Augmentations, Fillings, and Clusters
aa r X i v : . [ m a t h . S G ] A ug Augmentations, Fillings, and Clusters
Honghao Gao ∗ , Linhui Shen † , and Daping Weng ‡ Abstract
We prove that the augmentation variety of any positive braid Legendrian link carries a natu-ral cluster K structure. We present an algorithm to calculate the cluster seeds that correspondto the admissible Lagrangian fillings of the positive braid Legendrian links. Utilizing augmenta-tions and cluster algebras, we develop a new framework to distinguish exact Lagrangian fillings. Contents ∗ Department of Mathematics, Michigan State University, [email protected] † Department of Mathematics, Michigan State University, [email protected] ‡ Department of Mathematics, Michigan State University, [email protected] Reduction of Marked Points 65A Appendix: Basics of Cluster theory 67
A.1 Cluster Ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68A.2 Separation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A.3 (Quasi)-Cluster Morphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71Throughout this paper, we fix F to be an algebraically closed field of characteristic 2. Legendrian knots are central objects in low dimensional contact topology. Various associatedinvariants can be used to distinguish Legendrian knots and reveal properties of their ambientthree-manifolds. Augmentations are examples of “non-classical” Legendrian invariants, and theyform a moduli space called the augmentation variety. The aim of the paper is to establish aconnection between exact Lagrangian fillings of a Legendrian link, which are geometric objectsin the partially wrapped Fukaya category, and cluster seeds, which are building blocks to a rigidstructure in algebraic geometry, on the augmentation variety.
The classification of Legendrian links up to Legendrian isotopies is a fundamental problem in contactgeometry. Two classical invariants are the
Thurston-Bennequin number and the rotation number,through which one may distinguish Legendrian links that are topologically isotopic.Two decades ago, Chekanov constructed the first non-classical Legendrian invariant [Che02].The construction packages the Floer-theoretical data into the Chekanov-Eliashberg differentialgraded algebra (CE dga). The stable-tame isomorphism class of the CE dga is a Legendrian invari-ant but difficult to work with in practice. Instead, its “functors of points,” namely augmentations,and the “cotangent spaces” at these points, namely linearized contact homologies, are readily com-putable and have been used to distinguish many Legendrian links of the same Thurston-Bennequinnumber and rotation number. For example, the famous Chekanov pair consists of two distinctLegendrian representatives of the twist knot m [Che02].Augmentations, although algebraically defined, admit a symplecto-geometric interpretation[NRSSZ15, NZ09, Nad09, Syl19, GPS18, EL17]. Graded embedded exact Lagrangian fillings aregeometric objects in the Fukaya category. They form a particular family of cobordisms from theempty set to a Legendrian link. The cobordisms further induce morphisms between associated CEdgas and give rise to augmentations by definition [EGH00, EN18]. Many, but not all, augmentationscan be obtained this way.From the perspective of Floer theory with Novikov coefficients, we furnish every Lagrangianfilling with a GL -local system. The GL -character variety over a Lagrangian filling surface is analgebraic torus. The augmentation functoriality gives rise to a morphism from the algebraic torusto the augmentation variety, which potentially determines a cluster chart on the latter. In fact,a torus (2 , n )-link has at least a Catalan number worth of Lagrangian fillings up to Hamiltonianisotopy [Pan17, STWZ19], and the Catalan number is the number of cluster seeds in the Dynkintype A n − cluster algebra, suggesting that Lagrangian fillings should correspond to cluster seedson augmentation varieties. 2luster algebras were invented by Fomin and Zelevinsky [FZ02]. Cluster varieties were in-troduced by Fock and Goncharov [FG09] as geometric counterparts to cluster algebras. Clustervarieties come in pairs called cluster ensembles , each of which consists of a cluster K variety (alsoknown as a cluster A -variety) and a cluster Poisson variety (also known as a cluster X -variety).We mainly focus on cluster K varieties in this paper. A cluster K variety A is equipped with adistinguished set of cluster seeds, each of which contains the following ingredients: (1) a clusterchart: an open dense algebraic torus embedded in A , (2) cluster variables: a set of toric coordinateswhich are regular functions on A , (3) a quiver: each vertex of the quiver is labelled with a clustervariable. Cluster seeds are related to each other by a certain mutation procedure on their quiversand their cluster variables transform in a manner prescribed by the quiver mutation.Here are some “filling to cluster” heuristics:(1) Each torus chart comes from the GL -character variety supported on a Lagrangian fillingsurface. In this paper, we prove it for admissible fillings, which includes exact Lagrangianfillings arising from pinching sequences.(2) Cluster variables are holonomies of disk boundaries along the Lagrangian. They can beexpressed as polynomials of Reeb chords. Since Reeb chords are global regular functions overthe augmentation variety, so are their polynomials. In this paper, we prove that Reeb chordsare cluster variables.(3) The quiver is expected to come from the intersection form of a set of distinguished cycles inthe integral homology class of the Lagrangian surface [SW19, STWZ19].(4) The mutation is expected to come from the two Polterovich smoothings of a transverse doublepoint [Pol91], which produce a pair of Lagrangian surfaces which are smoothly isotopic butnot Hamiltonian isotopic. In variant settings, the geometric mutation induces wall-crossingtype formulae [Aur07, Via14, PT20, CZ20].There are a few reasons why we choose to work with positive braid links in this paper, asidefrom their combinatorial simplicity and their admission of natural Legendrian representatives. First,positive braid links possess an ample amount of embedded exact Lagrangian fillings, which otherLegendrian links may lack: for example, the figure eight knot 4 has an augmentation, but does nothave any embedded exact Lagrangian fillings; the same phenomenon also occurs for many hyperbolic2-bridge links. Most small knots are also obstructed from admitting fillings [Ng01]. Second, theyhave more symmetry compared to an arbitrarily presented Legendrian. This is crucial when weapply our results to the question of whether a Legendrian admits infinitely many Lagrangian fillings. Let β be a positive braid of n strands and length l . Let Λ β be the associated positive braidLegendrian link (see (3.1)), decorated by n marked points placed near the right cusps. Let Q β be aquiver constructed from β via the wiring diagram [BFZ05, FG06] (see Section 5.3, initial seed). Ourfirst main result is on the fully non-commutative CE dga in characteristic 2, denoted by A (Λ β ). Theorem 1.1. (a)
The homology H ( A (Λ β )) is a non-commutative Z -algebra generated by de-gree 0 Reeb chords b , · · · , b l and formal varaibles t ± , · · · , t ± n associated with marked points,modulo the relations t − k = M k for all ≤ k ≤ n, here M is an n × n matrix whose entries are polynomials in b , · · · , b l , and M k is theGelfand-Retakh quasideterminant of the upper-left k × k submatrix of M with respect to the(k,k)-entry (Corollary 3.8). (b) The abelianization H ( A (Λ β ) , F ) c is a cluster algebra associated with the quiver Q β . All degree Reeb chords b , . . . , b l are cluster variables in H ( A (Λ β ) , F ) c (Corollary 5.24). Our second main result establishes a natural correspondence between exact Lagrangian fillingsand cluster seeds. By generalizing a result of Ekholm-Honda-K´alm´an [EHK16] to our coefficient-enhanced setting, we obtain a contravariant functor Φ that takes every decorated exact Lagrangiancobordism L : Λ − → Λ + to a dga homomorphism Φ ∗ L : A (Λ + , P ) → A (Λ − , P ), where A (Λ ± , P ) isan enhancement of the CE-dga A (Λ ± ) with coefficients P taken from oriented marked curves on L (see Section 2.4). There are dga homomorphisms φ ∗± : A (Λ ± ) → A (Λ ± , P ). When L is an exactLagrangian filling of a Legendrian link Λ (i.e., a cobordism from the empty set to Λ), we obtain adga homomorphism A (Λ) φ ∗ + −→ A (Λ , P ) Φ ∗ L −→ A ( ∅ , P ) . The augmentation variety Aug (Λ β ) is the moduli space of F -valued augmentations of A (Λ β ).The dga A (Λ β ) is concentrated in non-negative degrees and Aug (Λ β ) is naturally identified withSpec H ( A (Λ β ) , F ) c . The above dga homomorphism Φ ∗ L ◦ φ ∗ + induces an algebraic variety morphism φ + ◦ Φ L : Aug ( ∅ , P ) −→ Aug (Λ) . (1.2)Every Hamiltonian isotopy of an exact Lagrangian cobordism L changes the dga homomorphismΦ ∗ L by a chain homotopy. Hence, the image of (1.2) is invariant under Hamiltonian isotopy.This paper focuses on a particular family of exact Lagrangian fillings, called admissible fillings. Definition 1.3.
An exact Lagrangian filling of a positive braid Legendrian link is admissible if itis a concatenation of the following exact Lagrangian cobordisms:(1) pinching, which is an exact Lagrangian cobordism that resolves a crossing inside the braid β ;(2) braid move, which changes a braid word via a Legendrian Reidemeister III move;(3) cyclic rotation, which changes Λ s i β ←→ Λ βs i by four Legendrian Reidemeister II moves;(4) minimum cobordism, which is the unique exact Lagrangian filling of a maximal tb unknot.By Theorem 1.1, Aug (Λ β ) is a cluster K variety associated with the quiver Q β . Theorem 1.4.
Let L be an admissible filling of Λ β decorated with a collection of oriented paths P . (a) The augmentation variety
Aug ( ∅ , P ) is isomorphic to the space of P -trivializations of rank-1local systems on L and is an algebraic torus of rank l ( β ) (Proposition 3.36 & Corollary 3.37) . (b) The morphism (1.2) is an open embedding of the algebraic torus
Aug ( ∅ , P ) into Aug (Λ β ) ,whose image coincides with a cluster chart in Aug (Λ β ) (Proposition 3.38 & Theorem 5.15) . (c) If two decorated admissible fillings L and L ′ correspond to cluster charts in different clusterseeds, then L ′ and L are not Hamiltonian isotopic (Corollary 5.16) . Throughout this paper we use ∗ symbols to denote algebra homomorphisms and reserve the plain symbols formorphisms between varieties. emark 1.5. Theorem 1.4 (c) allows us to distinguish non-Hamiltonian isotopic exact Lagrangianfillings. In a subsequent paper we will apply this method to recognize positive braid Legendrianlinks with infinitely many non-Hamiltonian isotopic but smoothly isotopic exact Lagrangian fillings.Below is a summary of how to obtain the cluster seed for an arbitrary admissible filling. SeeSection 5.3 for more details.
Initial Seed.
Let i = ( i , . . . , i l ) be a word for β ∈ Br + n . We construct an initial seed corre-sponding to the admissible filling that pinches the crossings in i one-by-one from left to right. Thequiver of the initial seed can be constructed using the wiring diagram [BFZ05, FG06]. Its unfrozenvertices are in bijection with the enclosed regions of a projection of the braid with respect to thebraid word i , and its frozen vertices are in bijection with the half open region on the right side ofthe projection. Note that each vertex lies to the right of a unique crossing i k . Accordingly we labelthe vertices by 1 , . . . , l . The cluster K coordinate associated with the k th vertex is A k = ∆ i k ( Z i ( b ) Z i ( b ) · · · Z i k ( b k ))where b j are the Reeb coordinates on Aug (Λ β ) (Definition 3.13), ∆ i is the i th principal minor ofa matrix (Definition 3.9), and the matrices Z i ( b ) are defined in (3.3). Example 1.6.
Let β ∈ Br +4 be presented by a word (1 , , , , , , , A = b , A = b , A = 1 + b b , A = b , A = b + b b + b b b ,A = b + b b , A = 1 + b b + b b b + b b + b b b b , A = b + b b b + b b . Cluster Seeds for admissible fillings.
Cluster seeds arisen from admissible fillings can beobtained from the initial seed via a sequence of cluster mutations. We list the sequences of mutationscorresponding to the first three moves described in Definition 1.3; the minimum cobordism merelyimposes a condition on the coefficients of the CE dga and does not change the cluster seed.(1)
Pinching . Suppose we pinch the crossing i k in the word i = ( i , . . . , i l ). For j < k , we set t j := { l | j < l ≤ k, i l = i j } . Let (cid:0) i (cid:1) , (cid:0) i (cid:1) , . . . denote the unfrozen quiver vertices on the i th level counting from left to right.We define the mutation sequences V j := µ ( ijtj ) ◦ µ ( ijtj − ) ◦ . . . µ ( ij ) , W j := µ ( ij ) ◦ µ ( ij ) ◦ . . . µ ( ijtj ) if i j = i k ,µ ( ij ) ◦ µ ( ij ) ◦ . . . µ ( ijtj ) if i j = i k . The mutation sequence for the pinching at the crossing i k is W ◦ W ◦ · · · W k − ◦ V k − ◦ V k − ◦ · · · V . Braid move . Let | i − j | = 1. The braid move from ( . . . , i, j, i, . . . ) to ( . . . , j, i, j, . . . ) corre-sponds to the mutation µ c at the vertex c associated with the region bounded by the crossings i, j, i .(3) Cyclic rotation . The mutation sequence for the rotation from βs i to s i β is R := µ ( i ) ◦ · · · ◦ µ ( ini ) , where (cid:0) in i (cid:1) be the last unfrozen quiver vertex on the i th level. The mutation sequence for thebackward rotation from s i β to βs i is R − . In Sections 2.1 – 2.3, we review the general theory of the Chekanov-Eliashberg dga and the augmen-tation variety of a Legendrian link. In Section 2.4, we introduce an enhancement of the Ekholm-Honda-K´alm´an functor for decorated exact Lagrangian cobordisms. This enhancement is necessaryto produce coordinates on the algebraic torus induced from a decorated admissible filling.Section 3 focuses on positive braid Legendrian links and their admissible fillings. In Section3.2, we relate the computation of dga differentials in A (Λ β ) to Gelfand-Retakh quasideterminantsand prove Theorem 1.1(a). In Section 3.3, we compute the morphism induced from a pinchingand prove it is an open embedding between augmentation varieties. The proof involves a “doubledipping” technique. In Section 3.4, we prove that admissible fillings induce algebraic open tori inthe augmentation variety. Each algebraic torus is interpreted as the space of trivializations of localsystems on an admissible filling. We prove the first two statements of Theorem 1.4.Section 4 focuses on double Bott-Samelson (BS) cells. In Section 4.1, we construct naturalisomorphisms between augmentation varieties and double BS cells. In Section 4.2, we reviewthe cluster K structure on double BS cells, from which we induce the cluster K structure onaugmentation varieties via the natural isomorphisms. In Section 4.3, we present an open embeddingbetween double BS cells compatible with their cluster K structures, which is the counterpart of apinching.In Section 5, we establish the connection between admissible fillings and cluster seeds. We detailthe effect of the pinching in Section 5.1 and the effect of the cyclic rotation in Section 5.2. We showin Section 5.3 that any admissible filling of a positive braid link gives rise to a cluster seed on itsaugmentation variety, completing the proof of Theorem 1.1 and Theorem 1.4. Section 5.4 containsan example that computes the cluster seed for a concrete Legendrian filling.In Section 6, we restrict the obtained cluster K structure on Aug (Λ β ) to the augmentationvarieties of positive braid Legendrian links with exactly one marked point on each link component. Acknowledgement
We would like to thank Roger Casals, Matthew Hedden, Cecilia Karlsson, Lenhard Ng, Dan Ruther-ford, Kevin Sackel, and Eric Zaslow for the helpful discussion on Floer theory, contact geometry,and Legendrian links. We would like to thank Alexander Goncharov, Bernhard Keller, and MichaelShapiro for the helpful discussion on cluster theoretical aspect of the paper. We would like to thankVladimir Retakh for the helpful discussion on quasi-determinants and non-commutative cluster al-gebras. We would also like to thank Roger Casals, Kevin Sackel, and Bernhard Keller for theirvaluable suggestions and comments on the first draft of this paper.6
The Chekanov-Eliashberg dga and Augmentation Variety
In this section, we briefly recall the definitions and basic properties of Chekanov-Eliashberg dif-ferential graded algebras and augmentation varieties for Legendrian links. We refer the readers to[Gei08, Etn05, EN18] for more details on Legendrian links.
