WWEAVE REALIZABILITY FOR D − TYPE
JAMES HUGHES
Abstract.
We study exact Lagrangian fillings of Legendrian links of D n -type in the standardcontact 3-sphere. The main result is the existence of a Lagrangian filling, represented by a weave,such that any algebraic quiver mutation of the associated intersection quiver can be realized as ageometric weave mutation. The method of proof is via Legendrian weave calculus and a constructionof appropriate 1-cycles whose geometric intersections realize the required algebraic intersectionnumbers. In particular, we show that in D -type, each cluster chart of the moduli of microlocal rank-1 sheaves is induced by at least one embedded exact Lagrangian filling. Hence, the Legendrian linksof D n -type have at least as many Hamiltonian isotopy classes of Lagrangian fillings as cluster seedsin the D n -type cluster algebra, and their geometric exchange graph for Lagrangian disk surgeriescontains the cluster exchange graph of D n -type. Introduction
Legendrian links in contact 3-manifolds [Ben83, Ad90] are central to the study of 3-dimensionalcontact topology [OS04, Gei08]. Recent developments [CZ20, CG20, CN21] have revealed newphenomena regarding their Lagrangian fillings, including the existence of (many) Legendrian linksΛ ⊆ ( S , ξ st ) with infinitely many (smoothly isotopic) Lagrangian fillings in the Darboux 4-ball( D , λ st ) which are not Hamiltonian isotopic. The relationship between cluster algebras and La-grangian fillings [CZ20, GSW20] has also led to new conjectures on the classification of Lagrangianfillings [Cas20]. In particular, [Cas20, Conjecture 5.1] introduced a conjectural ADE classificationof Lagrangian fillings. The object of this manuscript is to study D -type and prove part of theconjectured classification.The A -type was studied in [EHK16, Pan17], via Floer-theoretic methods, and in [STWZ19, TZ18]via microlocal sheaves. Their main result is that the A n -Legendrian link λ ( A n ) ⊆ ( S , ξ st ), whichis the max-tb representative of the (2 , n + 1)-torus link, has at least a Catalan number C n +1 = n +2 (cid:0) n +2 n +1 (cid:1) of embedded exact Lagrangian fillings, where C n +1 is precisely the number of clusterseeds in the finite type A n cluster algebra [FWZ20b]. We will show that the same holds in D -type,namely that D n -type Legendrian links have at least as many distinct Hamiltonian isotopy classesof Lagrangian fillings as there are cluster seeds in the D n -type cluster algebra. This will be aconsequence of a stronger geometric result, weave realizability in D − type, which we discuss below.By definition, the Legendrian link λ ( D n ) ⊆ ( S , ξ st ), n ≥ D n -type is the standard satel-lite of the Legendrian link defined by the front projection given by the 3-stranded positive braid σ n − ( σ σ σ )( σ σ ) , where σ and σ are the Artin generators for the 3-stranded braid group. Fig-ure 1 depicts a front diagram for λ ( D n ); note that the ( − σ n − ( σ σ σ )( σ σ ) is Legendrian isotopic to the rainbow closure of σ n − ( σ σ σ ), the latter being depicted. TheLegendrian link λ ( D n ) is also a max-tb representative of the smooth isotopy class of the link ofthe singularity f ( x, y ) = y ( x + y n − ). Since these are algebraic links, the max-tb representativegiven above is unique – e.g. [Cas20, Proposition 2.2] – and has at least one exact Lagrangian filling[HS15].The N -graph calculus developed by Casals and Zaslow in [CZ20] allows us to associate an exactLagrangian filling of a ( − a r X i v : . [ m a t h . S G ] J a n igure 1. The front projection of λ ( D n ) ⊆ ( S , ξ st ). The box labelled with an n − n − σ n − . When n is even, λ ( D n ) has3-components, while when n is odd, λ ( D n ) only has 2 components.satisfying certain properties. See Figure 2 (left) for an example of a particular 3-graph, denoted byΓ ( D ) and associated to the Legendrian link λ ( D ). In Section 3, we will show that the 3-graphΓ ( D ) generalizes to a family of 3-graphs Γ ( D n ), depicted in Figure 2 (right) for any n ≥ . In anutshell, a 3-fold branched cover of D , simply branched at the trivalent vertices of these 3-graphs,yields an exact Lagrangian surface in ( T ∗ D , λ st ), whose Legendrian lift is a Legendrian weave.One of the distinct advantages of the 3-graph calculus is that it combinatorializes an operation,known as Lagrangian disk surgery [Pol91, Yau17] that modifies the weave in such a way as to yieldadditional – non-Hamiltonian isotopic – exact Lagrangian fillings of the link. Figure 2. ( D ) (left) and Γ ( D n ) (right), each pictured with itsassociated intersection quiver Q (Γ ( D ) , { γ (0) i } ) (right). The basis { γ (0) i } for H (Λ(Γ ( D )); Z ) is depicted by the light green, dark green, orange, and purplecycles drawn in the graph. Note that the quivers corresponds to the D and D n Dynkin diagrams, usually depicted rotated 90 ◦ counterclockwise.If we consider a 3-graph Γ and a basis { γ i } for the first homology of the weave Λ(Γ), i ∈ [1 , b (Λ(Γ))],we can define a quiver Q (Γ , { γ i } ) whose adjacency matrix is given by the intersection form in H (Λ(Γ)). Quivers come equipped with a involutive operation, known as quiver mutation, thatproduces new quivers; see subsection 2.6 or [FWZ20a] for more on quivers. A key result of [CZ20] We use λ ( D ), i.e. n=4, as a first example because n = 3 would correspond to λ ( A ), which has been studiedpreviously [EHK16, Pan17]. The study of λ ( D ) is also the first instance where we require the machinery of 3-graphsrather than 2-graphs. ells us that Legendrian mutation of the weave induces a quiver mutation of the intersection quiver.Quivers related by a sequence of mutations are said to be mutation equivalent, and the quiversthat are of finite mutation type (i.e. the set of mutation equivalent quivers is finite) have anADE classification [FWZ20b]. This classification parallels the naming convention for the D n linksdescribed above: the intersection quiver associated to λ ( D n ) is a quiver in the mutation class ofthe D n -Dynkin diagram. See Figure 2 (left) for an example of a D quiver. For our 3-graphΓ ( D n ), n ≥
3, we will give an explicit basis { γ (0) i } for H (Λ(Γ ( D n )) , Z ), whose intersection quiver Q (Γ ( D n ) , { γ (0) i } ) is the standard D n -Dynkin diagram.By definition, a sequence of quiver mutations for Q (Γ ( D n ) , { γ (0) i } ) is said to be weave realizable ifeach quiver mutation in the sequence can be realized as a Legendrian weave mutation for a 3-graph.Our main result is the following theorem: Theorem 1.
Any sequence of quiver mutations of Q (Γ ( D n ) , { γ (0) i } ) is weave realizable. In other words, Theorem 1 states that in D -type, any algebraic quiver mutation can actually berealized geometrically by a Legendrian weave mutation. Weave realizability is of interest becauseit measure the difference between algebraic invariants – e.g. the cluster structure in the moduliof sheaves – and geometric objects, in this case Hamiltonian isotopy classes of exact Lagrangianfillings. If any sequence of quiver mutations was weave realizable, we would know that each clusteris inhabited by at least one embedded exact Lagrangian filling – this general statement remainsopen for an arbitrary Legendrian link. For instance, any link with an associated quiver that isnot of finite mutation type satisfying the weave realizability property would admit infinitely manyLagrangian fillings, distinguished by their quivers. Note that weave realizability was shown forA-type in [TZ18], and beyond A and D -types we currently do not know whether there are anyother links satisfying the weave realizability property.We can further distinguish fillings by studying the cluster algebra structure on the moduli of mi-crolocal rank-1 sheaves C (Γ) of a weave Λ(Γ), e.g. see [CZ20]. Specifically, sheaf quantization ofeach exact Lagrangian filling of λ ( D n ) induces a cluster chart on the coordinate ring of functionsof C (Γ ( D n )) via the microlocal mondromy functor [STZ17, STWZ19]. Describing a single clusterchart in the cluster variety requires the data of the quiver associated to the weave, and the microlo-cal monodromy around each 1-cycle of the weave. Crucially, applying the Legendrian mutationoperation to the weave induces a cluster transformation on the cluster chart, and the specific clus-ter chart defined by a Lagrangian fillings is a Hamiltonian isotopy invariant. Therefore, Theorem1 has the following consequence. Corollary 1.
