this Legendrian mutation, the 1-cycle µ ( γ ) is still representedby an embedded Y -tree, as depicted in yellow in Figure 100.These properties hold true as we now perform Legendrian mutations at the monochromaticedges γ , γ , γ , γ . The free 3-graph resulting from these four mutations is drawn in Figure101:
12 3 4 56 (x3)
Figure 101.
Mutated 3-graph µ µ µ µ µ ( G ) and its associated intersectionquiver µ µ µ µ µ ( Q ).The claim is that we can iterate the sequence of mutations µ µ µ µ µ geometrically asLegendrian mutation of the free 3-graph, and these two properties hold. That is, at anystage in the sequence of mutations µ s n we have that(i) the 1-cycles γ , γ , γ , γ are represented by monochromatic edges,(ii) the 1-cycle γ is represented by an embedded Y -tree, with no multiplicities.In fact, the Y -tree representing γ always has exactly four Y -pieces. These four pieces havebeen surrounded by a dashed pink circle in Figures 99 through 103. The two items abovecan be readily verified, as follows. The behavior of the mutated 3-graph near each of the themonochromatic edges is as depicted in Figure 102:It thus follows that γ i , i ∈ [2 , any iteration ofthe 3-graph mutation µ µ µ µ µ . Similarly, according to the Legendrian mutation rulesof Subsection 4.8, each Y -piece of the Y -tree representing γ itself mutates to a Y -piece,and mutating at γ , γ , γ , γ preserves this property. Thus the pattern persists upon any igure 102. The effect of the sequence of mutations µ µ µ µ µ near themonochromatic edges γ i , i ∈ [2 , γ i are µ and µ i .iteration. The two properties ( i ) and ( ii ) now allow us to perform the sequence of mutations µ s n up to any point in the sequence. For instance, the sequence of mutations µ µ µ µ µ µ applied to G lead to the 3-graph in Figure 103:
12 3 4 56 (x3)(x2) (x2)(x2)(x2)
Figure 103.
Mutated 3-graph µ µ µ µ µ µ ( G ) and its associated intersec-tion quiver µ µ µ µ µ µ ( Q ).In order to pairwise distinguish the exact Lagrangian fillings associated to the sequence of3-graphs ( µ s n µ s n − · · · µ )( G ), up to Hamiltonian isotopy, we use the microlocal monodromies { x , x , x , x , x , x } along the 1-cycles γ i , i ∈ [1 , { x , x , x , x , x , x } associated to the quiver Q mutates to the clusterseed associated to ( µ s n µ s n − · · · µ )( Q ) upon performing the Legendrian 3-graph mutations( µ s n µ s n − · · · µ )( G ). Since the quivers ( µ s n µ s n − · · · µ )( Q ) are distinct, and so are the as-sociated cluster seeds, it follows that the associated Lagrangian fillings are distinct. Thisconcludes the proof for s = t = 1.The general case s, t ∈ N is proven with the same argument. Indeed, the free 3-graph G , in Figure 99 generalizes to a 3-graph whose boundary is β s,t , just by adding s copies ofthe leftmost pattern in G , , to the left, and t copies of the rightmost pattern in G , , to he right. In this general case, it is still true that γ is represented by a Y -tree and theremaining { γ , γ , . . . , γ s , γ s +1 , . . . , γ s + t +3 } cycles are represented by monochromatic edges.The argument is then identical, with the infinite sequence of mutations given by s n = i, if n ≡ i (mod s + t + 3) , ≤ i ≤ s + t + 3 . The reader can directly verify that this is an infinite sequence of mutations, as the multiplicityof the arrows to the cycle γ s + t +4 – generalizing the green cycle γ in Figure 99 – increases aswe apply the mutations µ s + t +3 µ s + t +2 · · · µ µ . (cid:3) Remark 7.15.
For s = t = 1, note that the sequence µ s n never mutates at the 1-cycle γ ,i.e. at the sixth vertex v in Q . It is nevertheless crucial to include γ in the quiver as wellas the cluster variable x , with its subsequent mutations. Note that the 1-cycle γ is initiallyrepresented by an embedded curve in the 3-graph, but this curve develops immersed pointsas we iterate the sequence of mutations µ s n according to Subsection 4.8. This still allows usto define the cluster coordinate associated to it but we would not be able to mutate alongsuch a 1-cycle just with the rules developed in Subsection 4.8. (This is just a side remark,since the argument for Theorem 7.14 does not require mutating at v .) (cid:3) The Legendrian links in Theorem 7.14 are relatively simple. For instance, the Legendrianknot associated to β , is genus-4 two-component link. One of the components is an unknotand the other is the (2 , . Note that Λ( β , ) is (smoothly) distinct from the(3 , new method to construct infinitely many Lagrangian fillings,but it in fact provides new Legendrian links with infinitely many Lagrangian fillings. Remark 7.16.
Note that the L quiver that we used in Theorem 7.14 appears as a subquiverof the intersection quiver for several other positive braids. Following L. Lewark’s positivebraid table each of the following positive genus-6 braids, 14 n , 15 n , 16 n ,16 n , 16 n , 16 n and 16 n , to name a few, contain L in their intersectionquiver. We believe that an argument similar to Theorem 7.14 should prove that the maximal-tb representative of each of these links has infinitely many exact Lagrangian fillings. (cid:3) Finally, the contrast between Theorem 7.14 and [CG20, Corollary 1.5] is interesting. Theformer constructs an infinite family of Lagrangian fillings for a Legendrian link by directlyusing Legendrian mutations, which are themselves distinguished by their effect – as clustermutations – on the microlocal monodromies. The latter result [CG20] is entirely about con-structing infinite order Lagrangian concordances, coming from Legendrian loops of positivebraids, and the infinite family of Lagrangian fillings is a byproduct of such construction. Inparticular, N -graph calculus should apply to much more general Legendrian links, and doesnot require knowing about the existence of an infinite order element in their Lagrangianconcordance monoid.8. Moduli Space for N -triangles and Non-Abelianization In this final section, we focus on N -graphs associated to N -triangulations, as introduced inSection 3. This class of N -graphs G yields Legendrian weaves Λ( G ) whose Lagrangian projec-tions are related to the Goncharov-Kenyon conjugate Lagrangian surfaces [GK13, STWZ19].These Lagrangian surfaces have also appeared in the context of Gaiotto-Moore-Neitzke’sspectral networks [GMN13, Nei14]. In particular, we prove Theorem 1.9, which computesthe flag moduli space M ( G ) for G any N -triangle t N , matching the algebraic results in[GMN14, Section 8] and [FG06b, Section 9]. In this case the Y -tree has ( s + t + 2) Y -pieces, s + 1 to the left and t + 1 to the right of the base root. Lukas Lewark’s Positive Knots Table: Braids and Trees at “ http://lewark.de/lukas/braids.html ”. .1. Flag moduli space of the N -triangle. Let us compute the flag moduli space associ-ated to the N -graph G ( t N ) of an N -triangle t N , as we defined in Section 3.2 (see Figure 15).The result reads as follows: Theorem 8.1.