Let ( R xyz , ξ st ) be the standard tight contact manifold, where the contact structure ξ st is the kernelof the 1-form α = dz − ydx . A Legendrian knot is a closed embedding γ : S → R xyz such thatthe pull back γ ∗ α = 0. A Legendrian link is a disjoint union of Legendrian knots. Two Legendrianlinks are Legendrian isotopic if they are isotopic through a family of Legendrian links.Legendrian links can be visualized by two-dimensional projections, particularly • the front projection π F : R xyz → R xz , and • the Lagrangian projection π L : R xyz → R xy .A Legendrian link Λ in ( R , ξ st ) is projected to an immersed curve in either projection. In the frontprojection, the y -coordinate can be recovered by the slope y = dz/dx . No vertical tangencies areallowed and the immersed curve must turn via cusps (which can be locally modeled by z = ± x ).In the Lagrangian projection, π L (Λ) is an immersed Lagrangian curve with respect to dx ∧ dy andgenerically has finitely many crossings. One needs to specify the over/under-crossing to recoverthe Legendrian link. The contact manifold R xyz can be identified with the 1-jet space J R x = T ∗ R x × R z , which leads to another useful projection: • the base projection π b : R xyz → R x .The Reeb vector field R of a contact manifold ( M, ker α ) is the unique vector field such that dα ( R, · ) = 0 and α ( R ) = 1. In ( R , ξ st ), the Reeb vector field R = ∂ z . A Reeb chord for aLegendrian link Λ is a flow line of R with boundaries on Λ. A Reeb chord appears to be a double-crossing in the Lagrangian projection π L (Λ).There are three classical invariants for Legendrian links. The first one is the underlying smoothknot type. The second one is the Thurton-Bennequin number tb(Λ), which counts the differencebetween the contact framing and the Seifert framing. The third one is the rotation number rot(Λ),which can be realized as the winding number in a Lagrangian Grassmannian. These invariants are“classical” in the sense that the algebraic construction only involves singular homology. In thispaper, we investigate “non-classical” Legendrian invariants, which can distinguish Legendrians ofthe same underlying smooth knot types, Thurston-Bennequin numbers, and rotation numbers. The fully non-commutative Chekanov-Eliashberg differential graded algebra associated to a Leg-endrian Λ is a unital non-commutative algebra over F . For different Legendrian links representingthe same Legendrian isotopy class, their dgas are equivalent up to a sequence of stable tame iso-morphisms, which is a special case of dga quasi-isomorphisms.7 .2.1 CE dga for a Lagrangian Diagram The construction in this section is a fully non-commutative enhancement of Chekanov’s originalversion [Che02]. A partially non-commutative version of the dga is used in [Pan17].We decorate the Legendrian link Λ with a set of marked points T := { t i , · · · , t m } . We requirethat each component of Λ contains at least one marked point. Marked points are generic; that is,their front projections do not overlap with the cusps and crossings, and their Lagrangian projectionsare away from the immersed points.For a decorated Legendrian link (Λ , T ), we define the fully non-commutative Chekanov-Eliashbergdga ( A (Λ) , ∂ ) in three stages: algebra, grading, and differential. (I) Algebra. The algebra A (Λ) is freely generated over Z by the finite set of Reeb chords R = { c , . . . , c n } and the finite set of formal variables T ± = (cid:8) t ± , . . . , t ± m (cid:9) associated to themarked points, satisfying the relations t i t − i = t − i t i = 1 for 1 ≤ i ≤ m . By definition, A (Λ) has a Z -basis consisting of reduced words in R ∪ T ± . (II) Grading. The grading | · | is defined over generators and extended additively to words ofgenerators. For simplicity, we assume the rotation number rot = 0 for every link component, (whichholds for positive braid links). Under this assumption, all formal variables have degree 0.When Λ is a knot, the grading of a Reeb chord c i is defined from the winding number inthe Lagrangian Grassmannian. We perturb the Lagrangian diagram π L (Λ) is in a position thatall crossings locally look like the two diagonals in the plane. Let γ i be the path running alongorientated π L (Λ) from the overcrossing of c i to the undercrossing of c i . The counterclockwiserotation number of the tangent vector to γ i , denoted by rot( γ i ), is an odd multiple of 1 /
4. Wedefine | c i | = 2rot( γ i ) − /
2. When Λ is a link, the grading of Reeb chords is not well-defined, sincethere is no such a path γ i for a Reeb chord connecting different components of Λ. This can be fixedin several ways – by choosing capping paths in the Lagrangian diagram, or by choosing a Maslovpotential in the front diagram. We postpone it and explain in detail later in the front diagram. (III) Differential. Near each crossing of π L (Λ), the plane R xy is broken into four quadrants. Wedecorate each quadrant with a Reeb sign as follows. A quadrant is labeled with a + if traversingits boundary in the counterclockwise direction one moves from an understrand to an overstrand.Otherwise it is labeled with a − . ++ −− Reeb sign:The differential ∂ : A (Λ) → A (Λ) satisfies the graded Leibniz rule ∂ ( xy ) = ∂ ( x ) y + x ( ∂y )and ∂ ( t i ) = ∂ ( t − i ) = 0. For Reeb chords d and d , · · · , d n , let M ( d ; d , · · · , d n ) be the modulispace of immersed disks up to reparametrization. Each orientation-preserving immersed disks u : ( D , ∂D ) → ( R xy , π L (Λ)) has one convex corner mapped to a positive quadrant at d andconvex corners mapped to negative quadrants at d , · · · , d n , appearing in counterclockwise order.Given u ∈ M ( d ; d , · · · , d n ), the image of the disk boundary is a union of n + 1 paths η , · · · , η n in π L (Λ) where η i starts at d i and ends at d i +1 for i ∈ Z n +1 . Let t ( η i ) be the word of t ± , · · · , t ± m by the ordered counting of η i crossing marked points t , · · · , t m , where the exponent depends onwhether the disk orientation align with the knot orientation. Associated with u is a word8 ( u ) = t ( η ) d t ( η ) d · · · d n t ( η n ) , (2.1)The differential of a Reeb chord is defined as: ∂ ( d ) = X n ≥ , d , ··· ,d n Reeb chords ,u ∈M ( d ; d , ··· ,d n ) w ( u ) . The differential ∂ : A (Λ) → A (Λ) has the following properties [Che02, ENS02, EES05]: (1) ∂ has degree −
1, (2) ∂ = 0, and (3) the stable tame isomorphism type of ( A (Λ) , ∂ ) is an invariantof Λ under Legendrian isotopy and choice of base points. The symplectization R t × R xyz is equipped with an exact symplectic form ω = d ( e t α ). Let J be a generic compatible-tame almost-complex structure, the differential in the CE-algebra can beequivalently defined via counting J -holomorphic disks [ENS02].More specifically, R t × Λ ⊂ R t × R xyz is an exact Lagrangian submanifold. Any immerseddisk u : ( D , ∂D ) → ( R xy , π L (Λ)), u ∈ M ( d ; d , · · · , d n ), can be lift to a J -holomorphic curve˜ u ∈ M J ( d ; d , · · · , d n ), ˜ u : ( D n +1 , ∂D n +1 ) → ( R t × R xyz , R t × Λ) , where D n +1 is the disk with n + 1 boundary punctures, the positive double point d is lifted tothe corresponding Reeb chord at { + ∞} × Λ, the negative double points d , · · · , d n are lifted tocorresponding Reeb chords at {−∞} × Λ, the disk has asymptotic behavior near {±∞} × R xyz , andthe disk has finite energy R ˜ u ∗ ω . Finally, the lift is unique up to translation along the R t direction.From M J ( d ; d , · · · , d n ) to M ( d ; d , · · · , d n ) the map is given by the projection R txyz → R xy .Hence [ENS02] proves that M ( d ; d , · · · , d n ) = M J ( d ; d , · · · , d n ) / R . Consequently the CE dgacan be defined in the symplectization R t × R xyz . The dga is naturally defined in the Lagrangian projection, while Legendrian isotopies are easier todepict in the front projection. Ng’s resolution provides a combinatorial method to go between.
Theorem 2.2 (Ng’s resolution, [Ng03, Proposition 2.2]) . Suppose Λ has a generic front diagram π F (Λ) , then there is a Legendrian isotopic Λ ′ , such that π L (Λ ′ ) can be obtained from π F (Λ) by thefollowing local moves: With Ng’s resolution, one can migrate the definition of the CE dga to the front diagram.The preimage of cusps in Λ are called caustics . A
Maslov potential is a locally constant function9 : Λ \{ caustics } → Z / (2rot(Λ)) such that around each cusp, µ (upper strand) = µ (lower strand)+1.Let f , f be the local height functions of the strands bounding a Reeb chord c . The grading of c is | c | = µ (upper strand) − µ (lower strand) + ind c ( f − f ) − , where ind c ( f ) is the Morse index of f at c .The dga for the front projection can be generalized to a bordered version for Legendrain tangles ,which allow non-compact horizontal strands [Siv11, NRSSZ15]. The bordered algebra for a tangleincludes pairs of strands on the left boundary as additional generators, see loc.cit. for more details.
Remark 2.3.
Disk counting near a right cusp in π F (Λ) has the following cases after Ng’s resolution. Let ( A (Λ) , ∂ ) be the CE dga for a decorated Legendrian link (Λ , T ). For the rest of the paper, wealways assume that each component of Λ has rotation number 0. Definition 2.4. An augmentation of A (Λ) is a unital dga homomorphism ǫ : ( A (Λ) , ∂ ) → ( F , . The moduli space of augmentations of A (Λ) is called the augmentation variety of A (Λ), and wedenote it as Aug ( A (Λ)) or Aug (Λ) if the CE dga A (Λ) is clear from the context. Remark 2.5.
In general, if Λ has non-vanishing rotation number, its CE dga is periodic and theaugmentation should also have a periodic target.
Lemma 2.6.
Let ǫ be an augmentation of a CE dga ( A (Λ) , ∂ ) . Then for any homogeneous element a with | a | 6 = 0 , there is ǫ ( a ) = 0 .Proof. This follows from the fact that ǫ is degree-preserving. Corollary 2.7.
An augmentation ǫ of a CE dga ( A (Λ) , ∂ ) is uniquely determined by its evaluationsat the degree generators (both degree Reeb chords and formal variables), and the evaluations aresubject to the conditions that ǫ ◦ ∂ ( a ) = 0 for any degree 1 Reeb chord a .Proof. It follows from the fact that any algebra homomorphism is uniquely determined by theimages of the generators of the domain algebra.
Remark 2.8.
Instead of the CE dga A (Λ), one can consider its commutative dga A (Λ) c , definedto be the quotient of the two sided ideal generated by commutators. Since the target of an aug-mentation is a field, there is a natural bijection between augmentations of A (Λ) and augmentationsof A (Λ) c . Corollary 2.9.
Augmentation varieties are affine varieties. In particular, if A (Λ) is concentratedin non-negative degrees, then Aug( A (Λ)) = Spec H ( A (Λ) c , F ) . roof. Let n be the number of marked points and let m be the number of degree 0 Reeb chordsin Λ. Then the augmentation variety Aug( A (Λ)) is then a subvariety of ( F × ) n × F m cut out byequations ǫ ◦ ∂ ( a i ) = 0, one for each degree 1 Reeb chord a i .When A (Λ) is concentrated in non-negative degrees, then H ( A (Λ) c , F ) is the commutativealgebra over F generated by the invertible formal variables that are labels of the marked points andthe degree 0 Reeb chords, modulo the relations ∂a i = 0 for all degree 1 Reeb chords a i , which isprecisely the coordinate ring of Aug( A (Λ)).The number of Reeb chords is not invariant under Legendrian isotopies. Consequentially, theaugmentation variety in this context does depend on the geometric curve in its isotopy class. Interms of the CE dga defined via the front diagram, we can specify the effects of Reidemeister moves: • Reidemeister I moves – no change.The move creates a pair of Reeb chords with degrees 0 and 1. There is an extra dimensionfor the ambient space of the augmentation variety, and there is an extra polynomial to cutoff. Their effects cancel and the augmentation variety does not change. • Reidemeister II moves – it depends.The move creates a pair of Reeb chords with degrees i, i + 1. If i < − i ≥
1, then the set ofdegree 0 , { i, i +1 } = { , } ,then there is an extra generator and an extra relation – the augmentation variety does notchange. If { i, i + 1 } = {− , } , then there is one extra degree 0 chord, but the number of cutoff polynomial remains the same. The augmentation variety increases its dimension by 1. • Reidemeister III moves – no change.This move induces an identification between the dga of Legendrians before and after the move.The number of Reeb chords in each degree does not change. Therefore the augmentationvarieties before and after the move are isomorphic.On top of the Reidemeister moves, the decoration may also change the augmentation variety. • Adding a marked point – an extra dimension.There is an extra degree 0 generator but the same number of defining equations. Hence thedimension of the augmentation variety increases by one. • Moving a marked point through a crossing – no change.The move induces an isomorphism for the dga and the augmentation variety. ∗ t b ∗ ttb ∗ t b ∗ tbt − (2.10)The augmentation variety presented here is not the best moduli space for the purpose of defininga Legendrian isotopy invariant. Since the dga is only an invariant up to stable-tame isomorphisms,an invariant moduli space of augmentation should invoke dga homotopy and possibly more relations.The work [NRSSZ15] constructed a unital A ∞ category of augmentations, and the moduli space isa stack of objects. The augmentation variety here can be viewed as a framed version of the modulistack. It is an affine variety, and its algebraic geometry is easier to study.11 .4 Exact Lagrangian Cobordism, Coefficient Enhancement, and Functoriality The CE dga as well as its associated augmentation variety have a characterization from disk count-ing in the symplectization [EGH00, ENS02]. This perspective leads to the functoriality of theseLegendrian invariants with respect to an exact Lagrangian cobordism (with cylindrical ends).