Every cluster chart of the moduli of microlocal rank- sheaves C (Γ ( D n )) , whichis a cluster variety of D n -type, is induced by at least one embedded exact Lagrangian filling of λ ( D n ) ⊂ ( S , ξ st ) . In particular, there exist at least (3 n − C n − exact Lagrangian fillings of thelink λ ( D n ) up to Hamiltonian isotopy, where C n denotes the n th Catalan number Moreover, weave realizability implies a slightly stronger result. Specifically, we can consider the filling exchange graph associated to a link of D n -type, where the vertices are Hamiltonian isotopyclasses of embedded exact Lagrangians, and two vertices are connected by an edge if the two fillingsare related by a Lagrangian disk surgery. Then weave realizability implies that the filling exchangegraph contains a subgraph isomorphic to the cluster exchange graph for the cluster algebra of D n -type. Remark.
As of yet, we have no way of determining whether our method produces all possibleexact Lagrangian fillings of a type D n -link. This question remains open for A -type Legendrianlinks as well. In fact, the only known link for which we have a complete nonempty classification of This would be independent of the cluster structure defined by the microlocal monodromy functor, which weactually must use for D -type. agrangian fillings is the Legendrian unknot, which has a unique filling, and so thus the Legendrianunlink [EP96] . (cid:3) In summary, our method for constructing exact Lagrangian fillings will be to represent them usingthe planar diagrammatic calculus of N-graphs developed in [CZ20]. This diagrammatic calculusincludes a mutation operation on the diagrams that yields additional fillings. We distinguish theresulting fillings using a sheaf-theoretic invariant of our filling. From this data, we extract a clusteralgebra structure and show that every mutation of the quiver associated to the cluster can berealized by applying our Legendrian mutation operation to the 3-graph, thus proving that thereare at least as many distinct fillings as distinct cluster seeds of D n -type. The main theorem will beproven in Section 3 after giving the necessary preliminaries in Section 2. Acknowledgments.
Many thanks to Roger Casals for his support and encouragement throughoutthis project. Thanks also to Youngjin Bae for helpful conversations.
Relation to [ABL21] . While writing this manuscript, we learned that recent independent workby Byung Hee An, Youngjin Bae, and Eunjeong Lee also produces at least as many exact La-grangian fillings as cluster seeds for links of
ADE type [ABL21]. From our understanding, theyuse an inductive argument that relies on the combinatorial properties of the finite type generalizedassociahedron. Specifically, they leverage the fact that the Coxeter transformation in finite typeis transitive, if starting with a particular set of vertices, by finding a weave pattern that realizesCoxeter mutations. While both this manuscript and [ABL21] use the framework of N -graphs toapproach the problem of enumerating exact Lagrangian fillings, the proofs are different, indepen-dent, and our approach is able to give an explicit construction for realizing any sequence of quivermutations via an explicit sequence of mutations in the 3-graph. (cid:3) Preliminaries
In this section we introduce the necessary ingredients required for the proof of Theorem 1 andCorollary 1. We first discuss the contact topology needed to understand weaves and their homology.We then discuss the sheaf-theoretic material related to distinguishing fillings via cluster algebraicmethods.2.1.
Contact Topology and Exact Lagrangian Fillings.
A contact structure ξ on R is a2-plane field given locally as the kernel of a 1-form α ∈ Ω ( R ) satisfying α ∧ dα (cid:54) = 0. The standardcontact structure on ( R , ξ st ) is given by the kernel of α = dz − ydx . A Legendrian link λ in ( R , ξ )is an embedding of a disjoint union of copies of S that is always tangent to ξ . By definition, thecontact 3-sphere ( S , ξ st ) is the one point compactification of ( R , ξ st ) . Since a link in S can alwaysbe assumed to avoid a point, we will equivalently be considering Legendrian links in ( R , ξ st ) and( S , ξ st ) . The symplectization of ( R , ξ st ) is given by ( R × R t , d ( e t α )).Given two Legendrian links λ + and λ − in ( R , ξ ), an exact Lagrangian cobordism Σ from λ − to λ + is an embedded compact orientable surface in the symplectization ( R × R t , d ( e t α )) such that • Σ ∩ (cid:0) R × [ T, ∞ ) (cid:1) = λ + × [ T, ∞ ) • Σ ∩ (cid:0) R × ( −∞ , − T ) (cid:1) = λ − × ( −∞ , − T ] • Σ is an exact Lagrangian, i.e. e t α = df for some function f : Σ → R . The asymptotic behavior of Σ, as specified by the first two conditions, ensures that we can concate-nate Lagrangian cobordisms. By definition, an exact Lagrangian filling of λ + is an exact Lagrangiancobordism from ∅ to λ + . e can also consider the Legendrian lift of an exact Lagrangian in the contactization ( R s × R , ker { ds − d ( e t α ) } ) of ( R , d ( e t α )). Note that there exists a contactomorphism between ( R s × R , ker { ds − d ( e t α ) } ) and the standard contact Darboux structure ( R , ξ st ), ξ st = ker { dz − y dx − y dx } , and we will often work with the Legendrian front projection ( R , ξ st ) −→ R x ,x ,z for thelatter. This will be a useful perspective for us, as it allows us to construct Lagrangian fillings bystudying (wave)fronts in R = R x ,x ,z of Legendrian surfaces in ( R , ξ st ), and then projecting downto the standard symplectic Darboux chart R = R x ,y ,x ,y . In this setting, the exact Lagrangiansurface is embedded in R if and only if its Legendrian lift has no Reeb chords. The constructionwill be performed through the combinatorics of N -graphs, as we now explain.2.2. In this subsection, we discuss the diagrammatic method of con-structing and manipulating exact Lagrangian fillings of links arising as the ( − N -graphs. For this manuscript, it will suffice to take N = 3. Definition 1.
A 3-graph is a pair of embedded planar trivalent graphs
B, R ⊆ D such that, at anyvertex v ∈ B ∩ R , the six edges belonging to B and R incident to v alternate. (cid:3) Equivalently, a 3-graph is a edge-bicolored graph with monochromatic trivalent vertices and in-terlacing hexavalent vertices. Γ ( D ) , depicted in Figure 2 (left) contains two hexavalent verticesdisplaying the alternating behavior described in the definition. Remark. [CZ20] gives a general framework for working with N-graphs, where N − is the number ofembedded planar trivalent graphs. This allows for the study of fillings of Legendrian links associatedto N -stranded positive braids. This can also be generalized to consider N-graphs in a surface otherthan D . Here, the family of links we are interested in can be expressed as a family of 3-strandedbraids, hence our choice to restrict N to 3 in D . (cid:3) Given a 3-graph Γ ⊆ D , we describe how to associate a Legendrian surface Λ(Γ) ⊆ ( R , ξ st ). To doso, we first describe certain singularities of Λ(Γ) that arise under the Legendrian front projection π : ( R , ξ st ) → ( R , ξ st ). In general, such singularities are known as Legendrian singularities orsingularities of fronts. See [Ad90] for a classification of such singularities. The three singularitieswe will be interested in are the A , A and D − singularities, pictured in Figure 3 below. Figure 3. A (left), A (center), and D − (right) singularities represented in the3-graph by an edge, hexavalent vertex, and trivalent vertex, respectively.Before we describe our Legendrian surfaces, we must first discuss the ambient contact structurethat they live in. For Γ ⊆ D we will take Λ(Γ) to live in the first jet space ( J D , ξ st ) = ( T ∗ D × R z , ker( dz − λ st )), where λ st is the standard Liouville form on the cotangent bundle T ∗ D . Wecan view J D as a certain local model for a contact structure, in the following way. If we take( Y, ξ ) to be a contact 5-manifold, then by the Weinstein neighborhood theorem, any Legendrianembedding i : D → ( Y, ξ ) extends to an embedding from ( J D , ξ st ) to a small open neighborhoodof i ( D ) with contact structure given by the restriction of ξ to that neighborhood. In particular, a egendrian embedding of i : S → S gives rise to a contact embedding ˜ i : J S −→ Op( i ( S )) intosome open neighborhood Op( i ( S )) ⊆ S . Of particular note in our case is that, under a Legendrianembedding D ⊆ ( R , ξ st ), a Legendrian link λ in J ∂ D is mapped to a Legendrian link in thecontact boundary ( S , ξ st ) of the symplectic ( R , λ st ) given as the co-domain of the Lagrangianprojection ( R , ξ st ) → ( R , λ st ). See [NR13] for a description of the Legendrian satellite operation.To construct a Legendrian weave Λ(Γ) ⊆ ( J D , ξ st ) from a 3-graph Γ, we glue together the localgerms of singularities according to the edges of Γ. First, consider three horizontal wavefronts D × { } (cid:116) D × { } (cid:116) D × { } ⊆ D × R and a 3-graph Γ ⊆ D × { } . We construct the associatedLegendrian weave Λ(Γ) as follows. • Above each blue (resp. red) edge, insert an A crossing between the D × { } and D × { } sheets (resp D × { } and D × { } sheets) so that the projection of the A singular locusunder π : D × R → D × { } agrees with the blue (resp. red) edge. • At each blue (resp. red) trivalent vertex v , insert a D − singularity between the sheets D × { } and D × { } (resp. D × { } and D × { } ) in such a way that the projection ofthe D − singular locus agrees with v and the projection of the A crossings agree with theedges incident to v . • At each hexavalent vertex v , insert an A singularity along the three sheets in such a waythat the origin of the A singular locus agrees with v and the A crossings agree with theedges incident to v . (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 4.