Let G ( t N ) be the N -graph associated to an N -triangle t N . The flag modulispace of G ( t N ) is a (cid:0) N − (cid:1) -dimensional complex torus, i.e. M ( t N , G ( t N ); k ) ∼ = ( k ∗ )( N − ) . This rest of this subsection is devoted to the proof of Theorem 8.1. The statement ofTheorem 8.1 is an instance of how incidence geometry problems connect to the contacttopology of Legendrian surfaces. Indeed, although our proof is entirely within projectivegeometry, the conclusion from Theorem 8.1 ought to be read as the fact that the modulispace M ( t N , G ( t N ); k ) is parametrized by the toric coordinates provided by the holonomiesHom( H (Λ( G ( t N ) , Z )) , k ∗ ). For k = C , this complex torus should be related to the complextorus appearing in Fock-Goncharov [FG06b, FG06a] in their study of cluster varieties, see[Kuw20, Theorem 8.3].Theorem 8.1 can also be interpreted as follows. The triangle t N is topologically a disk D with boundary a circle ∂ D = S . The Legendrian weaveΛ( G ( t N )) ⊂ ( J ( D ) , ξ st )has a Lagrangian projection L := π (Λ( G ( t N ))), which is an exact Lagrangian submanifold,where π : J ( D ) −→ T ∗ D is the projection along the standard (vertical) Reeb flow. TheLagrangian L has boundary in T ∗ D | S ∼ = J ( S ) , and it is checked that ∂L is the cylindricalLegendrian braid ∆ , where ∆ is the half-twist positive braid corresponding to a longest wordin the Weyl group, i.e. the Garside element. Since G ( t N ) is free, L is an embedded exactLagrangian filling of ∂L . Now, by looking at the boundary circle S and considering themoduli space `a la [STZ17], we conclude that the moduli space of Lagrangian fillings shouldcarry a cluster structure: the flag moduli space M ( t N , G ( t N ); k ) is one such chart.In fact, by an argument akin to Lemma 5.6, the flags at two vertices of the triangle t N determine the flags along the edge they bound, and therefore the flags along the boundarycircle δ D = S must be determined by the flags at the vertices, themselves three mutuallycompletely transverse flags in the flag variety B . This space of triples of mutually transverseflags is one of the Richardson varieties R . Now, by the PGL N action, two totally transverseflags can be put in standard position B , B − , with residual symmetry the Cartan H ofdiagonal matrices up to scale. Then the moduli space R /H is a cluster variety and the exactLagrangian filling L provides a cluster chart via its moduli of local systems Loc ( L ) ∼ = Hom( H (Λ( G ( t N ) , Z )) , k ∗ ) ∼ = ( k ∗ )( N − ) . This torus can be checked to agree with that of 8.1. Note also that, following Section 7many other cluster charts and exact Lagrangian fillings can be found by performing N -graphmutations. Let us now prove our result: Proof of Theorem 8.1.
Let us argue by induction on N , where the base case N = 2 followsfrom the fact that PGL acts transitively on triples of distinct points. Let us assume that M ( t N , G ( t N ); k ) ∼ = ( k ∗ )( N − )for the N -graph of an N -triangle. Consider an ( N + 1)-triangle with one side being anarbitrary fixed preferred base, and thus the row associated to this base contains 2 N − t N +1 is infact an N -triangle t N , and thus we can construct t N +1 by adding such base row to t N . Thiscombinatorial splitting is translated into a containment of an N -graph G ( t N ) within G ( t N +1 ). We thank Ian Le for many discussions on the Richardson variety. et us describe such splitting in the ( N +1)-graph by providing its construction starting fromthe N -graph G ( t N ).Start with the N -graph G ( t N ) – see Section 3.2 – and consider the (cid:0) N (cid:1) edges intersectingthe base side of t N . The edges are depicted vertically and the base side horizontally – seeFigure 15. These are τ i -edges, i = 1 , . . . , N −
1, with exactly ( N − − k ) τ k -edges. The( N + 1)-graph G ( t N +1 ) can be described in the following N stages:1. First, insert an ( N − , N )-hexagonal point in the unique τ N − edge in the base sideof G ( t N ). The τ N -edge aligned with the previously existing τ N − -edge is continueddown vertically. The remaining two τ N -edges are extended horizontally, respectivelyto the left and to the right, and the remaining two τ N − edges are continued downdiagonally, in south-east and south-west direction respectively.2. Second, continue down the τ i -edges, i = 1 , . . . , N , until the two τ N − -edges intersectwith the two originally existing τ N − N − , N − τ N − and τ N − τ N − -edges adjacent tothe incoming τ N − -edges horizontally to the left and to the right.The remaining two pairs of three edges, each with two τ N − -edges and a τ N − -edge,are continued down, with the τ N − -edges continued vertically and the τ N − -edges con-tinued diagonally in the south-east or south-west directions, accordingly.3. Iteratively, we proceed as follows in the l th stage, 2 ≤ l ≤ N −
1. We continue downthe τ i -edges, i = 1 , . . . , N , and at this stage the only edges being continued diagonallydown are τ N − l +1 -edges. There are 2 l − τ N − l +1 -edges, respectively continuing south-west and south-east. For these two ex-ternal edges, we insert two ( N − l, N − l + 1)-hexagonal and describe the N -graph asdescribed in Stage 2. The 2 l − τ N − l +1 -edges, which continue down diago-nally, ought to intersect with τ N − l -edges, which continue down vertically.For the 2( l −
2) internal edges, there are ( l −
2) such intersections, since an intersectionoccurs for each pair. For each such an intersection, insert a ( N − l, N − l +1)-hexagonalvertex, and continue the outgoing three edges down as described by the local modelfor the hexagonal vertex. Hence, for each of these hexagonal vertices, the outgoing τ N − l +1 -edge continues vertically down whereas the two τ N − l -edges continue down di-agonally. Thus, at the l th stage we have inserted exactly l ( N − l, N − l +1)-hexagonalpoints.4. In the N th stage, all τ i -edges, i ≤ ≤ N continue down vertically and we are leftwith 2( N − τ -edges continuing diagonally. In line with the previous stages, thereare two external τ -edges and 2( N −
2) internal edges. Insert two τ -trivalent verticesat the end of the two external τ -edges. The internal edges will meet in consecutivepairs at N − τ -vertex in each ofthese intersection points, and continue the remaining τ -edge vertically down.Let us now compute the flag moduli space M ( t N , G ( t N +1 ); k ) using this inductive construc-tion of G ( t N +1 ). A crucial fact to be used is Lemma 5.6, i.e. at a hexagonal vertex, fourconsecutive flags uniquely determine the remaining two flags. Let us assume that we havechosen a point in M ( t N , G ( t N ); k ) and we thus have the data of a flag F in P N for each open egion in D \ G ( t N ). This data needs to be considered in the moduli space of flags, giventhat G ( t N +1 ) is an ( N + 1)-graph, and thus we fix an embedding i : P N −→ P N +1 and thecorresponding inclusion PGL N ⊆ PGL N +1 . Let us then start the construction G ( t N +1 ) from G ( t N ) by stages, as described above, and prove the statement in Theorem 8.1.In the first stage, the flag data at the inserted ( N − , N )-hexagonal point in the τ N − -edge is uniquely determined by a choice of a codimension-2 projective subspace H in P N ,transverse to i ( P N ). Note that the intersection of H and i ( P N ) is uniquely determined bythe flag data coming from M ( t N , G ( t N ); k ). We claim that this choice in the first stage canbe absorbed by the symmetry group PGL N +1 .In order to understand the symmetry group, it is convenient to represent an element inPGL N +1 via the projective matrix a , a , . . . a ,N a ,N +1 a , a , . . . a ,N a ,N +1 ... ... . . . ... ... a N, a N, . . . a N,N