Definition 2.11.
Let Λ + and Λ − be Legendrian links in standard contact ( R , ξ st ). A Lagrangiancobordism (with cylindrical ends) from Λ − to Λ + is a Lagrangian submanifold L of the symplecti-zation ( R × R , d ( e t α )) such that for some N > ( L ∩ (( −∞ , − N ] × R ) = ( −∞ , − N ] × Λ − ,L ∩ ([ N, ∞ ) × R ) = [ N, ∞ ) × Λ + . A Lagrangian cobordism is exact if there exists a function f : L → R such that ( e t α ) | L = df , andif f is constant on each end, ( −∞ , − N ) × Λ − and ( N, ∞ ) × Λ + .An exact Lagrangian filling of Λ is an exact Lagrangian cobordism from ∅ to Λ.In this paper, we assume an exact Lagrangian cobordism (or filling) always has cylindrical endsand is compatibly oriented with respect to Λ ± . In [EHK16], Ekholm-Honda-K´alm´an discussed an Z functoriality, which maps an exact Lagrangiancobordism L functorially to a dga homomorphism Φ ∗ L between the CE dga of the Legendrian atthe two ends. Briefly summarizing, • Let L : Λ − → Λ + be an exact Lagrangian cobordism. It induces an Z -dga morphismΦ ∗ L : ( A (Λ + ) , ∂ + ) → ( A (Λ − ) , ∂ − ) . Moreover, it sends Hamiltonian isotopy to stable tame equivalence, and it is functorial. • If L is decomposable, then Φ ∗ L can be computed explicitly over each generator.The dga morphism Φ ∗ L further induces a morphism between the (unenhanced) augmentation vari-eties via a pull-back: Φ L ( ǫ ) : ( A (Λ + ) , ∂ + ) Φ ∗ L −−→ ( A (Λ − ) , ∂ − ) ǫ −→ ( Z , . In this section, we will discuss in full detail how to enhance the morphism induced from aLagrangian cobordism so as to accommodate the non-commuting formal variables. To make thisconstruction, we need to equip the exact Lagrangian cobordism L with a decoration consisting ofa collection of oriented marked curves P . Definition 2.12.
Let L : Λ − → Λ + be an exact Lagrangian cobordism. View L as an embeddedsurface in R txyz as in Definition 2.11. A point on L is said to be a t -minimum if it is a localminimum for the coordinate function t . We denote the set of t -minima as T min . The superscript indicates the functor is contravariant. It is a consequence of the popular convention to align thearrows in the exact Lagrangian cobordism category with the direction of the Liouville vector field. t -minimum points are defined topologically. After a Morse type perturbation, we canassume these critical points are isolated, and hence finite. For the rest of the paper, we alwaysassume the exact Lagrangian cobordism L has finitely many t -minima. Definition 2.13.
Let (Λ ± , T ± ) be two decorated Legendrian links. A decorated exact Lagrangiancobordism ( L, P ) : (Λ − , T − ) → (Λ + , T + )is an exact Lagrangian cobordism L : Λ − → Λ + , together with a set of generic oriented markedcurves P = { p , . . . , p m } on L , such that(1) each p i is either a closed 1-cycle or an oriented curve that begins and ends at T + ∪ T − ∪ T min ;(2) intersections between marked curves are transverse and isolated;(3) each marked point in T + ⊔ T − is the restriction of a unique marked curve p i to Λ + ⊔ Λ − ;The genericity condition of P involves the configuration of holomorphic disks, which we will statelater. The collection of oriented curves P is called the decoration on L . If the decoration is clearfrom the context, we will omit the decoration and denote the decorated exact Lagrangian cobordismas L . Example 2.14.
Below is a candidate for the set of marked curves P for L . ∗∗∗∗ ∗∗∗ ∗ Recall that the algebra A (Λ + ) of the decorated Legendrian (Λ + , T + ) is a non-commutativealgebra over F generated by the set of Reeb chords R + , the set of formal variables T ± . Definition 2.15.
Given a decorated exact Lagrangian cobordism ( L, P ), we associate to the deco-ration P a collection of formal variables P ± = (cid:8) p ± i (cid:12)(cid:12) p i ∈ P (cid:9) . We define a fully non-commutativeunital F -algebra A (Λ + , P ), which is freely generated by R ∪ P ± , quotiented out by the followingrelations.(1) p i p − i = p − i p i = 1 for each oriented marked curve p i .(2) If two oriented curves p i and p j intersect, their associated formal variables commute.(3) Near each τ ∈ T min , let γ be a small oriented loop around τ , with orientation induced from L . It intersects a collection of oriented marked curves cyclically, say p , p , · · · , p l . Let h γ, p i i be the intersection number between ordered oriented curves. We impose the condition p h γ,p i p h γ,p i · · · p h γ,p l i l = 1 . xample 2.16. In the following example figure, suppose we label C i by p i . Then the imposedcondition is p − p − p = 1. τp p p γ Remark 2.17.
One can push a t -minimum below Λ − via a Hamiltonian flow and cut off a smallneighborhood with maximum-tb unknot boundary. The construction is unique up to Hamiltonianisotopy [EP96]. The marked curves connected to the minimum are now naturally connected to theunknot. In this way, all mark curves has boundary on Λ ± and the axioms can be simplified.However, the trade off is that we will have to require the cobordism L has no minima, andcompensate it by defining separately the dga for the Legendrian unknot with several marked pointsin an ad hoc fashion.Unfortunately, it seems impossible to use simplified marked curve axioms and allow all Lagrag-nian cobordism without restrictions at the same time — consider the unknot with a single markedpoint. It is our preference to scarifies the assumption of all marked curves having boundary on Λ ± for a more natural hypothesis on Lagrangians.Given a decorated exact Lagrangian cobordism ( L, P ) : (Λ − , T − ) → (Λ + , T + ), we define analgebra morphism φ ∗ + : A (Λ + ) → A (Λ + , P ) , (2.18)which sends each Reeb chord to itself and sends a marked point t to φ ∗ + ( t ) = (cid:26) p if t = source( p ) ,p − if t = sink( p ) , (2.19)Such a map φ ∗ + is well-defined because each marked point t is assumed to be connected to a uniquemarked curve p (Definition 2.13 (3)).We equip A (Λ + , P ) with a dg algebra structure. The degree of a Reeb chord is the same as itscounterpart in A (Λ + ) and the degree of the formal variables are 0. We define the differential overgenerators and then extend to the algebra via graded Leibniz rule: for each Reeb chord generator,define ∂ A (Λ + , P ) ( a ) := φ ∗ + ◦ ∂ A (Λ + ) ( a ) , and for a marked curve p , define ∂ A (Λ + , P ) ( p ) := 0. Note that this enhances the algebra homorphism φ ∗ + to a dga homorphism.The enhancement A (Λ − , P ) and the morphism φ ∗− : A (Λ − ) → A (Λ − , P ) are similarly defined.The only difference comparing with (2.19) appears in the convention of the exponent: φ ∗− ( t ) = (cid:26) p − if t = source( p ) ,p if t = sink( p ) . Let ( L, P ) : (Λ − , T − ) → (Λ + , T + ) be a decorated exact Lagrangian cobordism. Let J be a genericcompatible tame almost complex structure on the symplectization R × R , which is translationally14nvariant on the symplectization ends and intertwines the Liouville vector field and the Reeb vectorfield. For a ∈ R + and b , · · · , b n ∈ R − , we define the moduli space M ( a ; b , · · · , b n )to be the set modulo bi-holomorphic reparametrization of J -holomorphic curves having a positivepuncture asymptotic to the strip over the chord a at + ∞ , having a negative puncture asymptoticto the strip over the chord b i for each b i at −∞ , appearing in counterclock wise order along theboundary of the disk. For generic J , the moduli space is a transversely cut out manifold of dimension | a | − P ni =1 | b i | , [EHK16, Lemma 3.7].Given u ∈ M ( a ; b , · · · , b n ), the image of the disk boundary is the union of n +1 paths η , · · · , η n in the Lagrangian surface L . Let p ( η i ) be the word spelt out from alphabet p ± , · · · , p ± m accordingto the order in which η i crosses marked curves p , · · · , p m , where the exponents are the intersectionnumbers with respect to the orientation of the Lagrangian cobordism. Associated to u we have aword w ( u ) = p ( η ) b p ( η ) b · · · b n p ( η n ) . We define the dga homomorphism associated to the cobordism Φ ∗ L : A (Λ + , P ) → A (Λ − , P ).It is the identity on formal variables P ± and is given by the following formula on a Reeb chordgenerator: Φ ∗ L ( a ) = X dim M ( a ; b ··· ,b n )=0 u ∈M ( a ; b ··· ,b n ) w ( u ) . Note the finite set of holomorphic curves involved in the definition of Φ ∗ L depends only on thecobordism L but not the set of oriented marked curves P . We say P is generic if the intersectionof any marked curve p i and the boundary of any holomorphic disk ∂u is transverse. By followingthe proof in [EHK16], we arrive at the following theorem. Theorem 2.20.
Let ( L, P ) : (Λ − , T − ) → (Λ + , T + ) be a decorated exact Lagrangian cobordism. Theinduced map Φ ∗ L is a dga homomorphism Φ ∗ L : A (Λ + , P ) → A (Λ − , P ) . Definition 2.21.
Two decorations P and P ′ on an exact Lagrangian cobordism L : Λ − → Λ + are equivalent if P and P ′ are related by path homotopy and orientation reversing on some of themarked curves. Proposition 2.22.
Let L : Λ − → Λ + be an exact Lagrangian cobordism. Let P and P ′ be equivalentdecorations on L . Then there exist natural isomorphisms A (Λ ± , P ) ∼ = ←→ A (Λ ± , P ′ ) such that thefollowing diagram commutes. A (Λ + , P ) o o ∼ = / / Φ ∗ L (cid:15) (cid:15) A (Λ + , P ′ ) Φ ′∗ L (cid:15) (cid:15) A (Λ − , P ) o o ∼ = / / A (Λ − , P ′ ) Proof.
The only non-trivial part is orientation reversing. After path homotopies, we can assumethat P and P ′ have the same underlying paths, with potentially different orientations. If p ∈ P and p ′ ∈ P ′ have the same underlying path but with opposite orientations, then the natural isomorphismsends p to p ′− . 15ext we discuss the composition (concatenation) of two decorated exact Lagrangian cobordisms.Suppose ( L , P ) : (Λ , T ) → (Λ , T ) and ( L , P ) : (Λ , T ) → (Λ , T ) are two decorated exactLagrangian cobordisms. We would like to concatenate them into a decorated exact Lagrangiancobordism from (Λ , T ) to (Λ , T ). • Lagrangian. Note L := L ∪ Λ L is an exact Lagrangian cobordism from Λ to Λ becauseboth are cylindrical near Λ . • Decorations. The underlying unoriented marked curves in P and P can be concatenated.By reversing orientations of a subset in P and P if necessary, we can obtain orientatedmarked curves P on L . Note there is not a unique choice for orientation reversal, justifyingthe necessity of Definition 2.21.Following the construction above, any two induced decorations on L are equivalent, becausethe only possible difference is the orientation of marked curves in P . Definition 2.23.
We define the composition of exact Lagrangian cobordisms ( L , P ) ◦ ( L , P )to be the decorated exact Lagrangian cobordism ( L , P ), which is unique up to equivalence ofdecorations.Let ( L , P ) and ( L , P ) be a pair of decorated exact Lagrangian cobordisms defined asbefore. Let ( L , P ) := ( L , P ) ◦ ( L , P ). Without loss of generality, we assume that noorientation reversing takes place in the construction of P . There are natural maps between thesets of formal variables P ± , P ± , and P ± given by restrictions: P ± → P ± , P ± → P ± . These maps induce dga homomorphisms via coefficient enhancement A (Λ , P ) → A (Λ , P ) , A (Λ , P ) → A (Λ , P ) , A (Λ , P ) → A (Λ , P ) , A (Λ , P ) → A (Λ , P ) . These coefficient enhancements naturally lift Φ ∗ L and Φ ∗ L to dga homomorphisms among A (Λ , P ), A (Λ , P ), and A (Λ , P ). We abuse notation and write the lifts as Φ ∗ L and Φ ∗ L . A (Λ , P ) Φ ∗ L / / (cid:15) (cid:15) A (Λ , P ) (cid:15) (cid:15) A (Λ , P ) Φ ∗ L / / A (Λ , P ) A (Λ , P ) Φ ∗ L / / (cid:15) (cid:15) A (Λ , P ) (cid:15) (cid:15) A (Λ , P ) Φ ∗ L / / A (Λ , P )Next we observe that the coefficient enhancement A (Λ ) → A (Λ , P ) factors through A (Λ , P )and the coefficient enhancement A (Λ ) → A (Λ , P ) factors through A (Λ , P ); as for Λ , thecoefficient enhancement A (Λ ) → A (Λ , P ) factors through both A (Λ , P ) and A (Λ , P )in a commutative way, since any formal variable in T ± maps to the same image in P ± under16 ± → P ± → P ± and T ± → P ± → P ± . Therefore, the two commutative diagrams above canbe glued into the following bigger commutative diagram, proving that Φ ∗ L = Φ ∗ L ◦ Φ ∗ L . A (Λ ) (cid:15) (cid:15) A (Λ ) w w ♦♦♦♦♦♦♦♦♦♦♦ ' ' ❖❖❖❖❖❖❖❖❖❖❖ A (Λ ) (cid:15) (cid:15) A (Λ , P ) Φ ∗ L / / (cid:15) (cid:15) A (Λ , P ) ' ' ❖❖❖❖❖❖❖❖❖❖❖ A (Λ , P ) Φ ∗ L / / w w ♦♦♦♦♦♦♦♦♦♦♦ A (Λ , P ) (cid:15) (cid:15) A (Λ , P ) Φ ∗ L / / Φ ∗ L A (Λ , P ) Φ ∗ L / / A (Λ , P ) Proposition 2.24 (Associativity) . Suppose (Λ , T ) ( L , P ) / / (Λ , T ) ( L , P ) / / (Λ , T ) ( L , P ) / / (Λ , T ) is a sequence of decorated exact Lagrangian cobordisms. Then (( L , P ) ◦ ( L , P )) ◦ ( L , P ) = ( L , P ) ◦ (( L , P ) ◦ ( L , P )) up to equivalence of decorations; and Φ ∗ L ◦ (Φ ∗ L ◦ Φ ∗ L ) = (Φ ∗ L ◦ Φ ∗ L ) ◦ Φ ∗ L . Proof.