The weaving of the singularities pictured in Figure 3 along the edges ofthe N-graph. Gluing these local pictures together according to the 3-graph Γ yieldsthe weave Λ(Γ).If we take an open cover { U i } mi =1 of D ×{ } by open disks, refined so that any disk contains at mostone of these three features, we can glue together the resulting fronts according to the intersectionof edges along the boundary of our disks. Specifically, if U i ∩ U j is nonempty, then we defineΣ( U ∪ U ) to be the wavefront resulting from considering the union of wavefronts Σ( U ) ∪ Σ( U j )in ( U ∪ U ) × R . We define the Legendrian weave Λ(Γ) as the Legendrian surface contained in( J D , ξ st ) with wavefront Σ(Γ) = Σ( ∪ mi =1 U i ) given by gluing the local wavefronts of singularitiestogether according to the 3-graph Γ [CZ20, Section 2.3].The smooth topology of a Legendrian weave Λ(Γ) is given as a 3-fold branched cover over D withsimple branched points corresponding to each of the trivalent vertices of Γ. The genus of Λ(Γ) isthen computed using the Riemann-Hurwitz formula: g (Λ(Γ)) = 12 ( v (Γ) + 2 − χ ( D ) − | ∂ Λ(Γ) | ) here v (Γ) is the number of trivalent vertices of Γ and | ∂ Λ(Γ) | denotes the number of boundarycomponents of Γ. Example.
If we apply this formula to the 3-graph Γ ( D ) , pictured in Figure 2, we have trivalentvertices and 3 link components, so the genus is computed as g (Λ(Γ ( D ))) = (6 + 2 − −
3) = 1 . For Γ ( D n ) , we have three boundary components for even n and two boundary components for oddn. The number of trivalent vertices is n + 2 , so the genus g (Λ(Γ ( D n )) is (cid:98) n − (cid:99) , assuming n ≥ .This computation tells us that Λ(Γ ( D )) is smoothly a 3-punctured torus bounding the link λ ( D ) . Therefore, we can give a basis for H (Λ(Γ ( D )); Z ) in terms of the four cycles pictured in Figure2. For Γ ( D n ) , the corresponding weave Λ(Γ ( D n )) will be smoothly a genus (cid:98) n − (cid:99) surface witha basis of H (Λ(Γ); Z ) given by n cycles. By a theorem of Chantraine [Cha10] , our computationimplies that any filling of λ ( D n ) has genus (cid:98) n − (cid:99) . In the next section, we describe a general methodfor giving a basis { γ (0) i } , i ∈ [1 , n ] of the first homology H (Λ(Γ ( D n )); Z ) ∼ = Z n . Homology of Weaves.
We require a description of the first homology H (Λ(Γ)); Z ) in orderto apply the mutation operation to a 3-graph Γ. We first consider an edge connecting two trivalentvertices. Closely examining the sheets of our surface, we can see that each such edge correspondsto a 1-cycle, as pictured in Figure 5 (left). We refer to such a 1-cycle as a short I -cycle. Similarly,any three edges of the same color that connect a single hexavalent vertex to three trivalent verticescorrespond to a 1-cycle, as pictured in 6 (left). We refer to such a 1-cycle as a short Y -cycle. Seefigures 5 (right) and 6 (right) for a diagram of these 1-cycles in the wavefront Σ(Γ). We can alsoconsider a sequence of edges starting and ending at trivalent vertices and passing directly throughany number of hexavalent vertices, as pictured in Figure 7. Such a cycle is referred to as a long I -cycle. Finally, we can combine any number of I -cycles and short Y -cycles to describe an arbitrary1-cycle as a tree with leaves on trivalent vertices and edges passing directly through hexavalentvertices.In the proof of our main result, we will generally give a basis for H (Λ(Γ); Z ) in terms of short I -cycles and short Y -cycles. Indeed, Figure 8 gives a basis of H (Λ(Γ ( D n )); Z ) consisting of n − I -cycles and a single Y -cycle.
123 4561 2 3 4 5 6
Figure 5.
A short I -cycle γ ( e ) for the edge e ∈ G pictured in the wavefront Σ(Γ)(left) and a vertical slicing of Σ(Γ) (right).The intersection form (cid:104)· , ·(cid:105) on H (Λ(Γ)) plays a key role in distinguishing our Legendrian weaves.If we consider a pair of 1-cycles γ , γ ∈ H (Λ(Γ)) with nonempty geometric intersection in Γ, as
123 4 5 6
Figure 6.
A short Y -cycle γ ( e ) defined by the edges e , e , e ∈ G pictured in thewavefront Σ(Γ) (left) and a vertical slicing of Σ(Γ) (right). Figure 7.
A pair of long I -cycles, both denoted by γ . The cycle on the left passesthrough an even number of hexavalent vertices, while the cycle on the right passesthrough an odd number. Figure 8.
The 3-graph Γ ( D n ) and its associated intersection quiver. The blackdotted line represents n − I -cycles and the blue dotted line represents a totalof n − { γ (0) i } of H (Λ(Γ ( D n )); Z ) is given by the orange Y -cycle, the green I -cycles, and the n − I -cycles represented by the dotted blackline.pictured in Figure 9, we can see that the intersection of their projection onto the 3-graph differsfrom the intersection in Λ(Γ) . Specifically, we can carefully examine the sheets that the 1-cyclescross in order to see that γ and γ intersect only in a single point of Λ(Γ). If we fix an orientationon γ and γ , then we can assign a sign to this intersection based on the convention given in Figure9. We refer to the signed count of the intersection of γ and γ as their algebraic intersection anddenote it by (cid:104) γ , γ (cid:105) . Notation:
For the sake of visual clarity, we will represent an element of H (Λ(Γ); Z ) by a colorededge for the remainder of this manuscript. This also ensures that the geometric intersection moreaccurately reflects the algebraic intersection. The original coloring of the blue or red edges can bereadily obtained by examining Γ and its trivalent vertices. (cid:3) igure 9. Intersection of two cycles, γ and γ . The intersection point is indicatedby a black dot. We will set (cid:104) γ , γ (cid:105) = − Theorem 2 ([CZ20], Theorem 4.2) . Let Γ and Γ (cid:48) be two 3-graphs related by one of the movesshown in Figure 10. Then the associated weaves Λ(Γ) and
Λ(Γ (cid:48) ) are Legendrian isotopic relativeto their boundaries. (cid:3) Figure 10.