a N,N +1 a N +1 , a N +1 , . . . a N +1 ,N a N +1 ,N +1 , where the subgroup PGL N ⊆ PGL N +1 is defined byPGL N = { A ∈ PGL N +1 : a N +1 ,N +1 = 1 , a i,N +1 = a N +1 ,i = 0 , ≤ i ≤ N } . In these coordinates, we can assume that the subgroup K ⊆ PGL N +1 fixing our fixed hyper-plane i ( P n ) ⊆ P N − is cut out by the equations K := { A ∈ PGL N +1 : a N +1 ,i = 0 , ≤ i ≤ N } . As a result, the remaining PGL N +1 -symmetries (once the flag moduli space M ( t N , G ( t N ); k )is fixed, and thus the symmetries of PGL N have been used) consist of projective transforma-tions of the form a , a , . . . a ,N c a , a , . . . a ,N c ... ... . . . ... ... a N, a N, . . . a N,N c N . . . c N +1 , where a i,j are fixed, 1 ≤ i, j ≤ N , c i ∈ k , 1 ≤ i ≤ N , and c N +1 ∈ k ∗ are free. Then, in thiscoordinate system, we can assume that the choice of the codimension-2 projective subspace H uses the gauge provided by c , c ∈ k .In the second stage, two ( N − , N − H in each of these twovertices. Let us fix one of these choices by using the free coordinate c ∈ k and notice that he other choice has an a priori moduli of k . Nevertheless, the τ N − -edge that interacts withthe τ N − edges in the third stage forces that moduli to be k ∗ , since the two newly chosen flagsmust be τ N − -transverse. Thus in the second stage we have used the symmetry provided by c ∈ k and we are left with a k ∗ contribution to the flag moduli.In the l th stage, 3 ≤ l ≤ N −
1, we proceed inductively as follows. We partition the( N − l, N − l +1)-hexagonal vertices inserted in this stage into two groups: external, containingtwo of them, and internal, containing ( l −
2) of them. By definition, the two external ( N − l, N − l + 1)-hexagonal vertices are the leftmost and rightmost vertices. Each of these twoexternal vertices have flags fixed in three out of the six regions, by the process in the ( l − N − l, N − l + 1)-hexagonal vertices which determines each of their respective neighborhoods.This corresponds to a choice of codimension- l projective subspace H l +1 in accordance withthe incidence conditions imposed by the given flags. Proceeding as in the second stage, wefix one of these choices with the free variable c l +1 and the remaining choice contributes k ∗ to the flag moduli.The ( l −
2) internal ( N − l, N − l + 1)-hexagonal vertices have flags fixed in four out of thesix regions, given the process in the ( l − k ∗ contribution of oneof the external hexagonal vertices interacts with an internal vertex, no contributions to theflag moduli space come directly from the internal vertices.The argument then develops iteratively in the above manner until the ( N − N th stage consists of the insertion of N τ -trivalent vertices. Followingthe same pattern as before, only the two external trivalent vertices contribute to the flagmoduli, since each of the internal trivalent vertices have their three surrounding flags deter-mined at the ( N − c i , 1 ≤ i ≤ N have been fixedand the only remaining degree of free symmetry is c N +1 ∈ k ∗ . Let us use such symmetry tofix the choice in one of the two external trivalent vertices, and thus the contributions of thislast stage to the flag moduli space is the k ∗ choice of the remaining point coming from theremaining external trivalent vertex.The conclusion in the statement Theorem 8.1 now follows by gathering the contributions ofthe flag moduli space at each stage. Indeed, the first stage has no contribution, whereas eachof the ( N −
1) stages, from the second to the last N th stage, has a k ∗ flag moduli spacecontribution. By the inductive hypothesis, the desired flag moduli space is M ( t N +1 , G ( t N +1 ); k ) ∼ = M ( t N , G ( t N ); k ) × ( k ∗ ) N − ∼ = ( k ∗ )( N − ) × ( k ∗ ) N − ∼ = ( k ∗ )( N ) , which corresponds to the statement, as required. (cid:3) Remark 8.2.
Note that the inductive combinatorial description of G ( t N +1 ) in terms of G ( t N ) used in the proof of Theorem 8.1 can be used to provide a third alternative definitionof the local N -graph G ( t N ), in addition to the descriptions introduced in Subsection 3.2. (cid:3) Tetrahedral Triangulations at N = 3 and N = 4 . Let us study the Legendrian weavesΛ( G ( τ )) and flag moduli space M ( S , G ( τ )) associated to 3- and 4-graphs G ( τ ) for thetetrahedral 3- and 4- triangulations τ of the 2-sphere S . The case N = 2 has been discussedin Subsection 6.1 above, where Λ( G ) ∼ = T c is the Legendrian Clifford Torus and M ( G ) ∼ = P \ { , , ∞} . Let us denote the pair of pants P \ { , , ∞} by H .Let us consider the 3-graph G (3) = G ( τ (3) ) ⊆ S associated to the tetrahedral 3-triangulation τ (3) of the 2-sphere S , according to the construction in Section 3. We want to compute its flagmoduli space M ( G ). This will be done directly by using the N -graph calculus computationsin Section 4. Indeed, it is proven in Subsection 4.6 that in this case the (satellite of the) egendrian weave Λ( G ) is Legendrian isotopic to the four-fold connected sum of the Cliffordtorus T c . Hence, we obtain that M ( S , G (3) ) ∼ = H . From the description in Theorem 8.1,we are also giving a contact geometric proof of the following Corollary 8.3 ([FG06b]) . The moduli of four generic flags in C is isomorphic to H . (cid:3) The same argument, using N -graph calculus also allows us to study the flag moduli space M ( S , G (4) ), where G (4) = G ( τ (4) ) ⊆ S is the 4-graph associated to the tetrahedral 4-triangulation τ (4) of the 2-sphere S . It is left as an exercise for the reader to use Theorem8.1 and conclude that M ( S , G (4) ) is isomorphic, as an algebraic variety, to M ( S , G (4) ) = { ( z , w , . . . , z , w ) ∈ ( C ∗ ) : (1 − κ ) w i z i − z i + 1 = 0 , ≤ i ≤ } × ( H ) , where κ = 1 − z z z z z ∈ C ∗ . The exercise is solved in [DGG16, Section 6.3.2] in thelanguage of the 3 d N = 2 superconformal field theory T [∆ , Π].8.2.
A Computation of the Non-Abelianization Map.
We conclude the main bodyof the manuscript by exploring the relationship between Legendrian weaves and the works[FG06b, AV00, AV12, Pal15] in some explicit examples. In particular, we present a casein which the non-Abelianization map featured in [GMN13, GMN14] can be realized by themicrolocal monodromies associated to constructible sheaves microlocally supported alongLegendrian weaves.The context is described as follows. Let (
C, τ N ) be a polygon endowed with an ideal N -triangulation τ N , and choose a wavefront for Λ( G ( τ N )) with no Reeb chords, such that theLagrangian projection is a smooth exact Lagrangian L embedded in the cotangent bundle( T ∗ C, λ st ). This Lagrangian projection L has a sheaf quantization [NZ09] to a rank- N sheafon C with no singular support, i.e. a local system in C . Now, of course, all local systems onpolygons are trivial, but the crucial point is that the Lagrangian covering gives a preferred basis for the fibers of the local system, which can undergo changes `a la handle-slides inthe Morse context – see [GKS12]. Here, the Lagrangian covering is given by the restriction π | L : L −→ C of the projection π : T ∗ C −→ C onto the zero section. Now, the N -graphs andthe microlocal monodromies, as discussed in Section 7.2, precisely encode these changes. Inour context, the non-Abelianization map is the construction that recovers the constructiblesheaf from its microlocal monodromy.We illustrate this in the following example. Figure 104 shows the 3-graph G associated totwo adjacent 3-triangles. Suppose that we are given a local system on Λ( G ). Denote by x, y the two monodromies of the corresponding Legendrian weave around the two Y -cycles, and z, w the two microlocal monodromies along the two I -cycles represented by the two (red)monochromatic edges. Figure 104.