The proof will be left as an exercise for the readers.The functoriality respects Hamiltonian isotopy of decorated exact Lagrangian cobordisms. AHamiltonian isotopy of the decorated cobordism is one for the underlying Lagrangian submanifoldsuch that the associated decorations match accordingly. It uses the techniques from the Floertheory to see that a Hamiltonian isotopy induces a chain homotopy. The following statementfollows directly from [EHK16, Lemma 3.13].
Proposition 2.25.
Suppose
L, L ′ are Hamiltonian isotopic exact Lagrangian cobordisms from (Λ − , T − ) to (Λ + , T + ) , whose decorations are identified via the underlying isotopy. Denote bothdecorations by P . Let Φ ∗ L , Φ ∗ L ′ : A (Λ + , P ) → A (Λ − , P ) be the induced dga homomorphisms. Thenthe Hamiltonian isotopy induces a dga homotopy Φ ∗ L ∼ = Φ ∗ L ′ . When the Lagrangian cobordism admits some easy local description, the counting of holomorphicdisk can be explicitly enumerated via combinatorics. We will describe the morphism on augmenta-tion varieties induced from (1) a saddle corbordism of a simple contractible Reeb chords, and (2)a minimum cobordism.To facilitate the computation, we suppose the cobordism is a
Morse cobordism as an standingassumption. Heuristically, a Morse cobordism morsifies the cylindrical ends, in a way that theReeb chords of the one-jet lift of the cobordism are identified with the Reeb chords of Legendrian17inks on boundary, and that the total dg complex is the cone of the induced chain complex. In[EHK16, Lemma 1.4 and Section 2.4], it is explained how to convert back and forth between anexact Lagrangian Morse cobordism and an exact Lagrangian cobordism with cylindrical ends. (I) Saddle cobordism. A saddle cobordism (also known as a pinch move , or a pinching ) S :Λ − → Λ + has a local chart which is isomorphic to a certain saddle surface, and otherwise is thetrivial cobordism. Because the Lagrangian cobordism is exact, it can be lifted to the contactizationof the symplectization J ( R ). The one-jet lift of the saddle chart has the front projection surfaceas shown on the left below. b front projection b ∗ ∗ b Lagrangian projection ∗ ∗ (2.26)Different from a smooth cobordism, the exact Lagrangian saddle cobordism may not be inverted.The direction of the cobordism is controlled by the Liouville vector field. View the cobordismbackwards from Λ + to Λ − , it contracts a Reeb chord b , as depicted in the figure. Not every Reebchord can be contracted to create a saddle cobordism. A Reeb chord b ∈ R + is contractible if thereis a regular homotopy Λ s , s ∈ [0 ,
1] of the Legendrian immersions such that:(1) Λ = Λ; and(2) π L (Λ s ) has only transverse double points for all s ∈ [0 , has a transverse self-intersection which is obtained by sending E ( b s ) → s →
1, where b s is the Reeb chord corresponding to b .Suppose S is a saddle cobordism with respect to a contractible Reeb chord b . If the energy of b is sufficient small, there is a unique holomorphic disk u b with a positive puncture at b and nonegative punctures, which we term as the basic disk (associated to b ).There is a convenient choice for marked curves P . Away from the saddle point, the Lagrangianis cylinder like and we can choose R t families of marked points and orient them from Λ + to Λ − . Forconvenience, we label these oriented marked curves the same way as their corresponding markedpoints on Λ + .Near the saddle point, the closure of the unstable manifold is naturally a marked curve p with boundary points on Λ − . We orient this marked curve so that h ∂u b , p i = 1, where u b is thebasic disk. Note that both end points source( p ) and sink( p ) are on Λ − . In conclusion, we have T − = T + ∪ { source( p ) , sink( p ) } , and on the level of formal variables, P ± = T ± ∪ (cid:8) p ± (cid:9) . (2.27)Though a saddle cobordism is locally described, the holomorphic disks are global and canbehave wildly. With an additional assumption on the moduli space, the induced morphism can bepresented neatly [EHK16, Section 6.5]. We review the definition and the coefficient enhancementof the morphism here. 18 efinition 2.28. A contractible Reeb chord a is simple if ind( u ) ≥ k for all broken holomorhicdisks u : ( D , ∂D ) → ( R txyz , R t × Λ + ), with one positive puncture at c = a and k > a .Let S be the saddle cobordism of a simple contractible Reeb chord b . Its induced homomorphismmorphism Φ ∗ S acquires a concise presentation Φ ∗ S = (Φ ∗ S ) + (Φ ∗ S ) . Identifying R + ∼ = R − ∪ { b } , wedefine (Φ ∗ S ) : A (Λ + , P ) → A (Λ − , P )on generators by setting (Φ ∗ S ) ( c ) = (cid:26) c if c = b , p if c = b . (2.29) Definition 2.30.
For any Reeb chord c = b , define M ( c, b k ; d , · · · , d n )to be the moduli space of holomorphic disks mapped into ( R t × R xyz , R t × Λ + ), with one positivepuncture at c , k positive puncture at b , and one negative puncture at each d i Reeb chord.There is a translation action on M ( c, b k ; d , . . . , d n ) along R t . We define an algebra morphism(Φ ∗ S ) : A (Λ + , P ) → A (Λ − , P )on generators by setting (Φ ∗ S ) ( b ) = 0 and for any c = b ,(Φ ∗ S ) ( c ) = X dim M ( c,b ; d ··· ,d n )=1 u ∈M ( c,b ; d ··· ,d n ) / R w ( u ) | b = p − , (2.31)where w ( u ) is defined in the same way as in Equation (2.1). Proposition 2.32 ([EHK16]) . Let S be an simple contractible saddle cobordism, then Φ ∗ S = (Φ ∗ S ) + (Φ ∗ S ) . (2.33) Remark 2.34.
Note that in (Φ ∗ S ) , b is mapped to p , whereas in (Φ ∗ S ) , b is substituted by p − .The difference in the exponent can be understood by considering the holomorphic disk beforedegeneration. In the Legendrian surface description of the saddle cobordism, there is a cusp edgewhich is homotopic to the unstable manifold. The term in (Φ ∗ S ) comes from a negative degenerationof an end at the cusp edge, and the term in (Φ ∗ S ) comes from a positive degeneration of a switch at the cusp edge (these singular points were first introduced in [Ekh07], pictures are available in[EENS13, Figure 3]). Their marked curve contributions are reciprocal. (II) Minimum cobordism. The Legendrian unknot with tb = − the minumun cobordism . Topologically the filling is ahemisphere capping off the unknot. We discuss the induced dga morphism in the case when theLegendrian unknot is decorated with several marked point. Consider the following Lagrangianprojection of the maximal tb Legendrian unknot Λ O .19 ∗ t ∗ t ∗ t ∗ t ∗ t ∗ t k · · · ∗ t k +1 ∗ t k +2 ∗ t k +3 ∗ t k +4 ∗ t k +5 ∗ t n · · · Let M : ∅ → Λ O be the minimum cobordism with a unique t -minimum τ . We decorate M with aset of marked curves P , consisting of oriented curves p i : t i → τ for each 1 ≤ i ≤ n . The enhancedCE dga is then A (Λ O , P ) = Z h a, p ± , · · · , p ± n i / ( p i p − i = p − i p i = 1), with | a | = 1 and ∂a = p p · · · p k + p − n p − n − · · · p − k +1 . ∗∗ ∗∗ ∗∗ ∗· · · τ (2.35)By definition, A ( ∅ , P ) = Z (cid:10) p ± , . . . p ± n (cid:11)(cid:18) p i p − i = p − i p i = 1 ,p p . . . p n = 1 (cid:19) Since all formal variables p ± i have degree 0, the differential in A ( ∅ , P ) is zero. It then follows thatAug ( ∅ , P ) ∼ = Spec H ( A ( ∅ , P ) c ) ∼ = Spec A ( ∅ , P ) c ∼ = (cid:0) F × (cid:1) n − . Proposition 2.36.
Let M : ∅ → Λ O be a minimal cobordism from a tb = − unknot Λ O with n marked points. Then Φ ∗ M : A (Λ O , P ) → A ( ∅ , P ) induces an isomorphism between the th homology. Consequently, Aug(Λ O , P ) ∼ = Aug ( ∅ , P ) ∼ =( F × ) n − as algebraic varieties.Proof. It suffices to note that Φ ∗ M ( a ) = 0. Artin’s braid group on n strands is defined as Br n = h s ± , · · · , s ± n − | s i s i +1 s i = s i +1 s i s i +1 , and s j s k = s k s j if | j − k | ≥ i . positive braid semigroup Br + n is the sub-semigroup inside Br n generated by the s i ’s. Elementsof the braid group (resp. the positive braid semigroup) are called braids (resp. positive braids ).Geometrically, a braid corresponds to a diffeomorphism of a punctured disk. Let D n be an n -punctured disk. Let Diff + ( D n ) be the topological group of orientation preserving diffeomorphismsfrom D n to itself. The braid group Br n is isomorphic to the mapping class group MCG( D n ) := π (cid:0) Diff + ( D n ) (cid:1) . The symmetric group S n is the quotient of Br n by the relations s i = e for each i .The positive braid w = ( s n − · · · s s )( s n − · · · s ) · · · ( s n − s n − )( s n − ) is called the half twist ,and its square w is called the full twist . Under the quotient map from Br n to S n , the half twist w becomes the element of the longest Coxeter length in S n .Let β be a positive braid. By [EV18, Thoerem 3.4], the braid closure of β admits a uniqueLegendrian representative Λ β with maximal Thurston-Bennequin number. There are two ways toobtain this Legendrian link. • Combinatorially via the rainbow closure. We plot π F (Λ β ) by closing the braid using parallel,non-intersecting strands. The front diagram determines Λ β . β ... π F (Λ β ) β · · ·· · · ... π L (Λ β ) after Ng’s resolution (Theorem 2.2) (3.1) • Topologically via the satellite construction. Consider the positive braid word w βw and itscylindrical closure | w βw | is a Legendrian curves inside the contact one-jet space J ( S ).By the neighborhood theorem, the standard unknot inside ( R , ξ st ) has a small neighborhoodwhich is contactomophic to J ( S ). Composing the embeddings, we can place the cylindricalclosure | w βw | inside R , and it is Legendrian isotopic to Λ β . β w w ... w βw ... = β ...These two descriptions are equivalent. The difference come from the choice of a framing. Therainbow construction is given by the blackboard framing , while the satellite construction is given bythe ( − framing / contact framing (the framing determined by the Reeb vector field). The contactframing agrees with the blackboard framing except for a half twist at each cusp. Let i = ( i , . . . , i l ) be a word for a positive braid β ∈ Br + n . In this section, we define the CE-dga A (Λ β ) for Λ β using the rainbow closure construction. We plot the front projection π F (Λ β )21ccording to i and turn it into a Lagrangian projection π L (Λ β ) using Ng’s resolution (Theorem2.2). The braid β is oriented from left to right, and it induces an orientation for Λ β .Take the binary Maslov potential { , } . A Reeb chord of Λ β has degree 0 if it comes from acrossing, which we denote by b , and degree 1 if it comes from a right cusp, which we denote by a .We place a marked point t i next to each degree 1 Reeb chord a i . Example 3.2.
Below is a schematic picture of the Lagrangian projection π L (cid:0) Λ (1 , , , , , (cid:1) . ∗ ∗ ∗ b b b b b b a t a t a t The CE-dga A (Λ β ) is generated the Reeb chords, which are either in degree 0 or 1. To determinethe differentials, it suffices to compute ∂a i for degree 1 Reeb chords a i . Based on ideas from[K´al06, Siv11], we employ the following technique to compute ∂a i .For each 1 ≤ i < n , define an n × n matrix Z i ( b ) := b
11 0 . . . 1 , (3.3)where the 2 × i th and ( i + 1)st rows and columns. Proposition 3.4.