Legendrian Surface Reidemeister moves for 3-graphs. From left toright, a candy twist, a push-through, and a flop, denoted by I, II, and III respectively.See Figure 11 for a description of the behavior of elements of H (Λ(Γ); Z ) under these LegendrianSurface Reidemeister moves. In the pair of 3-graphs in Figure 11 (center), we have denoted apush-through by II or II − depending on whether we go from left to right or right to left.This helpsus to specify the simplifications we make in the figures in the proof of Theorem 1, as this move isnot as readily apparent as the other two. We will refer to the II − move as a reverse push-through.Note that an application of this move eliminates the geometric intersection between the light greenand dark green cycles in Figure 11. Figure 11.
Behavior of certain homology cycles under Legendrian Surface Reide-meister moves. emark. It is also possible to verify the computations in Figure 11 by examining the relativehomology of a cycle. Specifically, if we have a basis of the relative homology H (Λ(Γ) , ∂ Λ(Γ); Z ) ,then the intersection form on that basis allows us to determine a given cycle by Poincar´e-Lefschetzduality. (cid:3) Mutations of 3-graphs.
We complete our discussion of general 3-graphs with a descriptionof Legendrian mutation, which we will use to generate distinct exact Lagrangian fillings. Given aLegendrian weave Λ(Γ) and a 1-cycle γ ∈ H (Λ(Γ); Z ), the Legendrian mutation µ γ (Λ(Γ)) outputsa 3-graph and a corresponding Legendrian weave smoothly isotopic to Λ(Γ) but whose Lagrangianprojection is generally not Hamiltonian isotopic to that of Λ(Γ). Definition 2.
Two Legendrian surfaces Λ , Λ ⊆ ( R , ξ st ) with equal boundary ∂ Λ = ∂ Λ , aremutation-equivalent if and only if there exists a compactly supported Legendrian isotopy { ˜Λ t } rela-tive to the boundary, with ˜Λ = Λ and a Darboux ball ( B, ξ st ) such that (i) Outside the Darboux ball, we have ˜Λ | R \ B = Λ | R \ B (ii) There exists a global front projection π : R → R such that the pair of fronts π | B ∩ Λ and π | B ∩ Λ coincides with the pair of fronts in Figure 12 below. (cid:3) Figure 12.
Local fronts for two Legendrian cylinders non-Legendrian isotopic rel-ative to their boundary.We briefly note that these two fronts lift to non-Legendrian isotopic Legendrian cylinders in ( R , ξ st ),relative to the boundary, and that the 1-cycle we input for our operation is precisely the 1-cycledefined by the cylinder corresponding to Λ .Combinatorially, we can describe mutation as certain manipulations of the edges of our graph.Figure 13 (left) depicts mutation at a short I -cycle, while Figure 13 (right) depicts mutation at ashort Y -cycle. In the N = 2 setting, we can identify 2-graphs with triangulations of an n − gon, inwhich case mutation at a short I -cycle corresponds to a Whitehead move. In the 3-graph setting,in order to describe mutation at a short Y -cycle, we can first reduce the short Y -cycle case to ashort I -cycle, as shown in Figure 14, before applying our mutation. See [CZ20, Section 4.9] for amore general description of mutation at long I and Y -cycles in the 3-graph.The geometric operation above coincides with the combinatorial manipulation of the 3-graphs.Specifically, we have the following theorem. Theorem 3 ([CZ20], Theorem 4.2.1) . Given two 3-graphs, Γ and Γ (cid:48) related by either of the com-binatorial moves described in Figure 13, the corresponding Legendrian weaves Λ(Γ) and
Λ(Γ (cid:48) ) aremutation-equivalent relative to their boundary. (cid:3) igure 13. Mutations of a 3-graph. The pair of 3-graphs on the left depicts muta-tion at the orange I -cycle, while the pair of 3-graphs on the right depicts mutation atthe orange Y -cycle. In both cases, the dark green edge depicts the effect of mutationon any cycle intersecting the orange cycle. Figure 14.
Mutation at a short Y -cycle given as a sequence of Legendrian SurfaceReideister moves and mutation at a short I -cycle. The Y -cycle in the initial 3-graphis given by the three blue edges that each intersect the yellow vertex in the center.2.5. Lagrangian Fillings from Weaves.
We now describe in more detail how an exact La-grangian filling of a Legendrian link arises from a Legendrian weave. If we label all edges of Γ ⊆ D colored blue by σ and all edges colored red by σ , then the points in the intersection Γ ∩ ∂ D give us a braid word in the Artin generators σ and σ of the 3-stranded braid group. We canthen view the corresponding link β as living in ( J S , ξ st ). If we consider our Legendrian weaveΛ(Γ) as an embedded Legendrian surface in ( R , ξ st ), then according to our discussion above, ithas boundary Λ( β ) , where Λ( β ) is the Legendrian satellite of β with companion knot given by thestandard unknot. In our local contact model, the projection π : ( J D , ξ st ) → ( T ∗ D , λ st ) gives animmersed exact Lagrangian surface with immersion points corresponding to Reeb chords of Λ(Γ).If Λ(Γ) has no Reeb chords, then π is an embedding and Λ(Γ) is an exact Lagrangian filling ofΛ( β ) . Since ( S , ξ st ) minus a point is contactomorphic to ( R , ξ st ), we have that an embedding ofΛ(Γ) into ( R , ξ st ) gives an exact Lagrangian filling in ( R , ξ st ) of Λ( β ) ⊆ ( R , ξ st ), as it can beassumed – after a Legendrian isotopy – to be disjoint from the point at infinity. Remark.
We study embedded – rather than immersed – Lagrangian fillings due to the existence ofan h -principle for immersed Lagrangian fillings [EM02, Theorem 16.3.2] . In particular, any pairof immersed exact Lagrangian fillings is connected by a one-parameter family of immersed exactLagrangian fillings relative to the boundary. See also [Gro86] . Our desire for embedded Lagrangians motivates the following definition. efinition 3. A 3-graph Γ ⊆ D is free if the associated Legendrian front Σ(Γ) can be woven withno Reeb chords. (cid:3) Γ ( D n ), depicted in Figure 8, is an example of a free 3-graph of D n -type. Crucially, the mutationoperation described above preserves the free property of a 3-graph. Lemma 1 ([CZ20], lemma 7.4) . Let Γ ⊆ D be a free 3-graph. Then the 3-graph µ (Γ) obtained bymutating according to any of the Legendrian mutation operations given above is also a free 3-graph. (cid:3) Therefore, starting with a free 3-graph and performing the Legendrian mutation operation gives usa method of creating additional embedded exact Lagrangian fillings.At this stage, we have described the geometric and combinatorial ingredients needed for Theorem1. The two subsequent subsections introduce the necessary algebraic invariants relating Legendrianweaves and 3-graphs to cluster algebras. These will be used to distinguish exact Lagrangian fillings.2.6.
Quivers from Weaves.
Before we describe the cluster algebra structure associated to aweave, we must first describe quivers and how they arise via the intersection form on H (Λ(Γ); Z ) . A quiver is a directed graph without loops or directed 2-cycles. In the weave setting, the data ofa quiver can be extracted from a weave and a basis of its first homology. The intersection quiveris defined as follows: each basis element γ i ∈ H (Λ(Γ); Z ) defines a vertex v i in the quiver and wehave k arrows pointing from v j to v i if (cid:104) γ i , γ j (cid:105) = k . We will only ever have k either 0 or 1 for quiversarising from fillings of λ ( D n ). See Figure 2 (left) for an example of the quiver Q (Λ(Γ ( D )) , { γ (0) i } )defined by Λ(Γ ( D )) and the indicated basis for H (Λ(Γ ( D ); Z ).The combinatorial operation of quiver mutation at a vertex v is defined as follows, e.g. see[FWZ20a]. First, for every pair of incoming edge and outgoing edges, we add an edge startingat the tail of the incoming edge and ending at the head of the outgoing edge. Next, we reverse thedirection of all edges adjacent to v . Finally, we cancel any directed 2-cycles. If we started with thequiver Q , then we denote the quiver resulting from mutation at v by µ v ( Q ) . See Figure 15 (bottom)for an example. Under this operation, we can naturally identify the vertices of Q with µ v ( Q ), justas we can identify the homology bases of a weave before and after Legendrian mutation. Remark.