The 3-graph associated to an adjacent pair of 3-triangles. Themonodromies along the four 1-cycles are labeled x, y, z, w . In the Floer-theoretic languange of the Fukaya category, the basis elements are the intersections of theexact Lagrangian with the cotangent fibers. he Legendrian weave Λ( G ) is a thrice-punctured genus-one surface and these four 1-cyclesare a basis for H (Λ( G ) , Z ). We would like to reconstruct the flag data, specifying a con-structible sheaf, from the monodromies x, y, z, w of the local system. Indeed, this will realizethe Non-Abelianization map [GMN13] from rank-one local systems on the (spectral, or con-jugate) Lagrangian – parametrized by monodromies – to decorated rank-two local systemson the base surface. Since the base surface here is contractible, the only degrees of freedomare the choices of flags at vertices. The map is computed as follows.Let ( a, A ) , ( b, B ) , ( c, C ) be the flags at the vertices of the left triangle, and let ( d, D ) bethe remaining flag. We would like a birational map from the monodromies ( x, y, z, w ) tothe choice of flags. By using the PGL -action, we may assume ( a, A ) , ( b, B ) , ( c, C ) are as inSubsection 7.2.2 above, with triple product x . Then the flag ( c, D ) is determined by the crossratios z and w , and the triple product y . For instance, z is the cross ratio (cid:104) b, BC, AB, BD (cid:105) while we find w = (cid:104) a, AD, AB, AC (cid:105) . These determine D , whence the triple product y fixes d ∈ D . Direct computation shows d = − xq (1 + x ) x (1 + y ) − py (1 + x ) , D = ( pq, p (1 + x ) , x ) . This thus recovers [FG06b, Pal15] from the perspective of N -graphs. Example: Tetrahedron with 3-Triangulation.
Let us conclude this subsection by ana-lyzing the genus-4 Legendrian weave Λ in Example 4.6 from the microlocal perspective. Wealso compute, following [TZ18], the primitive which characterizes (a discrete cover of) M ( G )as an exact Lagrangian subvariety. Following [AV00, AV12], this primitive – the superpoten-tial of an effective 4d theory – is interpreted as a generating function of BPS numbers, andshould have integrality properties. We check this for this example.Consider the tetrahedron with its unique 3-triangulation, as in Example 4.6, which gives riseto a 3-graph G . An object in the category of simple constructible sheaves Sh G ) ( S × R , Λ) microlocally supported along Λ( G ) is defined by a four-tuple of transverse flags in V ∼ = C ,placed at the vertices of the tetrahedron, as in Figure 93.Note that there are 4 · g = 4.We therefore have 2 g = 8 cluster variables, specified by the monodromies around each of theeight loops, which themselves are a basis for H (Λ( G ) , Z ) ∼ = Z . Four of the monodromiesare the triple ratios along the faces. Let us label the faces by the three unordered verticesit contains, e.g. we write x for the monodromy of the loop deteremined by the minimaltriangle at the center of the face (123): it is the triple ratio of the three flags at vertices 1, 2and 3. There are 4 × · e we define a corresponding coordinate x e to be the negative of the cross ratio. Now there are two relations for each vertex: first, the productof the edge coordinates around the encircling triangular face is unity; second, the product ofedge and Y -monodromies encircling the vertex at a greater distance is unity. There are thus8 independent coordinates, and we can take two from each of the triangles surrounding thefour vertices. Let us then write x = − w ∧ v v ∧ v v ∧ v v ∧ w We believe the sign appears due to the fact that we should be considering twisted local systems, i.e. liftsto the circle bundle of the surface that have monodromy − or the coordinate associated to the edge of the triangle encircling vertex 1 and traversingthe one-simplex of the triangulation between vertices 1 and 2, where v i , i ∈ [1 , w i are generators for planes (thought of as anti-symmetric two-vectors)— and likewise for the other edges. Then the relation for the triangle encircling vertex 1 is x x x = 1, and likewise for the other vertices. Recall that we have similarly denoted by x the inverse of the coordinate corresponding to the Y -cycle in the face containing vertices1, 2 and 3 — and likewise for the coordinates x ijk , i, j, k ∈ [1 , x x x x x x = 1 , and likewise for the other three vertices. This expresses the flag moduli in terms of generators,given by x ij , x ijk , and relations, as above.Let us verify that these coordinates define a (holomorphic) Lagrangian embedding of flagmoduli space M ( G ) associated to the genus-4 Legendrian Λ( G ) into the moduli space offramed local systems for C = S . The symplectic 2-form is computed from the intersectionform to be ω = − d log x ∧ d log x + d log x ∧ d log x − d log x ∧ d log x + d log x ∧ d log x . We can directly compute the following four relations x = −
11 + x , x = −
11 + x , x = −
11 + x , x = −
11 + x , which readily imply that the embedding of the flag moduli space M ( G ) in each of thecluster charts for the moduli space of framed local systems is Lagrangian. This holomorphicLagrangian M ( G ) is in fact exact and we can compute a primitive function W for therestriction of the Liouville 1-form λ st . This would allow us to write M ( G ) as the graph Γ dW of the 1-form dW in this chart. This primitive encodes the BPS states associated to someLagrangian filling, given by the Lagrangian projection of Λ( G ), determined by a phase and aframing (implicit here) as in [TZ18, Section 4.8] – see also [AV00, AV12]. For that, we definethe variables U = − x , V = − x , U = − x , V = − x ,U = − x , V = − x , U = − x , V = − x . Also, recall that if we have U + V − = 1 with U = e u and V = e v , then we can write v = − log(1 − U ) = ∂ u Li ( U ) . Hence, since we have U i + V − i = 1 for all i , with symplectic 2-form ω = (cid:80) i du i ∧ dv i , weconclude that M ( G ) = Γ dW where W ( U , U , U , U ) = (cid:88) i =1 Li ( U i ) . This computation for the BPS potential is in line with the results in [TZ18, Section 5].Finally, let us review how geometric methods, as developed in Section 4, would lead tothis result. Instead of the algebraic computation above, we could have directly used thediagrammatic calculus, as in Example 4.6, and deduced that our Legendrian weave Λ( G ) ∼ = i =1 T c is the Legendrian connected sum of four Clifford 2-tori T c . Since the generatingfunction of BPS numbers for T c is given by one dilogarithm Li ( U ), by direct computation,and the potential W is additive under connected sum, we could have directly deduced that W ( U , U , U , U ) = (cid:80) i =1 Li ( U i ). This concludes that our algebraic computation above isconsistent with the contact topology of the underlying Legendrian weave. ppendix A. Soergel Calculus and Legendrian Weaves
In this appendix, we provide a construction and a concise speculation regarding the symplecticgeometrization of Soergel Calculus via Legendrian weaves. The following discussion owes agood deal to B. Elias and E. Gorsky, as explained in the introduction, to whom we arevery grateful. Soergel calculus, as developed by B. Elias, M. Khovanov and G. Williamson[EK10, EW16], provides a diagrammatic presentation of the category of Soergel bimodules,which itself categorifies the Hecke algebra. The similarities between Elias’ diagrammaticcalculus and our Legendrian weaves are apparent. Legendrian weaves can be understood asa geometric approach to the study of the algebra of certain complexes of Soergel bimodules.We explain this below.
Remark A.1.
Soergel bimodules are essential to categorifications of knot invariants [Rou06,Soe07, Kho07, EK10]. The link between these and moduli spaces of sheaves for Legendrianbraid closures was described in [STZ17, Section 6]. From this perspective, it is not unnaturalto seek a connection between planar Soergel structures and planar structure defined byLegendrian weaves, the two-dimensional version of braids. (cid:3)
The category of Soergel bimodules is the Karoubi completion of the subcategory of Bott-Samelson bimodules, arising as the equivariant cohomology of a closed Bott-Samelson vari-ety, and thus it suffices to understand the relation to this latter class of bimodules. The keyconnection between the present work and Soergel bimodules is that a subclass of Legendrianweaves yields exact Lagrangian cobordisms between Legendrian links, which are themselvesrepresented as positive braids. The moduli space of microlocal constructible sheaves sup-ported on a singular compactification of a positive braid is a closed Bott-Samelson variety,and our Legendrian weaves, understood as Lagrangian cobordisms – and singularly compact-ified – induce morphisms between these closed Bott-Samelson varieties.Thus, we are able to geometrize the diagrammatics of Soergel calculus by considering the D − -singularity for the trivalent vertices in [EK10, EW16], the A -swallowtail singularity forthe univalent vertex and the A -singularity for their hexagonal vertices. (The Soergel calcu-lus we geometrize corresponds to the m = 2 Coxeter exponent.) Remark A.2.