Let i = ( i , . . . , i l ) be a word for β ∈ Br + n . Set M (1) := Z i ( b ) · · · Z i l ( b l ) . Wedefine an ( n − k ) × ( n − k ) matrix M ( k +1) = (cid:16) M ( k +1) ij (cid:17) k
Following the bordered dga idea [Siv11], we consider the following diagram. β · · ·· · · ... n th ... n th ... M (1) M ( k )
22e label a dashed line between the braid region and right cusps. Any disk contributing to thedifferential can be divided into two parts. On the left part, Any disk boundary will travel alonga strand on the top, making many or no turns in the braid region, and then hit the bottom dashline. On the right part, the disk configurations near right cusps are listed in Remark 2.3, but onlyone out of five is allowed in our set up.We consider the left part. The ( i, j )-th entry of M (1) counts disks that are bounded by top level i and bottom level j near the dashed line. It can be computed inductively on crossings from leftto right. Before the braiding region, there is a unique pairing between strands, giving the identitymatrix. Next consider an arbitrary crossing i k . Let N (resp. N ′ ) be the disk counting matrixbefore (resp. after) scanning across i k . Then • N ′ i k +1 ,j = N i k ,j , due to (a); • N ′ i k ,j = N i k ,j b k + N i k +1 ,j , due to (b) and (c). j th( i k + 1)st i k th b k N N ′ (a) j th( i k + 1)st i k th b k N N ′ (b) j th( i k + 1)st i k th b k N N ′ (c)In other words, N ′ = N Z i k ( b k ). By induction, we have M (1) = Z i ( b ) Z i ( b ) . . . Z i l ( b l ).We can similarly place a dashed line between each pair of right cusps. Let M ( k ) the matrixassociated to the dashed line between a k − and a k . There is no disk between any two top strandsor between any two bottom strands near M (1) dashed line, and will be inductively true for anyother dash lines.Hence the only disk configurations from Remark 2.3 are the following: j th k th k th i th a k ∗ t k j th k th k th i th a k ∗ t k Therefore M ( k +1) ij = M ( k ) ij + M ( k ) ik t k M ( k ) kj . We are ready to compute ∂a k . It has two types of disks, one type consisting of those disks thathit the dashed line labeled by M ( k ) , the other type consists of only one disk which is the small loopwith no negative punctures. Hence ∂a k = M ( k ) kk + t − k . xample 3.6. Let β = s s s s ∈ Br +3 . Below is a Lagrangian projection of Λ β : b b b b a a a ∗ t ∗ t ∗ t Following Proposition 3.4, we have M (1) = b b + b b b b b b
11 0 0 , M (2) = (cid:18) b + b t + b t b b b t b t + t b b t b (cid:19) ,M (3) = t b + t t + t b b t + t t b t b + t b b t b t b . Therefore, the differentials are ∂a = b b + b + t − ,∂a = b + b t + b t b b + t − ,∂a = t b + t t + t b b t + t t b t b + t b b t b t b + t − . In [GR91], Gelfand and Retakh introduced the quasi-determinant as a replacement for thedeterminant for matrices with noncommutative entries. Let M , ,...,k − ,j , ,...,k − ,i denote the k × k submatrixof M = M (1) consisting of rows 1 , , . . . , k − , i and columns 1 , , . . . , k − , j . The following Lemmaestablishes a connection between M ( k ) ij and quasi-determinant. Proposition 3.7. If ∂a k = 0 for ≤ k ≤ n , then M ( k ) ij is the quasi-determinant (cid:12)(cid:12)(cid:12) M , ,...,k − ,j , ,...,k − ,i (cid:12)(cid:12)(cid:12) ij .Proof. The assumption 0 = ∂a k = M ( k ) kk + t − k implies that t k = − (cid:16) M ( k ) kk (cid:17) − . Then (3.5) becomes M ( k +1) ij = M ( k ) ij − M ( k ) ik (cid:16) M ( k ) kk (cid:17) − M ( k ) kj . By induction the RHS is equal to (cid:12)(cid:12)(cid:12) M ,...,k − ,j ,...,k − ,i (cid:12)(cid:12)(cid:12) ij − (cid:12)(cid:12)(cid:12) M ,...,k − ,k ,...,k − ,i (cid:12)(cid:12)(cid:12) ik (cid:12)(cid:12)(cid:12) M ,...,k − ,k ,...,k − ,k (cid:12)(cid:12)(cid:12) − kk (cid:12)(cid:12)(cid:12) M ,...,k − ,j ,...,k − ,k (cid:12)(cid:12)(cid:12) kj , whichyields (cid:12)(cid:12)(cid:12) M ,...,k,j ,...,k,i (cid:12)(cid:12)(cid:12) ij by the Sylvester’s identity for quasi-determinants [GR91, Proposition 1.5].Define M k := (cid:12)(cid:12)(cid:12) M , ,...,k , ,...,k (cid:12)(cid:12)(cid:12) kk = M ( k ) k . Then Lemma 3.7 automatically implies the follwoing. Corollary 3.8.
As non-commutative algebras over Z , H ( A (Λ β )) ∼ = Z (cid:10) b , . . . , b l , t ± , . . . , t ± n (cid:11)(cid:0) M k = t − k (cid:1) . Now let us focus on the augmentation variety A ug (Λ β ).24 efinition 3.9. Let N be an n × n matrix with commutative entries. The m th principal minor of N , denoted by ∆ m ( N ), is the determinant of the m × m submatrix of N formed by the first m rows and columns. Proposition 3.10.
Let β , i , and M (1) be as in Proposition 3.4. Let M be the matrix M (1) afterpassing to the commutative dga A (Λ β ) c . Then(1) Aug(Λ β ) ∼ = Spec (cid:0) F (cid:2) b , · · · , b l , t ± , · · · , t ± n (cid:3) / I (cid:1) , where I is generated by the equations: ∆ m ( M ) = m Y k =1 t − k , ≤ m ≤ n, (3.11) (2) Aug (Λ β ) ⊂ Spec F [ b , · · · , b l ] is the non-vanishing locus of the polynomial Q nm =1 ∆ m ( M ) .Proof. Note that A (Λ β ) is concentrated in non-negative degrees. By Proposition 2.9, we haveAug (Λ β ) = Spec H ( A (Λ β ) c , F ), where H ( A (Λ β ) c , F ) = F (cid:2) b i , t ± j (cid:3). ( ∂ c ( a k ) = 0) . Therefore the defining equations of Aug (Λ β ) are ∂ c ( a k ) = 0 for k = 1 , · · · , n .In the commutative setting, the quasi-determinant reduces to ratio of determinants: | N | ij = ( − i + j det N det N ij , (3.12)where N ij means delete the i th row and j th column from the matrix N . When working on thesetting of characteristic 2, we may further ignore the signs. Using Lemma 3.7 and (3.12) inductively,we see that ∂ c ( a k ) = 0 for all k is equivalent to the equations (3.11), which concludes the proof ofthe first statement. Now let us prove the second statement. Since t ± j are invertible, using (3.11)recursively, we can show that each t k is can be expressed in terms of principal minors, and hencein terms of Reeb chord coordinates. After eliminating all formal variables t ± k , we end up with thedesired equation. Definition 3.13.
Let β and i be as in Proposition 3.4. Note that Aug (Λ β ) is a Zariski opensubset of the affine space F lb ,...,b l and the Reeb chords b i can be regarded as coordinate functionson Aug (Λ β ). We call ( b , . . . , b l ) the Reeb coordinates on Aug (Λ β ) with respect to i . Corollary 3.14.
The augmentation variety
Aug (Λ β ) is smooth.Proof. By Proposition 3.10 (2), it is the non-vanishing locus of a polynomial function.
Corollary 3.15.
Any augmentation ǫ on A (Λ β ) satisfies Q nk =1 ǫ ( t k ) = 1 .Proof. By Proposition 3.10, the augmentation variety Aug (Λ β ) satisfies Q nk =1 t − k = ∆ n ( M ) =det( M ). On the other hand, since each (commutative) Z i k ( b k ) has determinant 1, it follows thatdet( M ) = 1. Therefore we have Q nk =1 t k = 1 . .3 Pinching Let β ∈ Br + n and let i = ( i , . . . , i l ) be a word for β . A degree 0 Reeb chord b k in Λ β is alwayscontractible. Let S k : Λ i , ˆ k → Λ β be the associated saddle cobordism. Rotating the Lagrangianprojection in Figure (2.26) gives the following figure. (The rotation is a compactly supportedHamiltonian isotopy.) b k Λ β S k ←− ∗∗ Λ i , ˆ k In general, b k may not be simple and we cannot directly apply formula (2.33) Φ ∗ L = (Φ ∗ L ) +(Φ ∗ L ) . To transform it into a simple contractible chord, [EHK16] suggests to invoke the dippeddiagram, which comprises the following Morse type perturbation. D ↑ Λ − Λ + ↑ x y front projection ↑ x y Lagrangian projectionGraphing the perturbation produces a cylindrical exact Lagrangian cobordism without t -minima.There is a convenient decoration — marked curves are traces of marked points which lie away fromthe dipping. The dipping creates two Reeb chords, x and y . Their degrees satisfy | x | = | y | + 1.Denote the dipping cobordism by D : Λ − → Λ + . In [Che02], Chekanov gave a recursiveformula for a tame dga isomomorphism ψ : A (Λ − ) → S A (Λ + ) where S A (Λ + ) stands for the dgastabilization of A (Λ + ) by adding two more generators e i and e i − with ∂e i = e i − , | e i | = | x | = i ,and | e i − | = | y | = i − ψ under the assumption | x | = | y | + 1 = 0; the constructionis similar for other degree combinations of x and y .Observe that R − = R + ⊔ { x, y } and T + = T − . Order Reeb chords in R − by their energy c r ≤ · · · ≤ c ≤ y < x ≤ d ≤ · · · ≤ d l . Because disks carry positive energy, we have ∂ − x = y + w, (3.16)where w is a polynomial involving only c i . 26n the other hand, we note that A (Λ + ) e − S A (Λ + ) is a direct summand of S A (Λ + ). Definea vector space endomorphism H on S A (Λ + ) to be the composition: H : S A (Λ + ) proj −−→ A (Λ + ) e − S A (Λ + ) → S A (Λ + ) ,ue − v ue v. Define a sequence of graded algebra homomorphisms ψ ( i ) : A (Λ − ) → S A (Λ + ) recursively for i = 0 , , . . . , l . We first define ψ (0) on the generating set R − ∪ T ± − = R + ∪ T ± ∪ { x, y } as ψ (0) ( b ) = e if b = x,e − + w if b = y,b otherwise . Here w is uniquely defined in Equation (3.16). We then define ψ ( i ) on the generating set as ψ ( i ) ( b ) = (cid:26) d i + H ◦ ψ ( i − ( ∂ − d i ) if b = d i ,ψ ( i − ( b ) otherwise . Let ψ := ψ ( l ) . By following the argument in loc. cit. , one can show that ψ : A (Λ − ) → S A (Λ + ) isa tame dga isomorphism.Note that A (Λ + ) naturally sits inside S A (Λ + ) as a dg subalgebra and there is a natural quotientmap S A (Λ + ) ։ A (Λ + ) defined by quotient the ideal generated by e and e − . We define two dgahomomorphisms Φ ∗ D : A (Λ + ) ֒ → S A (Λ + ) ψ − → A (Λ − ) , Φ ∗ D − : A (Λ − ) ψ → S A (Λ + ) ։ A (Λ + ) . (3.17) Remark 3.18.
The dga homomorphisms Φ ∗ D and Φ ∗ D − defined from the Lagrangian projectioncoincide with those induced from the concordance D and D − in the symplectization [EHK16,Lemma 6.7, 6.8, Remark 6.9]. Such identification also holds for other Legendrian isotopy moves[EK08, Section 6].In a positive braid legendrian link, any degree 0 Reeb chord is contractible, but may not besimple contractible. The following double dipping move makes it simple contractible. b k D ←− b k x L y L x R y R We denote the new Reeb chords from the left dipping x L and y L , and denote the new Reeb chordsfrom the right dipping x R and y R . Note that | x L | = | y L | + 1 =0 , ∂x L = y L , ∂y L = 0 , | x R | = | y R | + 1 =0 , ∂x R = y R , ∂y R = 0 . emark 3.19. The dipping construction was first introduced by Fuchs [Fuc03], then named“splash”. Later the construction evolved into several guises such as [FR11, EHK16, Sab05]. Thedipping localizes immersed disks within a confined region, with the trade-off creating many auxiliaryReeb chords. The double dipping/undipping construction implemented here is a variation, whichmake the chord simple contractible in the current setting while keeping the number of auxiliaryReeb minimum. The fully dipped diagram induces the same morphism on augmentation varieties.Next we compute the morphism induced from contracting a Reeb chord. Let p k : s p → t p bethe marked curve constructed from the unstable manifold. It follows from (2.27) that the set offormal variables for the enhanced dga is P ± := T ± ⊔ (cid:8) p ± k (cid:9) To understand the induced morphism between augmentation varieties, it suffices to compute fordegree 0 chords.
Theorem 3.20.
Let ( β, i ) ∈ Br + n with word length l . For any ≤ k ≤ l , Denote S k the saddlecobordism contracting Reeb chord b k . Then the functorial dga homomorphism Φ ∗ S k : A (Λ − , P ) →A (Λ + , P ) maps Φ ∗ S k ( b s ) = b s + X ∂ − b s = uy R v Φ ∗ S k ( u ) p − k v if s < k,p k if s = k,b s + X ∂ − b s = uy L v up − k Φ ∗ S k ( v ) if s > k. (3.21)This theorem will be proved via an explicit computation of Φ ∗ S k , which is broken down into thefollowing three steps: I. Φ ∗ D = Φ ∗ D L ◦ Φ ∗ D R = Φ ∗ D R ◦ Φ ∗ D L from the double dipping cobordism D ; II. Φ ∗ S from the simple contractible saddle cobordism associated to b k ; III. Φ ∗ D − = Φ ∗ D − L ◦ Φ ∗ D − R = Φ ∗ D − R ◦ Φ ∗ D − L from the double undipping cobordism D − . I. Double Dipping.
We compute the dga homomorphism Φ ∗ D R in detail. Let Λ + be the undippedLagrangian projection and let Λ − be the dipped Lagrangian projection. We observe that ∂ − b s = 0for all s ≤ k and therefore ψ R ( b s ) = b s for all s ≤ k . On the other hand, ∂ − b s may be non-trivialfor s > k . Nevertheless, non-trivial terms in ∂ − b s are all of the form uy R v , which come from disksof the following form; note that u and v are words in T ± ∪ { b k +1 , . . . , b s − } . y R b s uv − +Next we observe that H ◦ ψ ( s − R ( uy R v ) = H (cid:16) ψ ( s − R ( u ) e − ψ ( s − R ( v ) (cid:17) = H ( ψ R ( u ) e − ψ R ( v )) = ue ψ R ( v ) . ψ R ( u ) = u + { terms in the direct summand A (Λ + ) e S A (Λ + ) } byinduction hypothesis. Therefore other than terms of the form ue − ψ R ( v ), the remaining terms in ψ R ( u ) e − ψ R ( v ) do not survive under H .The above observation allows us to rewrite ψ R ( b s ) as ψ R ( b s ) = b s + X ∂ − b s = P uy R v ue ψ R ( v ) . Lemma 3.22.