The crucial difference between algebraic and geometric intersections is captured in thestep canceling directed 2-cycles. This cancellation is implemented by default in a quiver mutation, asthe arrows of the quiver only capture algebraic intersections. In contrast, the geometric intersectionof homology cycles after a Legendrian mutation will, in general, not coincide with the algebraicintersection. This dissonance will be explored in detail in Section 3. (cid:3)
The following theorem relates the two operations of quiver mutation and Legendrian mutation:
Theorem 4 ([CZ20], Section 7.3) . Given a 3-graph Γ , Legendrian mutation at an embeddedcycle γ induces a quiver mutation for the associated intersection quivers, taking Q (Γ , { γ i } ) to µ γ ( Q (Γ , { γ i } )) . (cid:3) See Figure 15 for an example showing the quiver mutation of Q (Γ ( D ) , { γ (0) i } ), i ∈ [1 , ( D )) . Microlocal Sheaves and Clusters.
To introduce the cluster structure mentioned above,we need to define a sheaf-theoretic invariant. We first consider the category of dg complexes ofsheaves of C − modules on D × R with constructible cohomology sheaves. For a given 3-graph Γand its associated Legendrian Λ(Γ), we denote by C (Γ) := Sh ( D × R ) the subcategory of igure 15. Mutation of Γ ( D ) and its associated intersection quiver at the short Y -cycle colored in orange.microlocal rank-one sheaves with microlocal support along Λ(Γ), which we require to be zero in aneighborhood of D × {−∞} . Here we identify the unit cotangent bundle T ∞ , − ( D × R ) with thefirst jet space J ( D ) . With this identification, the sheaves of C (Γ) are constructible with respect tothe stratification given by the Legendrian front Σ(Γ) . Work of Guillermou, Kashiwara, and Schapiraimplies that that C (Γ) is an invariant under Hamiltonian isotopy [GKS12].As described in [CZ20, Section 5.3], this category has a combinatorial description. Given a 3-graphΓ, the data of the moduli space of microlocal rank-one sheaves is equivalent to providing:(i) An assignment to each face F (connected component of D \ G ) of a flag F • ( F ) in the vectorspace C .(ii) For each pair F , F of adjacent faces sharing an edge labeled by σ i , we require that thecorresponding flags satisfy F j ( F ) = F j ( F ) , ≤ j ≤ , j (cid:54) = i, and F i ( F ) (cid:54) = F i ( F ) . Finally, we consider the moduli space of flags satisfying (i) and (ii) modulo the diagonal action of GL n on F • . The precise statement [CZ20, Theorem 5.3] is that the flag moduli space, denoted by M (Γ), is isomorphic to the space of microlocal rank-one sheaves C (Γ). Since C (Γ) is an invariantof Λ(Γ) up to Hamiltonian isotopy, it follows that M (Γ) is an invariant as well. In the I -cycle case,when the edges are labeled by σ , the moduli space is determined by four lines a (cid:54) = b (cid:54) = c (cid:54) = d (cid:54) = a ,as pictured in Figure 16 (left). If the edges are labeled by σ , then the data is given by four planes A (cid:54) = B (cid:54) = C (cid:54) = D (cid:54) = A. Around a short Y -cycle, the data of the flag moduli space is given by threedistinct planes A (cid:54) = B (cid:54) = C (cid:54) = A contained in C and three distinct lines a (cid:40) A, b (cid:40)
B, c (cid:40) C with a (cid:54) = b (cid:54) = c (cid:54) = a, as pictured in Figure 16 (right).To describe the cluster algebra structure on C (Γ), we need to specify the cluster seed associatedto the quiver Q (Λ(Γ) , { γ i } ) via the microlocal mondromy functor µ mon , which takes us from thecategory C (Γ) to the category of rank one local systems on Λ(Γ). As described in [STZ17, STWZ19], igure 16. The data of the flag moduli space given in the neighborhood of a short I -cycle (left) and a short Y -cycle (right). Lines are represented by lowercase letters,while planes are written in uppercase. The intersection of the two lines a and b iswritten as ab .the functor µ mon takes a 1-cycle as input and outputs the isomorphism of sheaves given by themonodromy about the cycle. Since it is locally defined, we can compute the microlocal monodromyabout an I -cycle or Y -cycle using the data of the flag moduli space in a neighborhood of the cycle.If we have a short I -cycle γ with flag moduli space described by the four lines a, b, c, d , as in Figure16 (left), then the microlocal monodromy about γ is given by the cross ratio a ∧ bb ∧ c c ∧ dd ∧ a Similarly, for a short Y -cycle with flag moduli space given as in Figure 16 (right), the microlocalmonodromy is given by the triple ratio B ( a ) C ( b ) A ( c ) B ( c ) C ( a ) A ( b )As described in [CZ20, Section 7.2], the microlocal monodromy about a 1-cycle gives rise to an X -cluster variable at the corresponding vertex in the quiver. Under mutation of the 3-graph, the crossratio and triple ratio transform as cluster X-coordinates. Specifically, if we start with a 3-graphwith cluster variables x j , then the cluster variables x (cid:48) j of the 3-graph after mutating at γ i are givenby the equation x (cid:48) j = x − j i = jx j (1 + x − i ) −(cid:104) γ i ,γ j (cid:105) (cid:104) γ i , γ j (cid:105) > x j (1 + x i ) −(cid:104) γ i ,γ j (cid:105) (cid:104) γ i , γ j (cid:105) < D n quiver as a mutationof the corresponding 3-graph. This will imply that there are at least as many exact Lagrangianfillings as cluster seeds of D n -type. There exists a complete classification of all finite mutation typecluster algebras, and in fact, the number of cluster seeds of D n -type is (3 n − C n − [FWZ20b]. Remark.
It is not known whether other methods of generating exact Lagrangian fillings for λ ( D n ) access all possible cluster seeds of D n -type. When constructing fillings of D by opening crossings,as in [EHK16, Pan17] , experimental evidence suggests that it is only possible to access at most 46out of the possible 50 cluster seeds by varying the order of the crossings chosen. Of note in thecombinatorial setting, we also contrast the 3-graphs Γ( D ) with double wiring diagrams for the toruslink T (3 , , which is the smooth type of λ ( D ) . The moduli of sheaves C (Γ( D )) for Γ( D ) embedsas an open positroid cell into the Grassmanian Gr (3 ,
6) [CG20] , so we can identify some clustercharts with double wiring diagrams. The double wiring diagrams associated to Gr (3 , only access igure 17. Prior to mutating at γ , we have (cid:104) γ , γ (cid:105) = −
1. Computing the crossratios for γ and µ ( γ ) we can see that the cross ratio transforms as µ ( γ ) = b ∧ cc ∧ e e ∧ aa ∧ b = x − under mutation. Similarly, computing the cross ratios for γ and µ ( γ ) and applying the relation e ∧ b · a ∧ c = b ∧ c · e ∧ a + a ∧ b · c ∧ e, we have µ ( x ) = e ∧ aa ∧ c c ∧ dd ∧ e (cid:0) a ∧ bb ∧ c c ∧ ee ∧ a (cid:1) .
34 distinct cluster seeds – out of 50 – via local moves applied to an initial double wiring diagram [FWZ20a] . (cid:3) Proof of Main Results
In this section, we state and prove Theorem 5, which implies Theorem 1. The following definitionsrelate the algebraic intersections of cycles to geometric intersections in the context of 3-graphs.
Definition 4.