Exact Lagrangian cobordisms are directed , due to the convexity directionalityin symplectic topology. The dissonance arises from the fact that, as of today, Soergel calculusonly considers closed Bott-Samelson varieties, whereas the moduli space of microlocal sheavessupported on a positive braid is an open
Bott-Samelson variety. Thus, the Soergel calculusis geometrized by singular compactifications of our Legendrian weaves, and our Legendrianweave calculus, without compactification, should naturally induce a Soergel calculus for open
Bott-Samelson varieties. (cid:3)
For instance, the A -Zamolodchikov relation from Soergel calculus corresponds to the A -Reidemeister move in Legendrian weave calculus, as depicted in Figure 105.Now, let us consider two positive braids β , β ∈ Br + n , n ∈ N , and their associated Leg-endrian (long) links Λ( β ) , Λ( β ) ⊆ ( J [0 , , ξ st ) [CG20, Section 2]. A Legendrian weaveΛ ⊆ ( J ([0 , × [1 , , ξ st ) with no Reeb chords and boundaries Λ( β ) at [0 , × { } , andΛ( β ) at [0 , × { } , yields an embedded and exact Lagrangian cobordism L (Σ) from Λ( β )to Λ( β ) in the symplectization of ( J [0 , , ξ st ), as in Section 7. In particular, each trivalentvertex Σ( G tri ) and hexagonal vertex Σ( G hex ) yield the following exact Lagrangian cobordism:(i) The Lagrangian projection L ( G tri ) of the Legendrian weave Λ( G tri ) is a Lagrangiancobordism from the Legendrian tangle Λ( β ) given by one crossing in two strands igure 105. Contact Isotopy among Legendrian weaves, relative to theboundaries. The lack of Reeb chords allows us to interpret these as exactLagrangian cobordisms between the positive braids σ i +1 σ i σ i − σ i +1 σ i σ i +1 and σ i − σ i σ i − σ i +1 σ i σ i − . The fact that these Lagrangian cobordisms are Hamil-tonian isotopic implies that the morphism induced between the associatedBott-Samelson bimodules must coincide. (cid:3) β = σ i , to the Legendrian tangle Λ( β ) given by two crossing in two strands β = σ i ,where i ∈ N is labeling the transposition τ i of the edges of G tri . Smoothly, this is asaddle cobordism obtained by an index-1 handle attachment to the Lagrangian coneΛ( β ) × [0 , ε ] in the symplectization, for ε ∈ R + small.(ii) The Lagrangian projection L ( G hex ) of the Legendrian weave Λ( G hex ) is a Lagrangianconcordance from the Legendrian tangle Λ( β ) given by three crossings in threestrands β = σ i σ i +1 σ i , to the Legendrian tangle Λ( β ) given by β = σ i +1 σ i σ i +1 ,where i ∈ N is labeling the transpositions τ i , τ i +1 in the edges of G hex . Smoothly,this is a Lagrangian surface obtained by graphing a Reidemeister three move.For simplicity, let us suppose that the relative homology H ( L, ∂ − L ; Z ), which we denote by H ( L ), is a free Z -module and the surface L is spin, as is verified for the two local cobordismsabove. An exact Lagrangian cobordism L ⊆ ( J [0 , , ξ st ) × [1 ,
2] from Λ( β ) to Λ( β ) yieldsan algebraic map Φ L : (cid:99) M (Λ( β )) −→ M (Λ( β )) , where (cid:92) M (Λ( β )) is an algebraic ( C ∗ ) b ( L ) -bundle over M (Λ( β )), and M (Λ( β )) denotes themoduli space of microlocal rank-1 objects in the dg-category of of microlocal sheaves in S × microlocally supported on Λ( β ), as described in [CG20, Section 3], [STWZ19, STZ17]. Remark A.3.
In the Floer-theoretic context, the map Φ L is obtained by applying thecontravariant functor Hom ( · , k ) in the category of dg-algebras to the morphismΦ F lL : A (Λ( β )) −→ A (Λ( β )) ⊗ Z Z [ H ( L )]of the Legendrian Contact dg-algebras A (Λ( β )) associated to Legendrian links Λ( β ). TheFloer theoretic map Φ F lL is described in [EHK16, Pan17b], and it is a count of holomorphicstrips whose boundary homology classes are encoded in Z [ H ( L )]. To ease the geometry, wehave tensored by the flag moduli space map Φ L above C [ H ( L )] ∼ = Z [ H ( L )] ⊗ Z C to basechange the Spec( Z [ H ( L )])-bundle to a complex variety (cid:99) M (Λ( β )). (cid:3) The relation to Soergel calculus now arises because the moduli space of simple microlocalsheaves M (Λ( β )) is (explicitly) isomorphic to the open Bott-Samelson variety associatedto β , also known as the Brou´e-Michel variety of β [STZ17, Tri19, CG20]. Let R = H ∗ ( B ) enote the cohomology of the complete flag variety for GL( N, C ), N ∈ N , and B s i the Bott-Samelson Soergel ( R ⊗ R )-bimodule associated to a permutation s i ∈ S N in the Weyl group S N of GL( N, C ). The Rouquier complex T i := [ B s i −→ R ] will be denoted by T i , for all i ∈ N . Consider a braid β = l (cid:89) j =1 σ i j , ≤ i j ≤ k − , where σ i is the leftmost crossing in the front diagram of the Legendrian braid, and thecrossings are read from left to right. Then the (singular) compactly supported cohomologyof algebraic variety M (Λ( β )) is described by the tensor product T β = T i ⊗ R T i ⊗ R · · · ⊗ R T i l , of Rouquier complexes. Remark A.4.
Should the reader be interested in the closure of the Legendrian Λ( β ) ⊆ ( J S , ξ st ), instead of the long link ( J [0 , , ξ st ), the cohomology of the corresponding modulispace M (Λ( β )) is obtained by applying Hochschild homology to the above complex T β . Inparticular, H ∗ ( M (Λ( β ))) coincides with the triply-graded homology of the knot associatedto β , equivalently, Khovanov-Rozansky link homology – see [STZ17, Theorem 6.14]. (cid:3) In conclusion, the geometric map Φ L : (cid:99) M (Λ( β )) −→ M (Λ( β )) functorially induces H ∗ c (Φ L ) : H ∗ c ( (cid:99) M (Λ( β ))) −→ H ∗ c ( M (Λ( β ))) , which is a map of (products of) Rouquier complexes (cid:98) T β −→ T β , where (cid:98) T β is the com-pactly supported cohomology of (cid:99) M (Λ( β )), which contains the information of the compactlysupported cohomology T β of the open Bott-Samelson variety for β .Now, applying this to the two Lagrangian cobordisms associated to the trivalent vertices G tri and the hexagonal vertices G hex , we obtain the following two maps:(i) The map Φ L ( G tri ) : T s i ⊗ H ∗ ( S ) −→ T s i ⊗ R T s i , where i labels the τ i -edges of G tri ,and we have identified the fiber bundle (cid:99) M (Λ( β )) ∼ = (Λ( β )) × C ∗ with the Cartesianproduct, as in this case the bundle is topologically trivial. The fact that there is one copy of C ∗ = S × R corresponds to the fact that the Lagrangian cobordism L ( G tri )has a unique index 1 critical point and its cocore carries the data C ∗ .(ii) The map Φ L ( G hex ) : T s i ⊗ T s i +1 ⊗ T s i −→ T s i +1 ⊗ T s i ⊗ T s i +1 , where in this case (cid:99) M (Λ( β )) ∼ = M (Λ( β )) as the Lagrangian L ( G hex ) is a cylinder and H ( L ) ∼ = { } istrivial.In conclusion, the above discussion can summarized according to the following tenet: Principle A.5.