Let ψ − R : S A (Λ + ) → A (Λ − ) be the inverse dga isomorphism to ψ R . Then ψ − R ( b s ) = b s + X ∂ − b s = P uy R v ψ − R ( u ) x R v. Proof.
It is not hard to see that ψ − R ( e ) = x R . Then b s = ψ − R ◦ ψ R ( b s )= ψ − R (cid:16) b s + X ue ψ R ( v ) (cid:17) = ψ − R ( b s ) + X ψ − R ( u ) x R ψ − R ◦ ψ R ( v )= ψ − R ( b s ) + X ψ − R ( u ) x R v, from which we can conclude the claimed formula.Combining Equation (3.17) and Lemma 3.22, we conclude thatΦ ∗ D R ( b s ) = b s if s ≤ k,b s + X ∂ − b s = P uy R v Φ ∗ D R ( u ) x R v if s > k. By a similar computation, one can find thatΦ ∗ D L ( b s ) = b s + X ∂ − b s = P uy L v ux L Φ ∗ D L ( v ) if s < k,b s if s ≥ k. where uy L v corresponds to immersed disks of the following form; note that u and v are now wordsin T ± ∪ { b s +1 , . . . , b k − } . b s y L vu + − By combining the two formulas above, we see thatΦ ∗ D ( b s ) = b s + X ∂ − b s = P uy L v ux L Φ ∗ D ( v ) if s < k,b k if s = k,b s + X ∂ − b s = P uy R v Φ ∗ D ( u ) x R v if s > k. (3.23)29 I. Simple Saddle.
After the double dipping, the Reeb chord b k is now simple. The local pictureof the simple saddle cobordism is the following. b k x L y L x R y R S ←− x L y L x R y R ∗∗ p k p − k Using (2.33), Φ ∗ L = (Φ ∗ L ) + (Φ ∗ L ) , we find the dga homomorphism Φ ∗ S isΦ ∗ S ( b k ) = p k , Φ ∗ S ( x L ) = x L + p − k , Φ ∗ S ( x R ) = x R + p − k . (3.24) III. Double Undipping.
Recall from (3.17) that Φ ∗ D − R is equal to the composition of the Chekanovdga isomorphism ψ R followed by a quotient by the ideal generated by e and e − . Since ψ R ( x R ) = e and ψ R ( y R ) = e − , it follows that Φ ∗ D − R maps all Reeb chords and marked points in Λ i , ˆ k tothemselves and annihilates the Reeb chords x R , y R . A similar observation can be made on Φ ∗ D − L . x L y L x R y R ∗∗ p k p − k D − ←− ∗∗ p k p − k Φ ∗ D − ( x L ) = Φ ∗ D − ( y L ) = Φ ∗ D − ( x R ) = Φ ∗ D − ( y R ) = 0 . (3.25) Proof of Theorem 3.20.
Let us now combine three steps (3.23), (3.24), and (3.25). For s = k , it isstraightforward to see Φ ∗ S k ( b k ) = Φ ∗ D − ◦ Φ ∗ S ◦ Φ ∗ D ( b k ) = p k . For s > k , in step I, we have Φ ∗ D ( b s ) = b s + P Φ ∗ D ( u ) x R v ; in step II, each letter x R in Φ ∗ D ( u ) x R v is replaced by x R + p − k (there might be more than one occurrence of x R in Φ ∗ D ( u ) x R v ); in stepIII, Φ ∗ D − annihilates all occurrences of x R . Therefore,Φ ∗ S k ( b s ) = b s + X ∂ − b s = uy R v Φ ∗ S k ( u ) p − k v. The argument for s < k similar. The proof is then complete.The recursive nature of formula (3.21) suggests a program to compute Φ ∗ S k ( b s ), termed matrixscanning , which inductively applies to the Reeb chords b s with s > k . The idea is to scan the braid( i k +1 , i k +2 , . . . , i l ) from left to right, computing Φ ∗ S k ( b s ) as we scan through i s , while “keepingtrack” of all possible incomplete disks of the form on the left below. y R uv − p − k Φ ∗ S k ( u ) v − ∗ S k , we need to replace y R with p − k and replace u with Φ ∗ S k ( u ),which turn these disks into the disks on the right above; the disks on the right are the actual disksof which we are keeping track.Consider a sequence of strictly upper triangular matrices U ( s ) = (cid:0) U ( s ) ij (cid:1) , where U ( s ) ij records thenumber of partial disks between the i th and j th strands with i < j before passing through thecrossing i s . The initial matrix is zero, except the ( i k , i k + 1)-entry being p − k . U ( k +1) = · · · · · · · · · p − k · · · · · · · · · · · · · · · Inductively for s > k , when scanning through a crossing i s , we perform two actions. First recordΦ ∗ S k ( b s ) = b s + U ( s ) i s ,i s +1 . (3.26)Second, we define U ( s +1) in terms of U ( s ) . These two matrices differ only at entries whose rows orcolumns are equal to i s or ( i s + 1): U ( s +1) i,i s = U ( s ) i,i s +1 + U ( s ) i,i s b s U ( s +1) i,i s +1 = U ( s ) i,i s U ( s +1) i s ,j = U ( s ) i s +1 ,j U ( s +1) i s +1 ,j = U ( s ) i s ,j + Φ ∗ S k ( b s ) U ( s ) i s +1 ,j .j th( i s + 1)st i s th i th b s To describe the matrix more compactly, we define the triangular truncations of a n × n matrix M + ij := (cid:26) M ij if i < j, M − ij := (cid:26) M ij if i > j, Z i s from (3.3). Then U ( s +1) is equivalently defined as U ( s +1) = (cid:16) Z i s (cid:0) Φ ∗ S k ( b s ) (cid:1) − · U ( s ) · Z i s ( b s ) (cid:17) + = (cid:18) Z i s (cid:16) b s + U ( s ) i s ,i s +1 (cid:17) − · U ( s ) · Z i s ( b s ) (cid:19) + . Similarly for Reeb chords b s with s < k , we scan the braid ( i , i , . . . , i k − ) from right to left,and compute Φ ∗ S k ( b s ) via a sequence of strictly lower triangular matrices L ( s ) with descending s = k − , k − , . . . ,
1. In particular, the entry L ( s ) ij records the number of such disks between the i th and j th strands with i > j before passing through the crossing i s .The initial matrix L ( k − is everywhere zero except for the ( i k + 1 , i k ) entry being p − k . Induc-tively for s < k , when scanning through a crossing i s , we recordΦ ∗ P k ( b s ) = b s + L ( s ) i s +1 ,i s , (3.28)31nd the matrix L ( s − can be obtained from L ( s ) by L ( s − := (cid:16) Z i s ( b s ) L ( s ) Z i s (cid:0) Φ ∗ S k ( b s ) (cid:1) − (cid:17) − = (cid:18) Z i s ( b s ) L ( s ) Z i s (cid:16) b s + L ( s ) i s +1 ,i s (cid:17) − (cid:19) − . Example 3.29.
Consider β ∈ Br +4 represented by i = (2 , , , , , , , S be the saddlecobordism contracting b . We compute Φ ∗ S ( b i ) using the matrices U ( s ) . Following the abovealgorithm, we get the following matrices. U (2) = p −
00 0 0 00 0 0 0 , U (3) = p −
00 0 b p −
00 0 0 00 0 0 0 , U (4) = p − b p − b p − b b p − ,U (5) = p − b p − b p − , U (6) = p − b b p b p − b p − b b p − ,U (7) = b p − p − b p − + b b p + p − b b b p − ,U (8) = b p −
00 0 p − + p − b b + b b p + p − b b b p − p − b Therefore, under Φ ∗ S , b p , b b , b b , b b + b p − b , b b ,b b + p − b b , b b + b b p − , b b + p − + p − b b + b b p − + p − b b b p − . Observe that in Φ ∗ S ( b ), there is a term with two factors of p − in it. Remark 3.30.
Formula (3.21) leads to an alternative description of Φ ∗ S k ( b s ):Φ ∗ S k ( b s ) = ∞ X m =0 (Φ ∗ S k ) m ( b s ) . (3.31)Here (Φ ∗ S k ) , (Φ ∗ S k ) remain the same. For m ≥
2, define (Φ ∗ S k ) m ( b k ) := 0 and(Φ ∗ S k ) m ( b s ) := X u ∈M ( b s ,b mk ; c ,...,c t ) / R m − w ( u ) | b k = p − k , where M ( b s , b mk ; c , . . . , c t ) is the moduli space of disks with positive quadrants at b s and b k , andremaining negative quadrants. In particular, a negative quadrant is allowed to be a ( − + − ) triple32uadrant. There is a total number of m − So far we are only able to prove the validity of (3.31) for degree 0 Reeb chords in positivebraid legendrian links. We notice from examples that this formula fails for degree 1 Reeb chords inpositive braid legendrian links. However, it remains unclear whether this formula holds for degree0 Reeb chords in other Legendrian links.Let Λ i , ˆ k be the decorated Legendrian link induced from the saddle cobordism. We writeAug (cid:0) Λ i , ˆ k (cid:1) for the augmentation variety of the dga A (cid:0) Λ i , ˆ k , P (cid:1) enhanced over the decorated sad-dle cobordism ( S k , P ).Let Λ β i , ˆ k be the decorated Legendrian of the positive braid β i , ˆ k = ( i , . . . , i k − , i k +1 , . . . , i l ). Incomparison, Λ i , ˆ k and Λ β i , ˆ k have the same underlying Legendrian link, but Λ i , ˆ k has two more markedpoints coming from the decorated saddle cobordism. Proposition 3.32.
The augmentation varieties of decorated Legendrian links Λ i , ˆ k and Λ β i , ˆ k satisfy Aug (cid:0) Λ i , ˆ k (cid:1) ∼ = Aug (cid:0) Λ β i , ˆ k (cid:1) × F × p k . Proof.
First we recall from Figure (2.10) that we can move the marked points p ± k on Λ i , ˆ k alonga strand, and such movement induces isomorphisms on both the CE dga and the augmentationvariety.By using these moves, we can move the pair of marked points p ± k all the way to the right intothe resolved right cusp region of Λ ˆ β, k . Since each resolved right cusp region has a marked pointon it already, we can record the value p k separately on the extra F × factor and then absorb themarked points p ± k into their adjacent marked points respectively. This defines the isomorphismAug (cid:0) Λ i , ˆ k (cid:1) ∼ = Aug (cid:0) Λ β i , ˆ k (cid:1) × F × p k .In fact, we can keep track of the effect of such movement of marked points on the augmentationvariety using matrices. Define a diagonal matrix D i ( p k ) = p k p − k . . . 1 . (3.33)where the entries p k and p − k are the i th and ( i + 1)st along the diagonal. Then by following thesame idea of Propositions 3.4 and 3.10, it is not hard to see that Aug (cid:0) Λ i , ˆ k (cid:1) is the open subvarietyof F l − b ,...,b k − ,b k +1 ,...,b l × F × p k defined by the common non-vanishing locus of the principal minors ofthe matrix product Z i ( b ) · · · Z i k − ( b k − ) D i k ( p k ) Z i k +1 ( b k +1 ) · · · Z i l ( b l ) . b ′ j be the Reeb coordinate defined by rescaling b j according to Figure (2.10) after movingthe pair of marked points p ± k across b j . Then we have Z i ( b ) · · · Z i k − ( b k − ) D i k ( p k ) Z i k +1 ( b k +1 ) · · · Z i l ( b l ) = Z i ( b ) · · · Z i k − ( b k − ) Z i k +1 (cid:0) b ′ k +1 (cid:1) · · · Z i l ( b ′ l ) D ′ , where D ′ is the diagonal matrix recording which levels the pair of marked points p ± k end up onthe far right.Recall the dga homomorphism φ ∗ + : A (Λ β ) → A (Λ β , P ) from the coefficient enhancement. Itinduces a map between augmentation varieties φ + : Aug ( A (Λ β , P )) → Aug (Λ β ) . Proposition 3.34.
The map φ + ◦ Φ S k : Aug (cid:0) Λ i , ˆ k (cid:1) → Aug (Λ β ) is an open embedding.Proof. It suffices to prove thatΦ ∗ S k ◦ φ ∗ + : O (Aug (Λ β )) → O (cid:16) Aug (cid:0) Λ i , ˆ k (cid:1)(cid:17) is a localization of at the Reeb coordinate function b k . First, Φ ∗ S k ◦ φ ∗ + is injective, because for anygiven values of p k and all the b i with i = k in the image, we can use Equations (3.26) and (3.28)to regenerate all the Reeb chord coordinate functions O (Aug (Λ β )). Next, because Φ ∗ S k ◦ φ ∗ + ( b k ) = p k is an invertible element in O (cid:16) Aug (cid:0) Λ i , ˆ k (cid:1)(cid:17) , we find that O (Aug (Λ β )) h b k i ⊂ O (cid:16) Aug (cid:0) Λ i , ˆ k (cid:1)(cid:17) .Meanwhile, Equations (3.26) and (3.28) yields that the generators of O (cid:16) Aug (cid:0) Λ i , ˆ k (cid:1)(cid:17) are contained in O (Aug (Λ β )) h b k i , and hence we have O (Aug (Λ β )) h b k i ⊃ O (cid:16) Aug (cid:0) Λ i , ˆ k (cid:1)(cid:17) . The mutual inclusionsyields the desired localization. Let β ∈ Br + n of length l . Recall from Definition 1.3 that admissible fillings of Λ β is an exactLagrangian filling L that is a composition of pinchings, braid moves, cyclic rotations, and minimumcobordisms.To define the decoration P on L , it suffices to describe the decoration on each possible con-stituent piece: we will use the decoration described in Section 2.4.2 for pinchings (saddle cobor-disms) and minimum cobordisms; on the other hand, since braid moves and cyclic rotations areLegendrian isotopies and hence topologically a cylinder, each marked point naturally traces out anoriented marked curve from top to bottom, and we use these oriented marked curves as decorationson braid moves and cyclic rotations. Below is a demonstration of the decoration on a cyclic rotationΛ s i β → Λ βs i . ∗∗ ∗∗∗∗ t ′ i +1 t i t ′ k t k t ′ i t i +1 Λ βs i Λ s i β · · · ∅ , P ). Lemma 3.35.