A 3-graph Γ with associated basis { γ i } , i ∈ [1 , b (Λ(Γ)] of H (Λ(Γ); Z ) is sharp at acycle γ j if, for any other cycle γ k ∈ { γ i } , the geometric intersection number of γ j with γ k is equalto the algebraic intersection (cid:104) γ j , γ k (cid:105) . Γ is locally sharp if, for any cycle γ ∈ { γ i } , there exist a sequence of Legendrian Surface Reide-meister moves taking Γ to some other 3-graph Γ (cid:48) such that Γ (cid:48) is sharp at the corresponding cycle γ (cid:48) ∈ H (Λ(Γ (cid:48) ); Z ) .A 3-graph Γ with a set of cycles Γ is sharp if Γ is sharp at all γ i ∈ { γ i } . (cid:3) For 3-graphs that are not sharp, it is possible that a sequence of mutations will cause a cycle tobecome immersed. This is the only obstruction to weave realizability. Therefore, sharpness is adesirable property for our 3-graphs, as it simplifies our computations and helps us avoid creatingimmersed cycles. We will not be able to ensure sharpness for all Γ( D n ) that arise as part of ourcomputations, (e.g., see the type III.i normal form in Figure 19) but we will be able to ensure thateach of our 3-graphs is locally sharp.3.1. Proof of Theorem 1.
The following result is slightly stronger than the statement of Theorem1, as we are able to show that each 3-graph in our sequence of mutations is locally sharp.
Theorem 5.
Let µ v , . . . , µ v k be a sequence of quiver mutations, with initial quiver Q (Γ ( D n ) , { γ (0) i } ) .Then, there exists a sequence Γ ( D n ) , . . . , Γ k ( D n ) of 3-graphs such that i. Γ j − ( D n ) is related to Γ j ( D n ) by mutation at a cycle γ j and by Legendrian Surface Reide-meister moves I, II and III. The cycle γ j represents the vertex v j in the intersection quiverand it is given by one of the cycles in the initial basis { γ (0) i } after mutation and Reidemeis-ter moves. ii. Γ j ( D n ) is sharp at γ j . ii. Γ j ( D n ) is locally sharp. iv. The basis of cycles for Γ j ( D n ) , obtained from the initial basis { γ (0) i } by mutation and Rei-demeister moves, consists entirely of short Y -cycles and short I -cycles. The conditions ii-iv allow us to continue to iterate mutations after applying a small number ofsimplifications at each step. Theorem 1 thus follows from Theorem 5.
Proof.
We proceed by organizing the 3-graphs arising from any sequence of mutations of Γ ( D n )into four types, in line with the organization scheme introduced by Vatne for quivers of D n -type[Vat10]. Vatne’s classification of quivers in the mutation class of D n -type uses the configurationof a certain subquiver to define the different types. Outside of that subquiver, there are a numberof disjoint subquivers of A n -type that are referred to as A n tail subquivers. We will refer to thecorresponding cycles in the 3-graph as A n tail subgraphs, or simply A n tails when it is clear fromcontext whether we are referring to the quiver or the 3-graph. For each type, Vatne describesthe results of quiver mutation at different vertices, which can depend on the existence of A n tailsubquivers. See Figures 20, 26, 30, and 34 for the four types and their mutations. Notation.
As mentioned in the previous section, cycles are pictured as colored edges for the sakeof visual clarity. Throughout this section, we denote all of the pink cycles by γ , purple cycles by γ , orange cycles by γ , dark green cycles by γ , light green cycles by γ and light blue cycles by γ . With this notation, γ i will correspond to the vertex labeled by v i in the quivers given below. A n Tails.
We briefly describe the behavior of the A n tail subquivers, as given in [Vat10], in termsof weaves. Any of the n vertices in an A n subquiver can have valence between 0 and 4. Cycles inthe quiver are oriented with length 3. If a vertex v has valence 3, then two of the edges form partof a 3-cycle, while the third edge is not part of any 3-cycle. If v has valence 4, then two of theedges belong to one 3-cycle and the remaining two edges belong to a separate 3-cycle.Any A n tail of the quiver can be represented by a sharp configuration of n I -cycles in the 3-graph.See Figure 18 for an identification of I -cycles with quiver vertices of a given valence. Mutation atany vertex v i in the quiver corresponds to mutation at the I -cycle γ i in the 3-graph, so it is readilyverified that mutation preserves the number of I -cycles and requires no application of LegendrianSurface Reidemeister moves to simplify. As a consquence, any sequence of A n tail mutations isweave realizable, and a sharp 3-graph remains sharp after mutation at A n tail I -cycles that onlyintersect other A n tail I -cycles. Figure 18. I -cycles in an A n tail of the 3-graph and the corresponding A n tail subquiver. ormal Forms. For each of the four types of D n quivers described in [Vat10], we give one or twospecific subgraphs of Γ( D n ), which we refer to as normal forms. These normal forms are picturedin Figure 19. We indicate the possible existence of A n tail subgraphs by a black circle. We will saythat an edge of the 3-graph carries a cycle if it is part of a homology cycle. We will generally usethis terminology to specify which edges cannot carry a cycle. Figure 19.
Normal forms of types I-IV. In the top row, pictured from left toright, are the normal forms for Types I, II, III.i, and III.ii. In the bottom row, arenormal forms for Types IV.i, IV.ii, and IV ( k > I -cyclescorresponding to A n tails of the quiver are represented by black circles. For TypeIV, k represents the length of the directed cycle of edges in the quiver that remainsafter deleting all of the circle vertices. The dotted lines in the k > k − I -cycles that, together with the two Y -cycles and single I -cycle pictured, forma k + 2-gon with a single blue diagonal.For each possible quiver mutation, we describe the possible mutations of the 3-graph and showthat the result matches the quiver type and retains the properties listed in Theorem 5 above. Inaddition, the Legendrian Surface Reidemeister moves we describe ensure that the A n tail subgraphscontinue to consist solely of short I -cycles. If the mutation results in a long I -cycle or pair of long I -cycles connecting our A n tail to the rest of the 3-graph, we can simplify by applying a sequence of n push-throughs to ensure that these are all short I -cycles. It is readily verified that we can alwaysdo this and that no other simplifications of the A n tails are required following any other mutations.We include A n tail cycles only where relevant to the specific mutation. In our computations below,we generally omit the final steps of applying a series of push-throughs to make any long I or Y -cyclesinto short I or Y -cycles. Figure 25 provides an example where these push-throughs are shown forboth an I -cycle and a Y -cycle. emark. The Type I normal form does not cover every possible arrangement of the 3-graph corre-sponding to a Type I quiver. Mutating at either of the short I -cycles γ or γ produces one of fourpossible arrangements of the cycles γ , γ , and γ in a 3-graph corresponding to a Type I quiver.Since these mutations are somewhat straightforward, we simplify our calculations by giving a singlenormal form rather than four, and describing the relevant mutations of two of the four possible3-graphs in figures 21, 22, 23, and 24. The remaining cases can be seen by swapping the cycle(s)to the left of the short Y -cycle with the cycle(s) to the right of it. This symmetry corresponds toreversing all of the arrows in the quiver. In general, we will implicitly appeal to similar symmetriesof the normal form 3-graphs to reduce the number of cases we must consider. (cid:3) Type I.
We start with 3-graphs, always endowed with a homology basis, whose associated inter-section quivers are a Type I quiver. See Figure 20 for the relevant quiver mutations.
Figure 20.