Let T β , T β be the Rouquier complexes associated to positive braids β , β and Ψ : T β −→ T β the morphism given by a graph G Ψ with only (upwards) trivalentand hexagonal morphisms in (open) Soergel calculus. Then the Lagrangian projection of theLegendrian weave Λ( G Ψ ) yields an embedded exact Lagrangian cobordism L from Λ( β ) to Λ( β ) and a geometric map Φ L : (cid:99) M (Λ( β )) −→ Λ( β ) such that H ∗ c (Φ L ) = Ψ . (cid:3) The difference between the principle above and a theorem lies on the correct definition of open
Soergel calculus, of which we are not aware at this stage. That said, since the trivalentand the hexagonal vertices are two of the main building blocks for closed Soergel calculus, theabove construction provides a potential symplectic geometrization of open
Soergel calculus, ssociated to Rouquier complexes, instead of Soergel bimodules. In particular, in the contextof open
Bott-Samelson varieties, the Lagrangian cobordisms above indicate the need foradditional data from H ( L ) = Z | V | in specifying a morphism, where | V | is the number oftrivalent vertices. The development of open Soergel calculus, the computations establishingthat our geometric maps induce the expected algebraic maps, as well as the Lagrangiandescription of the univalent vertex, will be the subject of upcoming and more algebraic work. References [AB27] J. W. Alexander and G. B. Briggs. On types of knotted curves.
Ann. of Math. (2) , 28(1-4):562–586,1926/27.[Ad75] Vladimir Igorevich Arnol (cid:48) d. Critical points of smooth functions.
Proceedings of the InternationalCongress of Mathematicians (Vancouver, B. C., 1974), Vol. 1 , pages 19–39, 1975.[Ad76] Vladimir Igorevich Arnol (cid:48) d. Wave front evolution and equivariant Morse lemma.
Comm. PureAppl. Math. , 29(6):557–582, 1976.[Ad79] Vladimir Igorevich Arnol (cid:48) d. Indexes of singular points of 1-forms on manifolds with boundary,convolutions of invariants of groups generated by reflections, and singular projections of smoothsurfaces.
Uspekhi Mat. Nauk , 34(2(206)):3–38, 1979.[Ad90] Vladimir Igorevich Arnol (cid:48) d. Singularities of caustics and wave fronts , volume 62 of
Mathematicsand its Applications (Soviet Series) . Kluwer Academic Publishers Group, Dordrecht, 1990.[AdG01] V. I. Arnol (cid:48) d and A. B. Givental (cid:48) . Symplectic geometry [ MR0842908 (88b:58044)]. In
Dynamicalsystems, IV , volume 4 of
Encyclopaedia Math. Sci. , pages 1–138. Springer, Berlin, 2001.[Aur07] Denis Auroux. Mirror symmetry and T -duality in the complement of an anticanonical divisor. J.G¨okova Geom. Topol. GGT , 1:51–91, 2007.[Aur09] Denis Auroux. Special Lagrangian fibrations, wall-crossing, and mirror symmetry. In
Surveysin differential geometry. Vol. XIII. Geometry, analysis, and algebraic geometry: forty years ofthe Journal of Differential Geometry , volume 13 of
Surv. Differ. Geom. , pages 1–47. Int. Press,Somerville, MA, 2009.[AV00] Mina Aganagic and Cumrun Vafa. Mirror Symmetry, D-Branes and Counting Holomorphic Discs.2000.[AV12] Mina Aganagic and Cumrun Vafa. Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots. 2012.[Ben83] Daniel Bennequin. Entrelacements et ´equations de Pfaff. In
Third Schnepfenried geometry confer-ence, Vol. 1 (Schnepfenried, 1982) , volume 107 of
Ast´erisque , pages 87–161. Soc. Math. France,Paris, 1983.[Ben86] Daniel Bennequin. Caustique mystique (d’apr`es Arnol (cid:48) d et al.). Number 133-134, pages 19–56.1986. Seminar Bourbaki, Vol. 1984/85.[Bir74] Joan S. Birman.
Braids, links, and mapping class groups . Princeton University Press, Princeton,N.J.; University of Tokyo Press, Tokyo, 1974. Annals of Mathematics Studies, No. 82.[BM08] J. A. Bondy and U. S. R. Murty.
Graph theory , volume 244 of
Graduate Texts in Mathematics .Springer, New York, 2008.[BS15] Tom Bridgeland and Ivan Smith. Quadratic differentials as stability conditions.
Publ. Math. Inst.Hautes ´Etudes Sci. , 121:155–278, 2015.[BST15] Fr´ed´eric Bourgeois, Joshua M. Sabloff, and Lisa Traynor. Lagrangian cobordisms via generatingfamilies: construction and geography.
Algebr. Geom. Topol. , 15(4):2439–2477, 2015.[CE12] Kai Cieliebak and Yakov Eliashberg.
From Stein to Weinstein and back , volume 59 of
AmericanMathematical Society Colloquium Publications . American Mathematical Society, Providence, RI,2012. Symplectic geometry of affine complex manifolds.[CE14] Kai Cieliebak and Yakov Eliashberg. Stein structures: existence and flexibility. In
Contact andsymplectic topology , volume 26 of
Bolyai Soc. Math. Stud. , pages 357–388. J´anos Bolyai Math.Soc., Budapest, 2014.[CG20] Roger Casals and Honghao Gao. Infinitely many Lagrangian fillings.
ArXiv e-prints , 2020.[Che02] Yuri Chekanov. Differential algebra of Legendrian links.
Invent. Math. , 150(3):441–483, 2002.[CM18] Roger Casals and Emmy Murphy. Differential algebra of cubic planar graphs.
Adv. Math. , 338:401–446, 2018.[CM19] Roger Casals and Emmy Murphy. Legendrian fronts for affine varieties.
Duke Math. J. , 168(2):225–323, 2019.[CMP19] Roger Casals, Emmy Murphy, and Francisco Presas. Geometric criteria for overtwistedness.
J.Amer. Math. Soc. , 32(2):563–604, 2019.[Con84] Marston D. E. Conder. Some results on quotients of triangle groups.
Bull. Austral. Math. Soc. ,30(1):73–90, 1984.
DGG16] Tudor Dimofte, Maxime Gabella, and Alexander B. Goncharov. K-decompositions and 3d gaugetheories.
J. High Energy Phys. , (11):151, front matter+144, 2016.[Die17] Reinhard Diestel.
Graph theory , volume 173 of
Graduate Texts in Mathematics . Springer, Berlin,fifth edition, 2017.[DR11] Georgios Dimitroglou Rizell. Knotted Legendrian surfaces with few Reeb chords.
Algebr. Geom.Topol. , 11(5):2903–2936, 2011.[DR16] Georgios Dimitroglou Rizell. Legendrian ambient surgery and Legendrian contact homology.
J.Symplectic Geom. , 14(3):811–901, 2016.[EENS13a] Tobias Ekholm, John Etnyre, Lenhard Ng, and Michael Sullivan. Filtrations on the knot contacthomology of transverse knots.
Math. Ann. , 355(4):1561–1591, 2013.[EENS13b] Tobias Ekholm, John B. Etnyre, Lenhard Ng, and Michael G. Sullivan. Knot contact homology.