Let ( L, P ) be a decorated admissible filling. The complement of the oriented markedcurves P i ,σ is topologically a disjoint union of pieces of the following form, . . . where the lowest points of dashed curve are the t -minima on L .Proof. From the construction we know that the oriented marked curves in P do not intersecteach other. Since each constituent piece of L is by definition a composition of (1) pinchings, (2)braid moves, (3) cyclic rotations, and (4) minimum cobordisms, we can decompose the connectedcomponents of the complement of the oriented marked curves into a union of connected componentsfrom the complements of oriented marked curves on each constituent piece of L .Note that the gluing of two exact Lagrangian cobordisms always takes place over a disjointunion of circles, which are in bijection with connected components in a Legendrian link Λ. Sincein the original construction of Λ β we have put a marked point t i at each level, it is now guaranteedthat each connected component of Λ carries at least one marked point and hence the gluing ofcomplements of oriented marked curves always takes place over an open interval.The proposition then follows from the three claims below.(1) Within each pinch, each connected component of the complement of the oriented markedcurves is of the form or .(2) Within each braid move or cyclic rotation, each connected component of the complement ofthe oriented curves is of the form .(3) Within each minimum cobordism, each connected component of the complement of the ori-ented marked curves is of the form .Note that (2) and (3) obvious. For (1), note that S k is topologically a disjoint union of cylindersand exactly one pair of pants. On each cylinder the oriented marked curves all go from top to bottomand there is at least one such oriented marked curve t i ; therefore the connected components oforiented marked curves on each cylinder is a rectangle of the form .Now for the pair of pants, there are two possible configurations as below. Note that except forthe oriented marked curve p k all other oriented marked curves go from the top boundary to thebottom boundary, and each connected boundary component has at least one such oriented marked35urve. ··· · · · ··· · · · It is not hard to see that the connected components of oriented marked curves are all of the formexcept for two, which are of the form .
Proposition 3.36.
Let ( L, P ) be a decorated admissible filling of Λ β . There is a natural isomor-phism Aug ( ∅ , P ) ∼ = Triv ( L, P ) , the space of P -trivializations of rank-1 local systems on L .Proof. Note that by definition, each element of Aug ( ∅ , P ) is a map ǫ : P → F × . We think of such an assignment as assigning transition functions to the oriented curves P . To showthat such assignment is a trivialization of a rank 1 local system on L , it suffices to prove that themonodromy around the boundary of any generic embedded disk D ⊂ L is trivial (recall that forrank 1 local systems, we can replace the fundamental group π with the 1st homology group H ).Moreoever, we may cut the disk into small pieces and reduce the problem to the following threecases.(1) D is contained inside a connected component of the complement of the marked curves.(2) Only one marked curve goes across D .(3) D contains a t -minimum in the interior.The first two cases are trivial; case (3) only happens at the bottom of a minimum cobordism,and the trivial monodromy condition is equivalent to the coefficient enhancement condition (3) inDefinition 2.15.Now we have a natural map Aug ( ∅ , P ) → Triv ( L, P ), which is injective by definition. To seesurjectivity, it suffices to note that the only conditions we put on Aug ( ∅ , P ) are from t -minima in L and they have to be satisfied by any trivialization. This finishes the proof that Aug ( ∅ , P ) ∼ =Triv ( L, P ).Note that the boundary of L is precisely the Legendrian link Λ β , and any trivialization of arank 1 local system on L can be restricted to Λ β , which gives a trivialization of a rank 1 localsystem on Λ β as well. Note that Λ β defines a 1-cycle γ on L , and it is not hard to see that γ ishomologous to the sum of all oriented boundary components of L i ,σ , which is trivial in H ( L i ,σ ).Therefore the monodromy around γ is trivial for any trivialization ǫ of a rank 1 local system on L .On the other hand, we observe that among the oriented marked curves P i ,σ , only those definedby the initial marked points t , . . . , t n on Λ β intersect γ . Therefore the trivial monodromy conditionbecomes n Y i =1 ǫ ( t i ) = 1 , which gives a geometric proof of Corollary 3.15.36 orollary 3.37. Let ( L, P ) be a decorated admissible filling. Then Aug ( ∅ , P ) is an algebraic torusof rank l ( β ) .Proof. It suffices to see that the conditions from the n number of t -minima can be used to fixthe transition functions (values) of the oriented marked curves t i based on the transition functions(values) of the oriented marked curves p i . Therefore there are in total l ( β ) number of free F × -parameters.Let ( L, P ) be a decorated admissible filling. Recall from (2.18) that we have a morphism φ + : Aug (Λ β , P ) → Aug (Λ β ) induced by the CE dga homomorphism φ ∗ + . Consider the composition φ + ◦ Φ L : Aug ( ∅ , P ) → Aug (Λ β ) . Proposition 3.38.
Let ( L, P ) be an admissible filling for Λ β with β ∈ Br + n . Then φ + ◦ Φ L :Aug ( ∅ , P ) → Aug (Λ β ) is an open embedding.Proof. We again break L down into decorated exact Lagrangian cobordism pieces that are (1)pinchings, (2) braid moves, (3) cyclic rotations, and (4) minimum cobordisms. Note that since (2)and (3) are both Legendrian isotopies, their induced morphisms between augmentation varietiesare isomorphisms. We know from Proposition 2.36 that Φ M is an isomorphism for any minimumcobordism M as well. Therefore the only kind of non-trivial constituent pieces are pinchings, andhence we may assume without loss of generality that L is a composition of pinchings followed byfilling up a disjoint union of unknots by minimum cobordisms.Fix a word i = ( i , . . . , i l ) for β . Let σ ∈ S l be a permutation recording the order of pinchingsthat defines the admissible filling L . Let β ( k ) denote the Legendrian link obtained by pinching thecrossings i σ (1) , i σ (2) , . . . , i σ ( k ) and absorbing all new marked points p ± j into the marked points t i as described in Proposition 3.32. Now by Propositions 3.32 and 3.34 repetitively we get an openembeddingAug (cid:0) Λ β ( l ) (cid:1) × (cid:0) F × (cid:1) l f l ֒ → Aug (cid:0) Λ β ( l − (cid:1) × (cid:0) F × (cid:1) l − f l − ֒ → . . . f ֒ → Aug (cid:0) Λ β (1) (cid:1) × F × f ֒ → Aug (Λ β ) , Note that Aug (cid:0) Λ β ( l ) (cid:1) ∼ = Y n Aug (Λ O ) , where each Λ O is a maximal tb Legendrian unknot with a single marked point. Then by Proposition2.36 we know that Aug (cid:0) Λ β ( l ) (cid:1) is in fact a singleton space. Therefore we obtain an open embedding τ L : ( F × ) l → Aug(Λ β ) . Now it suffices to prove the commutativity of the following diagram.( F × ) l o o ∼ = / / r(cid:18) τ L $ $ ❏❏❏❏❏❏❏❏❏❏ Aug ( ∅ , P i ,σ ) φ + ◦ Φ L w w ♣♣♣♣♣♣♣♣♣♣♣ Aug (Λ β )To see this, note that the only difference between τ L and φ + ◦ Φ L is the movement of marked points.Let us analyze how such movements affect the coordinates between ( F × ) l and Aug ( ∅ , P ).37irst recall from the proof of Proposition 3.34 that for each map f i in the above factorization,the coordinate of the new F × factor appears in the domain is a Reeb coordinate of the codomain.Let us name the Reeb coordinate which we need to localize to get img ( f i ) as B i . Then it is nothard to see that B = p σ (1) .However, B may or may not be equal to p σ (2) . Note that in the process of moving the pair ofmarked points p ± to the right to get Aug (cid:16) Λ i , d σ (1) (cid:17) ∼ = Aug (cid:0) Λ β (1) (cid:1) × F × , as described in the proofof Proposition 3.32, certain Reeb coordinate will be rescaled by certain power of p σ (1) . Therefore B in general is equal to a product of p σ (2) with a power of p σ (1) .Proceed by induction, we see that B in general is a product of p σ (3) with powers of p σ (2) and p σ (1) , and B in general is a product of p σ (4) and powers of p σ (3) , p σ (2) , and p σ (1) , and so on. Fromthis analysis we see that the transformation of the Reeb coordinates ( B i ) on ( F × ) l and the orientedmarked curve coordinates ( p i ) on Aug ( ∅ , P ) can be recorded by an upper triangular matrix with 1down the diagonal on the character lattice. Therefore the map between ( F × ) l and Aug ( ∅ , P ) is analgebraic torus isomorphism. We constructed the augmentation variety for a positive braid in the last section. In this section,we prove that it admits a cluster K structure by relating it to the double Bott-Samelson cells.Double Bott-Samelson (BS) cells, introduced in [SW19], are moduli spaces of flags with pre-scribed relative positions encoded by positive braids. In this section we briefly recall their definitionand basic properties following loc. cit . Theorem 4.10 establishes natural isomorphisms between theaugmentation varieties of positive braid closures and the double BS cells associated with SL n . InSection 4.3, we study an open embedding relating different double BS cells, which corresponds toa single pinch in Section 5.1. Remark 4.1.
To topologists, it is helpful to keep in mind that non-isomorphic algebraic varietiesmay carry the same cluster type. There are four different versions of double BS cells constructedin [SW19], however, the augmentation variety is not isomorphic to any of them. The main theoremin this section yields the augmentation variety can be viewed as the fifth version of the double BScell.
Remark 4.2.
Throughout Section 4, we consider double BS cells in the most general setting, i.e.,as a Z -scheme defined for a Kac-Moody group G and a pair of generalized positive braids ( β, γ )associated to the group G . The only place where we need to restrict to G = SL n while simultaneouslybase-change to F is Theorem 4.10. However, after this section, we will keep G = SL n and the F base-change to simplify our argument, since this is the most relevant set-up for this paper. Let B ± be a pair of opposite Borel subgroups of a Kac-Moody group G and let U ± := [ B ± , B ± ] bethe maximal unipotent subgroups. There are flag varieties B + := G / B + and B − := B − \ G . Notethat B + and B − are canonically isomorphic when G is semisimple (particularly when G = SL n ). Byreplacing B ± with U ± we define decorated flag varieties A + := G / U + and A − := U − \ G . There arenatural projections π : A ± → B ± . If π ( A ) = B then we say that A is a decoration over B .38e denote elements in B + as B i and elements in B − as B i . The same convention is applied to A ± with the letter B replaced by A .Let T := B + ∩ B − and let W := N ( T )/ T be the Weyl group. Consider the Bruhat decomposi-tions and Birkhoff decomposition G = G w ∈ W B + w B + = G w ∈ W B − w B − = G w ∈ W B − w B + . We adopt the convention of writing x B + w / / y B + if x − y ∈ B + w B + , B − x w / / B − y if xy − ∈ B − w B − , B − x w y B + if xy ∈ B − w B + . We often omit w in the notation when it is the identity. Moreover, when decorated flags are involved,the notations only concern the underlying flags; for example, B i w / / A j means B i w / / π (cid:0) A j (cid:1) .Let i = ( i , . . . , i m ) be a word for a positive braid β . A chain B s i / / · · · s im / / B m will beabbreviated as B β / / ❴❴❴ B m . By Theorem 2.18 of [SW19], the chains of flags associated to differentwords of β have a natural one-to-one correspondence. In this sense, the chain B β / / ❴❴❴ B m doesnot depend on the word i chosen (see Definition 2.19 of loc. cit. ). Definition 4.3.
Let β and γ be two positive braids. The half decorated double BS cell Conf γβ ( C )is a moduli space parametrizes G -orbits of the collections B A m B B l βγ If one forgets to choose a decoration A m over B m , then the resulting space is denoted by Conf γβ ( B ).Denote by π the forgetful map from Conf γβ ( C ) to Conf γβ ( B ). Remark 4.4.
This version of double BS cells is slightly different from those in [SW19]: first, thetwo chains of flags swap places with the B + -chain at the bottom and the B − -chain at the top now;second, there is only one decoration A m over B m and the flag B is no longer decorated. The symbol C is chosen because one needs to induce decorations in a C-shaped fashion when defining the clusterK structure, as we will see in the next section.Given a triple B B s i / / B , there is a unique flag B − such that B s i B − s i / / B .It then follows from B − s i B s i / / B that B − B . This construction gives rise to thefollowing reflection maps. 39 efinition 4.5. The left reflection map i r : Conf γs i β ( C ) → Conf s i γβ ( C ) is an isomorphism that maps B ⑥⑥⑥⑥⑥⑥⑥ γ / / ❴❴❴ A m B s i / / B β / / ❴❴❴ B n B − s i / / ❈❈❈❈❈❈❈❈ B γ / / ❴❴❴ A m B β / / ❴❴❴ B n Its inverse map i r : Conf s i γβ ( C ) → Conf γs i β ( C ) is defined by an analogous process.Let ϕ i : SL → G be the group homomorphism associated to the simple root α i . Define e i ( q ) = ϕ i (cid:18) q (cid:19) , e − i ( q ) = ϕ i (cid:18) q (cid:19) , s i = ϕ i (cid:18) −
11 0 (cid:19) , s i = ϕ i (cid:18) − (cid:19) . We further set R i ( q ) = e i ( q ) s i = ϕ i (cid:18) q −
11 0 (cid:19) . (4.6) Lemma 4.7.
Fix a flag B j . The space of flags B k such that B j s i / / B k is isomorphic to A . Inparticular, if B j = B + , then B k = R i ( q ) B + for some unique q ∈ A .Proof. It suffices to prove the lemma for B j = B + . Let U i := (cid:8) e i ( t ) (cid:12)(cid:12) t ∈ A (cid:9) be the 1-dimensionalunipotent subgroup associated to the simple root α i and let Q i := B + ∩ s i B + s i . By Lemma 6.1.3of [Kum02], we know that B + = U i Q i . Therefore B + s i B + / B + = U i Q i s i B + / B + = U i s i Q i B + / B + = U i s i B + / B + . Hence B k = R i ( q ) B + for some unique q ∈ A .We prove important properties of the double BS cells, following [SW19, § Proposition 4.8.