From top to bottom, Type I to Type I, Type I to Type II, and TypeI to Type IV quiver mutations. The arrow labeled by µ v i indicates mutation atthe vertex v i . In each line, the first quiver mutation shows the case where v isonly adjacent to one A n tail vertex, while the second quiver mutation shows thecase where v is adjacent to two A n tail vertices. Note that reversing the directionof all of the arrows simultaneously before mutating gives additional possible quivermutations of the same type.i. (Type I to Type I) There are two possible Type I to Type I mutations of 3-graphs depictedin Figure 21 (left) and (right). The second 3-graph in the first sequence is the result ofmutating at γ . As shown there, mutation does not create any new additional geometricor algebraic intersections. Instead, it takes positive intersections to negative intersectionsand vice versa. This is reflected in the quivers pictured underneath the 3-graphs, as theorientation of edges has reversed under the mutation. As explained above, we could simplifythe resulting 3-graph by applying a push-through move to each of the long I -cycles to get asharp 3-graph where the homology cycles are made up of a single short Y -cycle and some igure 21. Type I to Type I mutation. Arrows labeled by µ indicate mutation ata cycle of the same color.number of short I -cycles.ii. (Type I to Type I) For the second possible Type I to Type I mutation, we proceed aspictured in Figure 21 (right). There we can see that mutation at γ only affects the signof the intersection of γ with the γ . This reflects the fact that the corresponding quivermutation has only reversed the orientation of the edge between v and v . Mutating at anyother I -cycle is equally straightforward and yields a Type I to Type I mutation as well.iii. (Type I to Type II) In Figure 22 we consider the cases where the Y -cycle γ intersects one I -cycle (top) or two I -cycles (bottom) in the A n tail subgraph. Mutation at γ introducesan intersection between γ and γ that causes the second 3-graph in of each mutation se-quences to no longer be sharp. Applying a push-through to γ resolves this intersectionso that the geometric intersection between γ and γ matches their algebraic intersection.This simplification ensures that the result of µ γ is a sharp 3-graph that matches the TypeII normal form. If we compare the mutations in the sequence on the left and the sequenceon the right of the figure, we can see that the presence of the A n tail cycle γ does not affectthe computation. Figure 22.
Type I to Type II mutations. Arrows labeled by I , II, or III indicatea twist, push-through, or flop involving a cycle of the same color.iv. (Type I to Type IV.i) We now consider the first of two Type I to Type IV mutations,shown in Figure 23. Starting with the configuration of cycles at the left of each sequenceand mutating at γ causes γ and γ to cross. Applying a push-through to γ or to γ (notpictured) simplifies the resulting intersection and yields a Type IV.i normal form made up f the cycles γ , γ , γ , and γ . The sequences on the left and right of Figure 23 differ onlyby the presence of the A n tail cycle γ . Figure 23.
Type I to Type IV.i mutations.v. (Type I to Type IV.ii) In Figure 24, we consider the cases where γ intersects one I -cycle(left) or two I -cycles (right) in the A n tail subgraph, as we did in the Type I to Type IIcase. As in the Type I to Type II case, we must apply a push-through to resolve the newintersections between that cause the second 3-graph in each sequence to fail to be sharp.When we include both γ and γ in the sequence on the right, we get two new intersectionsafter mutating, and therefore require two push-throughs. Note that in the IV.ii case, wemust first apply the push-through to γ and γ in order to ensure that we can apply apush-through to any additional cycles in the A n tail subgraph. This causes the Y -cyclesof the graph to correspond to different vertices in the quiver than in the Type IV.i normalform, which is the main reason we distinguish between the normal forms for Type IV.i andType IV.ii. Figure 24.
Type I to Type IV.ii mutations.In Figure 25 we show how to apply push-throughs to completely simplify the long I and Y -cyclespictured in the Type I to Type IV.ii graph. As mentioned above, these push-throughs are identicalto any other computation required to simplify our resulting 3-graphs to a set of short I -cycles andshort Y -cycles. igure 25. Push-through examples. The first push-through move simplifies thepink long I -cycle γ , while the second simplifies the dark green long Y -cycle γ The above cases describe all possible mutations of the Type 1 normal form. Each of these mutationsyields a sharp 3-graph with short I -cycles and Y -cycles, as desired. Type II.
We now consider mutations of our Type II normal form. See Figure 26 for the relevantquivers. As shown in the figure, performing a quiver mutation at the 2-valent vertices labeled by v or v yields a Type III quiver, while a quiver mutation at the vertices labeled v or v yieldseither another Type II quiver or a Type I quiver, depending on the intersection of v or v withany A n tail subquivers. Figure 26.
From top to bottom, Type II to Type I mutations, Type II to Type II,and Type II to Type III quiver mutations.i. (Type II to Type I) We first consider the sequence of 3-graphs pictured in Figure 27. Muta-tion at γ results in a new geometric intersection between γ and γ even though (cid:104) γ , γ (cid:105) = 0. igure 27. Type II to Type I mutations.We can resolve this by applying a reverse push-through at the trivalent vertex where γ and γ meet. The resulting 3-graph is sharp, as γ and γ no longer have any geometricintersection. This computation is identical if γ were to intersect a single A n tail cycle andwe mutated at γ instead. Note that here we require the red edge adjacent to the trivalentvertex where we applied our push-through not carry a cycle, as specified by our normal form. Figure 28.
Type II to Type II mutations.ii. (Type II to Type II) We now consider the sequence shown in Figure 28. After mutating at γ , we have the same intersection between γ and γ as in the previous case, which we againresolve by reverse push-through at the same trivalent vertex. In this case, we also have anintersection between γ and γ , which we resolve via push through of γ . As a result, γ becomes a Y -cycle, and the Type II normal form is now made up of the cycles γ , γ , γ , and γ , while γ becomes an A n tail cycle.iii. (Type II to Type III.i) Mutation at γ or γ in the Type II normal form yields either of theType III normal forms. In the sequence on the left of Figure 29, mutation at γ leads to ageometric intersection between γ and γ at two trivalent vertices. Since the signs of thesetwo intersections differ, the algebraic intersection (cid:104) γ , γ (cid:105) is zero, so the resulting 3-graphis not sharp. However, it is sharp at γ and γ , and applying a flop to the 3-graph removesthe geometric intersection between γ and γ at the cost of introducing the same inter-section between γ and γ . Therefore, applying the flop does not make the 3-graph sharp, igure 29. Type II to Type III mutations.but it does show that the 3-graph resulting from our mutation is locally sharp at every cycle.iv. (Type II to Type III.ii) In the sequence on the right of Figure 29, mutation at γ yields asharp 3-graph that matches the Type III.ii normal form. Type III:
Figure 30 illustrates the Type III quiver mutations. Figures 31, 32, and 33 depict thecorresponding Legendrian mutations of the Type III normal forms.
Figure 30.
Type III quiver mutationsi. (Type III.i to Type II) We first consider the sequence of 3-graphs in Figure 31 (left). Mu-tating at γ or γ immediately yields a Type II normal form. Mutating at γ and γ insuccession yields a Type III.ii normal form. Note that if the 3-graph were not sharp at γ or γ we would first need to apply a flop. We can always apply this move because the 3-graphis locally sharp at each of its cycles. See the Type III.i to Type IV.i subcase below for anexample where we demonstrate this move. igure 31. Type III.i to Type II mutations (left) and Type III.ii to Type II mu-tations (right).ii. (Type III.ii to Type II) In the sequence on the right of Figure 31, mutation at either γ or γ yields a Type II normal form. Mutation at γ and γ in succession yields a Type III.inormal form. Therefore, applying these two moves in succession can take us between bothof our Type III normal forms. Figure 32.