Geom. Topol. , 17(2):975–1112, 2013.[EES05a] Tobias Ekholm, John Etnyre, and Michael Sullivan. The contact homology of Legendrian sub-manifolds in R n +1 . J. Differential Geom. , 71(2):177–305, 2005.[EES05b] Tobias Ekholm, John Etnyre, and Michael Sullivan. Non-isotopic Legendrian submanifolds in R n +1 . J. Differential Geom. , 71(1):85–128, 2005.[EGH00] Y. Eliashberg, A. Givental, and H. Hofer. Introduction to symplectic field theory.
Geom. Funct.Anal. , (Special Volume, Part II):560–673, 2000. GAFA 2000 (Tel Aviv, 1999).[EHK16] Tobias Ekholm, Ko Honda, and Tam´as K´alm´an. Legendrian knots and exact Lagrangian cobor-disms.
J. Eur. Math. Soc. (JEMS) , 18(11):2627–2689, 2016.[EK10] Ben Elias and Mikhail Khovanov. Diagrammatics for Soergel categories.
Int. J. Math. Math. Sci. ,pages Art. ID 978635, 58, 2010.[Eli90] Yakov Eliashberg. Topological characterization of Stein manifolds of dimension > Internat. J.Math. , 1(1):29–46, 1990.[Eli93] Yakov Eliashberg. Legendrian and transversal knots in tight contact 3-manifolds. In
Topologicalmethods in modern mathematics (Stony Brook, NY, 1991) , pages 171–193. Publish or Perish,Houston, TX, 1993.[EM02] Y. Eliashberg and N. Mishachev.
Introduction to the h -principle , volume 48 of Graduate Studiesin Mathematics . American Mathematical Society, Providence, RI, 2002.[ENS18] Tobias Ekholm, Lenhard Ng, and Vivek Shende. A complete knot invariant from contact homology.
Invent. Math. , 211(3):1149–1200, 2018.[Etn05] John B. Etnyre. Legendrian and transversal knots. In
Handbook of knot theory , pages 105–185.Elsevier B. V., Amsterdam, 2005.[EV18] John Etnyre and Vera V´ertesi. Legendrian satellites.
Int. Math. Res. Not. IMRN , (23):7241–7304,2018.[EW16] Ben Elias and Geordie Williamson. Soergel calculus.
Represent. Theory , 20:295–374, 2016.[FG06a] V. V. Fock and A. B. Goncharov. Cluster x-varieties, amalgamation, and Poisson-Lie groups.In
Algebraic geometry and number theory , volume 253 of
Progr. Math. , pages 27–68. Birkh¨auserBoston, Boston, MA, 2006.[FG06b] Vladimir Fock and Alexander Goncharov. Moduli spaces of local systems and higher Teichm¨ullertheory.
Publ. Math. Inst. Hautes ´Etudes Sci. , (103):1–211, 2006.[FOOO09] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.
Lagrangian intersection Floer the-ory: anomaly and obstruction. Part II , volume 46 of
AMS/IP Studies in Advanced Mathematics .American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.[FZ02] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Foundations.
J. Amer. Math. Soc. ,15(2):497–529, 2002.[Gei08] Hansj¨org Geiges.
An introduction to contact topology , volume 109 of
Cambridge Studies in Ad-vanced Mathematics . Cambridge University Press, Cambridge, 2008.[GK13] Alexander B. Goncharov and Richard Kenyon. Dimers and cluster integrable systems.
Ann. Sci.´Ec. Norm. Sup´er. (4) , 46(5):747–813, 2013.[GKS12] St´ephane Guillermou, Masaki Kashiwara, and Pierre Schapira. Sheaf quantization of Hamiltonianisotopies and applications to nondisplaceability problems.
Duke Math. J. , 161(2):201–245, 2012.[GLPY17] Maxime Gabella, Pietro Longhi, Chan Y. Park, and Masahito Yamazaki. BPS graphs: fromspectral networks to BPS quivers.
J. High Energy Phys. , (7):032, front matter+47, 2017.[GMN10] Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke. Four-dimensional wall-crossing viathree-dimensional field theory.
Comm. Math. Phys. , 299(1):163–224, 2010.[GMN13] Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke. Spectral networks.
Ann. HenriPoincar´e , 14(7):1643–1731, 2013.[GMN14] Davide Gaiotto, Gregory W. Moore, and Andrew Neitzke. Spectral networks and snakes.
Ann.Henri Poincar´e , 15(1):61–141, 2014.[Gom98] Robert E. Gompf. Handlebody construction of Stein surfaces.
Ann. of Math. (2) , 148(2):619–693,1998.
Gon17] A. B. Goncharov. Ideal webs, moduli spaces of local systems, and 3d Calabi-Yau categories. In
Algebra, geometry, and physics in the 21st century , volume 324 of
Progr. Math. , pages 31–97.Birkh¨auser/Springer, Cham, 2017.[GPS19a] Sheel Ganatra, John Pardon, and Vivek Shende. Microlocal Morse theory of wrapped Fukayacategories. 2019.[GPS19b] Sheel Ganatra, John Pardon, and Vivek Shende. Covariantly functorial wrapped Floer theory onLiouville sectors.
Publ. Math. Inst. Hautes ´Etudes Sci. (to appear) , 2019.[GPS19c] Sheel Ganatra, John Pardon, and Vivek Shende. Sectorial descent for wrapped Fukaya categories.2019.[Gro86] Mikhael Gromov.
Partial differential relations , volume 9 of
Ergebnisse der Mathematik und ihrerGrenzgebiete (3) . Springer-Verlag, Berlin, 1986.[GS14] St´ephane Guillermou and Pierre Schapira. Microlocal theory of sheaves and Tamarkin’s non dis-placeability theorem. In
Homological mirror symmetry and tropical geometry , volume 15 of
Lect.Notes Unione Mat. Ital. , pages 43–85. Springer, Cham, 2014.[GS18] Alexander Goncharov and Linhui Shen. Donaldson-Thomas transformations of moduli spaces ofG-local systems.
Adv. Math. , 327:225–348, 2018.[Hur92] A. Hurwitz. Ueber algebraische Gebilde mit eindeutigen Transformationen in sich.
Math. Ann. ,41(3):403–442, 1892.[K´05] Tam´as K´alm´an. Contact homology and one parameter families of Legendrian knots.
Geom. Topol. ,9:2013–2078, 2005.[Kel94] Bernhard Keller. Deriving DG categories.
Ann. Sci. ´Ecole Norm. Sup. (4) , 27(1):63–102, 1994.[Kho07] Mikhail Khovanov. Triply-graded link homology and Hochschild homology of Soergel bimodules.
Internat. J. Math. , 18(8):869–885, 2007.[KS85] Masaki Kashiwara and Pierre Schapira. Microlocal study of sheaves.
Ast´erisque , (128):235, 1985.Corrections to this article can be found in Ast´erisque No. 130, p. 209.[KS90] Masaki Kashiwara and Pierre Schapira.
Sheaves on manifolds , volume 292 of
Grundlehren derMathematischen Wissenschaften . Springer-Verlag, Berlin, 1990. With a chapter in French byChristian Houzel.[KS10] Maxim Kontsevich and Yan Soibelman. Motivic Donaldson-Thomas invariants: summary of re-sults. In
Mirror symmetry and tropical geometry , volume 527 of
Contemp. Math. , pages 55–89.Amer. Math. Soc., Providence, RI, 2010.[KS14] Maxim Kontsevich and Yan Soibelman. Wall-crossing structures in Donaldson-Thomas invariants,integrable systems and mirror symmetry. In
Homological mirror symmetry and tropical geometry ,volume 15 of
Lect. Notes Unione Mat. Ital. , pages 197–308. Springer, Cham, 2014.[Kuw20] Tatsuki Kuwagaki. Sheaf quantization from exact WKB analysis. 2020.[Law17] John W. Lawson. Minimal mutation-infinite quivers.
Exp. Math. , 26(3):308–323, 2017.[LO08] Yves Laszlo and Martin Olsson. The six operations for sheaves on Artin stacks. I. Finite coeffi-cients.