Type III.i to Type IV mutationsiii. (Type III.i to Type IV) We now consider the sequence of 3-graphs in Figure 32. Since theinitial 3-graph is not sharp at γ , we must first apply a flop before mutating. After applyingthis flop, γ is a short I -cycle and the 3-graph is sharp at γ . Mutating at γ then yields aType IV.i normal form. The short I -cycles γ and γ are included to indicate where any A n tail cycles would be sent under this mutation.iv. (Type III.ii to Type IV) In Figure 33, mutation at γ causes γ and γ to cross while stillintersecting γ and γ at either end. We resolve this by first applying a push-through to γ and then applying a reverse push-through to the trivalent vertex where γ and γ intersecta red edge. This results in a sharp 3-graph with γ , γ , γ , and γ making up the TypeIV normal form. We again include γ and γ as cycles belonging to a potential A n tail igure 33. Type III.ii to Type IV mutationssubgraph in order to show where the A n tail cycles are sent under this mutation. Type IV:
Figure 34 illustrates all of the relevant Type IV quivers and their mutations. In general,the edges of a Type IV quiver have the form of a single k − cycle with the possible existence of3-cycles or outward-pointing “spikes” at any of the edges along the k − cycle. At the tip of each ofthese spikes is a possible A n tail subquiver. We will refer to a vertex at the tip of any of the spikes(e.g., the vertex v in Figure 34) as a spike vertex and any vertex along the k − cycle will be referredto as a k − cycle vertex. A homology cycle corresponding to a spike vertex will be referred to as aspike cycle. Mutating at a spike vertex increases the length of the internal k − cycle by one, whilemutating at a k − cycle vertex decreases the length by 1, so long as k >
3. Figures 35, 36, 37, and38 illustrate the corresponding mutations of 3-graphs for Type IV to Type I and Type IV to TypeIII when k = 3.i. (Type IV.i to Type I) We first consider the sequence of 3-graphs in Figure 35. Mutation at γ causes γ and γ to cross. Application of a reverse push-through at the trivalent vertexwhere γ and γ intersect a red edge removes this crossing and yields a Type I normal formwhere γ is the sole Y -cycle.ii. (Type IV.ii to Type I) Mutation at γ in Figure 36 yields a 3-graph with geometric in-tersections between γ and γ , and γ and γ . The application of reverse push-throughsat the trivalent vertex intersections of γ with γ and γ with γ removes these geometricintersections, resulting in a Type I normal form where γ is the sole Y -cycle. We also applya candy twist (Legendrian Surface Reidemeister move I) to simplify the intersection at thetop of the resulting 3-graph.iii. (Type IV.i to Type III) We now consider the two sequences of 3-graphs in Figure 37. Muta-tion at any of γ , γ , γ , or γ in the Type IV.i normal form yields a Type III normal form.Specifically, mutation at γ yields a Type III.i normal form that requires no simplification,while mutation at γ (not pictured) yields a Type III.ii normal form that also requires nosimplification. The computation for mutation at γ is pictured in the sequence on the rightand is identical to the computation for mutation at γ . The first step of the simplificationis the same as the Type IV.i to Type I subcase described above. However, we require the igure 34. Type IV quiver mutations
Figure 35.
Type IV.i to Type I mutationsapplication of an additional push-through to remove the geometric intersection between γ and γ . This makes γ into a Y -cycle and results in a Type III normal form.iv. (Type IV.ii to Type III) Mutation at γ in our Type IV.ii normal form, depicted in Figure38, results in a pair of geometric intersections between γ and γ . Application of a flopremoves these geometric intersections and results in a sharp 3-graph with Y -cycles γ and γ , which matches our Type III.ii normal form. Note that the computations for mutations igure 36. Type IV.ii to Type I mutations
Figure 37.
Type IV.i to Type III mutations
Figure 38.
Type IV.ii to Type III mutationsof the two possible Type IV.ii 3-graphs given in Figure 24 (left) and (right) are identical.The remaining three subcases are all Type IV to Type IV mutations.v. (Type IV.ii to Type IV) Figure 39 depicts mutation of a Type IV.ii normal form at a spikecycle. Mutating at γ results in an additional geometric intersection between γ and γ .We first apply a reverse push-through at the trivalent vertex where γ , γ and γ meet.This introduces an additional geometric intersection between γ and γ , that we resolveby applying a push-through to γ . Application of a reverse push-through to the trivalentvertex where γ and γ intersect a red edge resolves the final geometric intersection be-tween γ and γ . The Y -cycles of the resulting 3-graph correspond to k − cycle vertices ofthe quiver. As shown below, none of the other Type IV to Type IV mutations result in Y -cycles corresponding to spike vertices. Therefore, assuming we have simplified after each f our mutations in the manner described above, the only possible way a Type IV.ii 3-grapharises is by mutating from the initial Type I graphs in Figure 24. Hence, all other Type IV3-graphs only have Y -cycles corresponding to k − cycle vertices in the quiver. The computa-tions for the different Type IV.ii 3-graphs given in Figure 24 (top right and bottom right)are again identical. Figure 39.
Type IV.ii graph mutation at a spike cycle.vi. (Type IV to Type IV) Figure 40 depicts Type IV to Type IV mutations when the length ofthe quiver k − cycle is greater than 3. When mutating at a homology cycle correspondingto a k − cycle vertex of the quiver, we have two possibilities. Figure 40 (top) shows the casewhere γ intersects another Y -cycle γ , which corresponds to a k − cycle vertex in the quiver.Figure 40 (bottom) considers the case where γ only intersects I -cycles. In both of thesecases we must apply a reverse push-through to the trivalent vertex where γ and γ intersecta red edge in order to simplify the 3-graph. This particular simplification requires that nei-ther of the two edges adjacent to the leftmost edge of γ carry a cycle before we mutate. Asimilar computation involving the purple Y -cycle (not pictured) also requires that neither ofthe two edges adjacent to the bottommost edge of γ carry a cycle. Crucially, our computa-tions show that Type IV to Type IV mutation preserve this property, i.e., that both of the Y -cycles have an edge that is adjacent to a pair of edges which do not carry a cycle. When k = 4 , the resulting 3-graph in the top line will have a short I -cycle adjacent to γ and γ ,while the resulting 3-graph in the middle line will have a short Y -cycle adjacent to γ and γ .vii. (Type IV to Type IV) Figure 41 depicts mutation at a spike cycle. Since we have alreadydiscussed the Type IV.ii spike cycle subcase above, we need only consider the case whereeach of the spike cycles is a short I -cycle. The navy short I -cycle and γ are included to helpindicate where A n tail cycles are sent under this mutation. The computation for mutatingat a spike edge for Type IV.i (i.e. the k = 3 case) is identical to the k > I -cycle, butthe computation is again a straightforward mutation of a single I -cycle that requires nosimplification.In each of the Type IV to Type IV subcases above, mutating at a Y -cycle or an I -cycle andapplying the simplifications as shown preserves the number of Y -cycles in our graph. Therefore,our computations match the normal form we gave in Figure 19 with k − I -cycles in thenormal form 3-graph not belonging to any A n tail subgraphs. igure 40. Type IV to Type IV mutations at homology cycles corresponding to k − cycle vertices in the quiver. Mutating at γ , γ , or γ (corresponding to k − cyclevertices in the quiver) in the 3-graphs on the left decreases the length of the k − cyclein the quiver by 1. Figure 41.
Type IV to Type IV mutations at spike cycles. Mutating at the spikecycles γ or γ in the 3-graphs on the left increases the length of the k − cycle in theintersection quiver by 1.This completes our classification of the mutations of normal forms. In each case, we have produceda 3-graph of the correct normal form that is locally sharp and made up of short Y -cycles and I -cycles. Thus, any sequence of quiver mutations for the intersection quiver Q (Γ ( D n ) , { γ (0) i } ) of ourinitial Γ ( D n ) is weave realizable. Hence, given any sequence of quiver mutations, we can apply asequence of Legendrian mutations to our original 3-graph to arrive at a 3-graph with intersectionquiver given by applying that sequence of quiver mutations to Q (Γ ( D n ) , { γ (0) i } ), as desired. (cid:3) Having proven weave realizability for Γ ( D n ), we conclude with a proof of Corollary 1.3.2. Proof of Corollary 1.
We take our initial cluster seed in C (Γ) to be the cluster seed asso-ciated to Γ ( D n ). The cluster variables in this initial seed exactly correspond to the microlocal onodromies along each of the homology cycles of the initial basis { γ (0) i } . The intersection quiver Q (Γ ( D n ) , { γ i } ) is the D n Dynkin diagram and thus the cluster seed is D n -type. By definition,any other cluster seed in the D n -type cluster algebra is obtained by a sequence of quiver muta-tions starting with the quiver Q (Γ ( D n ) , { γ i } ) and its associated cluster variables. Theorem 1implies that any quiver mutation of Q (Γ ( D n ) , { γ i } ) can be realized by a Legendrian mutation inΛ(Γ ( D n )) , so we have proven the first part of the corollary. The remaining part of the corollaryfollows from the fact that the D n -type cluster algebra is known to be of finite mutation type with(3 n − C n − distinct cluster seeds. (cid:3) References [ABL21] Byung Hee An, Youngjin Bae, and Eunjeong Lee. Lagrangian fillings for legendrian links of finite type.arXiv:2101.01943, 2021.[Ad90] V. I. Arnol (cid:48) d. Singularities of caustics and wave fronts , volume 62 of
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