Publ. Math. Inst. Hautes ´Etudes Sci. , (107):109–168, 2008.[LO09] Yves Laszlo and Martin Olsson. Perverse t -structure on Artin stacks. Math. Z. , 261(4):737–748,2009.[LO10] Valery A. Lunts and Dmitri O. Orlov. Uniqueness of enhancement for triangulated categories.
J.Amer. Math. Soc. , 23(3):853–908, 2010.[LT99] A. Lucchini and M. C. Tamburini. Classical groups of large rank as Hurwitz groups.
J. Algebra ,219(2):531–546, 1999.[Mar35] A.A. Markov. ¨uber die freie aquivalenz der geschlossen zopfe.
Recueil Math. Moscou , 1:73–78,1935.[MGOT12] Sophie Morier-Genoud, Valentin Ovsienko, and Serge Tabachnikov. 2-frieze patterns and the clus-ter structure of the space of polygons.
Annales de l’Institut Fourier , 62(3):937–987, 2012.[Mur12] Emmy Murphy. Loose Legendrian Embeddings in High Dimensional Contact Manifolds. 2012.[Nad09] David Nadler. Microlocal branes are constructible sheaves.
Selecta Math. (N.S.) , 15(4):563–619,2009.[Nad17a] David Nadler. Arboreal singularities.
Geom. Topol. , 21(2):1231–1274, 2017.[Nad17b] David Nadler. A combinatorial calculation of the Landau-Ginzburg model M = C , W = z z z . Selecta Math. (N.S.) , 23(1):519–532, 2017.[Nei14] Andrew Neitzke. Cluster-like coordinates in supersymmetric quantum field theory.
Proc. Natl.Acad. Sci. USA , 111(27):9717–9724, 2014.[Ng03] Lenhard L. Ng. Computable Legendrian invariants.
Topology , 42(1):55–82, 2003.[Ng11] Lenhard Ng. Combinatorial knot contact homology and transverse knots.
Adv. Math. , 227(6):2189–2219, 2011.[NR13] Lenhard Ng and Daniel Rutherford. Satellites of Legendrian knots and representations of theChekanov-Eliashberg algebra.
Algebr. Geom. Topol. , 13(5):3047–3097, 2013.
NRS +
15] Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow. Augmentations areSheaves.
To appear in Geom. Top. , 2015.[NZ09] David Nadler and Eric Zaslow. Constructible sheaves and the Fukaya category.
J. Amer. Math.Soc. , 22(1):233–286, 2009.[OST13] Valentin Ovsienko, Richard Evan Schwartz, and Serge Tabachnikov. Liouvillearnold integrabilityof the pentagram map on closed polygons.
Duke Math. J. , 162(12):2149–2196, 2013.[Pal15] Frederic Palesi. Introduction to positive representations and Fock-Goncharov Coordinates. 2015.[Pan17a] Yu Pan.
Augmentations and Exact Lagrangian Cobordisms . ProQuest LLC, Ann Arbor, MI, 2017.Thesis (Ph.D.)–Duke University.[Pan17b] Yu Pan. Exact Lagrangian fillings of Legendrian (2 , n ) torus links.
Pacific J. Math. , 289(2):417–441, 2017.[Pol91] L. Polterovich. The surgery of Lagrange submanifolds.
Geom. Funct. Anal. , 1(2):198–210, 1991.[PS97] V. V. Prasolov and A. B. Sossinsky.
Knots, links, braids and 3-manifolds , volume 154 of
Transla-tions of Mathematical Monographs . American Mathematical Society, Providence, RI, 1997.[Rei27] Kurt Reidemeister. Elementare Begr¨undung der Knotentheorie.
Abh. Math. Sem. Univ. Hamburg ,5(1):24–32, 1927.[Rol76] Dale Rolfsen.
Knots and links . Publish or Perish Inc., Berkeley, Calif., 1976. Mathematics LectureSeries, No. 7.[Ros98] Dennis Roseman. Reidemeister-type moves for surfaces in four-dimensional space. In
Knot theory(Warsaw, 1995) , volume 42 of
Banach Center Publ. , pages 347–380. Polish Acad. Sci. Inst. Math.,Warsaw, 1998.[Rou06] Rapha¨el Rouquier. Categorification of sl and braid groups. In Trends in representation theoryof algebras and related topics , volume 406 of
Contemp. Math. , pages 137–167. Amer. Math. Soc.,Providence, RI, 2006.[RS19a] Daniel Rutherford and Michael Sullivan. Cellular Legendrian contact homology for surfaces, partII.
Internat. J. Math. , 30(7):1950036, 135, 2019.[RS19b] Daniel Rutherford and Michael Sullivan. Cellular Legendrian contact homology for surfaces, partIII.
Internat. J. Math. , 30(7):1950037, 111, 2019.[RSTZ14] Helge Ruddat, Nicol`o Sibilla, David Treumann, and Eric Zaslow. Skeleta of affine hypersurfaces.
Geom. Topol. , 18(3):1343–1395, 2014.[Sch53] Horst Schubert. Knoten und Vollringe.
Acta Math. , 90:131–286, 1953.[She19] Vivek Shende. The conormal torus is a complete knot invariant.
Forum Math. Pi , 7:e6, 16, 2019.[Siv11] Steven Sivek. A bordered Chekanov-Eliashberg algebra.
J. Topol. , 4(1):73–104, 2011.[Smi15] Ivan Smith. Quiver algebras as Fukaya categories.
Geom. Topol. , 19(5):2557–2617, 2015.[Soe07] Wolfgang Soergel. Kazhdan-Lusztig-Polynome und unzerlegbare Bimoduln ¨uber Polynomringen.
J. Inst. Math. Jussieu , 6(3):501–525, 2007.[SS16] Joshua M. Sabloff and Michael G. Sullivan. Families of Legendrian submanifolds via generatingfamilies.
Quantum Topol. , 7(4):639–668, 2016.[Sta18] Laura Starkston. Arboreal singularities in Weinstein skeleta.
Selecta Math. (N.S.) , 24(5):4105–4140, 2018.[STW16] Vivek Shende, David Treumann, and Harold Williams. On the combinatorics of exact Lagrangiansurfaces. 2016.[STWZ19] Vivek Shende, David Treumann, Harold Williams, and Eric Zaslow. Cluster varieties from Leg-endrian knots.
Duke Math. J. , 168(15):2801–2871, 2019.[STZ17] Vivek Shende, David Treumann, and Eric Zaslow. Legendrian knots and constructible sheaves.
Invent. Math. , 207(3):1031–1133, 2017.[Tab05] Goncalo Tabuada. Une structure de cat´egorie de mod`eles de Quillen sur la cat´egorie des dg-cat´egories.
C. R. Math. Acad. Sci. Paris , 340(1):15–19, 2005.[Toe07] Bertrand Toen. The homotopy theory of dg-categories and derived morita theory.
Invent. Math. ,167(3):615–667, 2007.[Tri19] Minh-Tam Q. Trinh. Annular Homology of Artin Braids I. 2019.[TZ18] David Treumann and Eric Zaslow. Cubic planar graphs and Legendrian surface theory.
Adv.Theor. Math. Phys. , 22(5):1289–1345, 2018.[Via14] Renato Vianna. On exotic Lagrangian tori in CP . Geom. Topol. , 18(4):2419–2476, 2014.[Wei71] Alan Weinstein. Symplectic manifolds and their Lagrangian submanifolds.
Advances in Math. ,6:329–346 (1971), 1971. niversity of California Davis, Dept. of Mathematics, Shields Avenue, Davis, CA 95616, USA
E-mail address : [email protected] Northwestern University, Department of Mathematics, Evanston, IL 60208-2730, USA
E-mail address : [email protected]@math.northwestern.edu