The space
Conf γβ ( C ) is the non-vanishing locus of a polynomial in A l ( β )+ l ( γ ) .Consequently, it is a smooth affine variety.Proof. It suffices to prove the lemma for Conf eβ ( C ); the general case will follow by using the reflec-tions to shift the top γ to the bottom. Suppose β is of length l . Every point of Conf eβ ( C ) admits aunique representative as follows U − B + B B · · · B l s i s i s i s i l (4.9)Using Lemma 4.7 recursively, we obtain parameters ( q , . . . , q l ) ∈ A l such that B k = R i ( q ) · · · R i k ( q k ) B + , k = 1 , . . . , l. By definition, we require that the rightmost pair ( U − , B l ) is in general position.40et ω i be the i th fundamental weight. The i th principal minor ∆ i : G → A is a regular functionuniquely determined by the following two conditions: (1) ∆ i ( u − gu + ) = ∆ i ( g ), where u ± ∈ U ± ; (2)∆ i ( h ) = h ω i for h ∈ T . When G = SL n , the function ∆ i coincides with (3.9). Note that g ∈ B − B + if and only if ∆ i ( g ) = 0 for all i . Therefore the pair ( U − , B l ) is in general position if and only if f ( q , . . . , q l ) := Y ≤ i ≤ rk G ∆ i ( R i ( q ) · · · R i l ( q l )) = 0 . Note that Proposition 4.8 presents Conf γβ ( C ) as a scheme over Z . The advantange of definingit as a Z -scheme is that we can perform a base-change to any field k , which yields a k -schemeSpec (cid:18) O (cid:0) Conf eβ ( C ) (cid:1) ⊗ Z k (cid:19) . In particular, when G = SL n and k = F , the factor R i ( q ) is equal to the matrix Z i ( q ) introducedin (3.3). Furthermore, we prove the following theorem. Theorem 4.10.
Let G = SL n and let β ∈ Br + n . After a base-change of Conf eβ ( C ) to F , there is anatural isomorphism (as F -varieties) Aug (Λ β ) γ −→ Conf eβ ( C ) . Proof.
Fix a word i = ( i , i , . . . , i l ) for β . Let ( b , . . . , b l ) be the Reeb coordinates of Aug (Λ β )(Definition 3.13) and let ( q , . . . , q l ) be the affine coordinates of Conf eβ ( C ) as described in Proposition4.8. Define an isomorphism of ambient affine spaces γ : F lb ,...,b l → F lq ,...,q l by γ ∗ ( q k ) = b k . We claim that γ restricts to an isomorphism between Aug (Λ β ) and Conf eβ ( C ). To see this,note that after identifying q k = b k for all 1 ≤ k ≤ l , we have R i k ( q k ) = Z i k ( b k ) (over F ).It then follows that the non-vanishing locus of Q ≤ i ≤ n ∆ i ( R i ( q ) · · · R i l ( q l )) coincides with thenon-vanishing locus of Q ≤ i ≤ n ∆ i ( Z i ( b ) · · · Z i l ( q l )). Note that these two non-vanishing loci areprecisely Conf eβ ( C ) (Proposition 4.8) and Aug (Λ β ) (Proposition 3.10 respectively. Therefore γ doesrestrict to an isomorphism between the two F -varieties.Now we prove that γ c does not depend on the choice of i . Since any two words of the samebraid can be transformed into one another via a sequence of braid moves, it suffices to prove thelocal case when β = s s s = s s s . We claim that the following square commutes F b ,b ,b γ / / φ (cid:15) (cid:15) F q ,q ,q ψ (cid:15) (cid:15) F b ′ ,b ′ ,b ′ γ ′ / / F q ′ ,q ′ ,q ′ (4.11)where the isomorphism γ is obtained from the word (1 , ,
1) and γ ′ is obtained from the word(2 , , φ and ψ as follows: φ ∗ (cid:0) b ′ (cid:1) = b , φ ∗ (cid:0) b ′ (cid:1) = b + b b , φ ∗ (cid:0) b ′ (cid:1) = b ,ψ ∗ (cid:0) q ′ (cid:1) = q , ψ ∗ (cid:0) q ′ (cid:1) = q + q q , ψ ∗ (cid:0) q ′ (cid:1) = q . (4.12)These formulas then imply the commutativity of (4.11) directly.41 .2 Cluster Structures on Double Bott-Samelson Cells In this section we briefly recall the cluster K structure on Conf γβ ( C ) from [SW19].A pair of positive braids ( β, γ ) can be regarded as a single braid in the product Br × Br . Weshall prove that every word of ( β, γ ) gives rise to a cluster seed of Conf γβ ( C ). First, each worddetermines a labeling of arrows and a triangulation on the configuration diagram. Then we requirethat every pair of flags that are connected by a diagonal in the triangulation are in general position.The subspace of Conf γβ ( C ) that satisfy these general position conditions is an algebraic torus. Thealgebraic tori obtained from all words of ( β, γ ) form a subset of the atlas of cluster charts.In detail, let t be a word of ( β, γ ). We label the arrows and draw the triangulation on theconfiguration diagram according to t as shown in Example 4.16. On top of the triangulation, wedraw rank( G ) many parallel lines, each of which represents a simple root of G . Triangles in thetriangulation are either upward pointing or downward pointing (as shown below), and dependingon the orientation and the labeling of the base, each triangle places a node at one of the lines,cutting it into segments called strings . The segments from such cutting become the vertices ofthe quiver Q t , and the arrows in Q t are drawn according to the pictures below, where the dashedarrows between different levels i = j are weighted by weights that are related to Cartan numbers(see loc. cit. for more details). In particular, in the simply-laced cases (which include SL n ), thedashed arrows all have weight 1 /
2. In the end, we delete the left most vertices (together with allincident arrows) and freeze the right most vertices on each level. s i − ij th i th − ij th i th • •• • •• s i ij th i th ij th i th • •• • •• To define the cluster K coordinates, we first need to decorate the flags. Definition 4.13.
Two decorated flags x U + w / / y U + (resp. U − x w / / U − y ) are said to be compatible if x − y ∈ U + w U + (resp. xy − ∈ U − w U − ). Two decorated flags U − x y U + iscalled a pinning if xy ∈ U − U + . Lemma 4.14.
Given B w / / B ′ (resp. B ′ w / / B or B B ′ ), for every decoration A over B ,there exists a unique decoration A ′ over B ′ , such that A w / / A ′ are compatible (resp. A ′ w / / A are compatible or A A ′ is a pinning). Using the above lemma, we can begin with the decoration A m over B m and induce decorations42ne-by-one over the rest flags following the C -shape path illustrated by the dashed circles below B B B · · · B m − A m B B B · · · B l − B l s i s i s i s i l − s i l s j s j s j s j m − s j m The next proposition presents a standard representative for every point of Conf γβ ( C ). Proposition 4.15.
Fix words i and j for the positive braids β and γ respectively. Every point in Conf γβ ( C ) admits a unique representative in the following form: U − U − y · · · U − y m U + x U + · · · x l U + s j s j s j m s i s i s i l where x k = R i ( q ) R i ( q ) . . . R i k ( q k ) ,y k = R j k ( p k ) . . . R j ( p ) R j ( p ) . This gives an open embedding
Conf γβ ( C ) ֒ → A mp ,...,p m × A lq ,...,q l .Proof. Let us first verify that adjacent decorated flags along the top chain and the bottom chainare compatible. Let x = y = e . Note that U + x − k − x k U + = U + e i k ( q k ) s i k U + = U + s i k U + , U − y k − y − k U − = U − ( e j k ( p k ) s j k ) − U − = U − e − j k ( − p k ) s j k U − = U − s j k U − . Since a compatible decoration on one end of any adjacent pair of flags along either of thehorizontal chains can be uniquely determined by the decoration on the other end of the pair, theexistence of uniqueness of such representative automatically follows from the fact that G acts freelyand transitively on the space of pinnings.Now for a fixed word t of ( β, γ ), we get a quiver Q t with vertices corresponding to strings,which necessarily cross certain diagonals (possibly more than one) in the triangulation. i th ax U + U − y coordinate associated to the string a is defined to be the i th principal minor of xy : A a = ∆ i ( yx ) . The function A a is independent of the choice of diagonals if a crosses more than one diagonals. Example 4.16.
Let G = SL , β = s s s s , and γ = s s . For Br × Br , we use negative numbersfor letters in the second factor. The word t = (2 , − , , , − ,
1) for ( β, γ ) gives rise to the followingtriangulation, string diagram, and quiver A A A A A A A A s s s s s s A A A A A A A A s s s s s s − − − − • • (cid:3) • • (cid:3) Remark 4.17.
In [SW19] a cluster K structure is constructed on the decorated double BS cellConf γβ ( A sc ) for a simply-connected group G , which has frozen vertices on both sides of the quiver.The cluster K structure on Conf γβ ( C ) is essentially obtained from that of Conf γβ ( A sc ) by settingall the frozen variables on the left to be 1 due to the pinning condition on A A .The next Proposition provides an interpretation of left reflections in terms of standard repre-sentatives in Proposition 4.15. It implies that the left reflections are cluster transformations. Proposition 4.18.
Fix words i and j for the positive braids β and γ respectively. Then the leftreflection Conf γs i β ( C ) → Conf s i γβ ( C ) can be expressed in terms of standard representatives as U − U − y · · · U − y m U + R i ( q ) U + R i ( q ) x U + · · · R i ( q ) x l U + s j s j s j m s i s i s i s i l + x U + · · · x l U + U − U − R i ( q ) U − y R i ( q ) · · · U − y m R i ( q ) s i s i s i l s i s j s j s j m Proof.
The left reflection does the following. U − ✈✈✈✈✈✈✈✈✈✈ U + s i / / R i ( q ) U + U − s i s i / / s i U − U + s i / / R i ( q ) U + U − s i s i / / ❏❏❏❏❏❏❏❏❏ U − R i ( q ) U + To restore to the standard representative, we need to act on the resulting configuration by ( R i ( q )) − .Note that under the such action, x U + ( R i ( q )) − x U + and U − y U − yR i ( q ). It is not hard tosee that such action will give the standard configuration as claimed in the proposition. In this section we construct an open embedding φ i : Conf γβ ( C ) × G m ֒ → Conf γs i β ( C ) whose image isthe localization (freezing) at a cluster variable of the latter. In terms of fillings, φ i corresponds toa single pinch in Section 5.1. We prove that φ i is a quasi-cluster isomorphism onto its image.Recall from Lemma 4.7 that the moduli space of B − that fits into the triangle in the picture onthe left below is parametrized by the multiplicative group scheme G m . Note that the base changeof G m to any field k is isomorphic to k × as affine schemes over k . B − B − B + s i B − · · · A m B + · · · B l | {z } ij z }| { On the other hand, consider a standard representative and let us temporarily forget about thedecorations on the pinning and the bottom chain, as shown in the picture on the right above. Bygluing these two figures along the pinning B − B + , we end up with a point in Conf γs i β ( C ),which defines a morphism φ i : Conf γβ ( C ) × G m → Conf γs i β ( C ) , It is easy to see that φ i is an open embedding and does not depend on the choice of words i and j . Proposition 4.19.
The image of φ i in Conf γs i β ( C ) is the distinguished open subset corresponding o the localization (freezing) at the left most cluster variable A c in the picture below. B · · · γ A m B − B · · · β B l s i i th c (4.20) Proof.
There is a unique representative of (4.20) such that B = B − , B − = B + , and B = R i ( d ) B + .The principal minors of R i ( d ) are ∆ k ( R i ( d )) = (cid:26) d if i = k ;1 if i = k. Hence, the left cluster variable A c = d . By definition, (4.20) is in the image of φ i when B and B are in general position, or equivalently when d = 0. In other words, the image of φ i is precisely thenon-vanishing locus of the cluster variable A c . In cluster theory, localization of a cluster K varietyat a cluster variable A c is again a cluster K variety, which can be obtained by freezing the vertex c . Therefore the image of φ i is also a cluster K variety.Now we make Conf γβ ( C ) × G m into a cluster K variety by adding an extra frozen variable d corresponding to the G m factor. There should not be no arrows connecting c and the unfrozenvariables of Conf γβ ( C ) because the extra G m factor will not affect their mutations. However, thereis freedom of adding arrows connecting c and the frozen variables of Conf γβ ( C ). The next propo-sition shows that these arrows can be uniquely determined by requiring φ i to be a quasi-clusterisomorphism onto its image. Proposition 4.21.
The space
Conf γβ ( C ) × G m can be equipped with a unique cluster K structurewhich extends the cluster K structure on Conf γβ ( C ) by adding one extra frozen vertex c and possiblyarrows between c and the original frozen part, such that φ i becomes a quasi-cluster isomorphismonto its image.Proof. Suppose we start with a standard representative in the image of φ i as follows U − U − y · · · U − y m U + x U + x U + · · · x l U + s j s j s j m s i s i s i l s i i th c From the last proposition we know that x U + = R i ( d ) U + for some non-zero d .To obtain the preimage of this representative under φ i , we need to delete the flag U + at the46ower left corner and re-scale the decorations along the bottom chain as follows U − U − y · · · U − y m x h U + x h U + · · · x l h l U + s j s j s j m s i s i s i l Here h k ∈ T are such that ( U − , x h U + ) is a pinning and ( x k − h k − U + , x k h k U + ) are compatible.Set λ ∨ = − α ∨ i . We recursively define co-characters λ ∨ k of T for 1 ≤ k ≤ l by the relation λ ∨ k := s i k (cid:0) λ ∨ k − (cid:1) . Note that x = R i ( d ). An easy calculation shows that x h ∈ U − U + if and only if h = d λ ∨ . Since( x k − U + , x k U + ) is a compatible pair, by definition we get x − k − x k ∈ U + s i k U + . Therefore, U + ( x k − h k − ) − · x k h k U + = U + s i k · s i k ( h − k − ) h k U + . The pair ( x k − h k − U + , x k h k U + ) is compatible if and only if h k = s i k ( h k − ). By induction we get h k = d λ ∨ k for 0 ≤ k ≤ l .Next let us investigate the pull-back of cluster K coordinates of Conf γs i β ( C ) under φ i . Fix aword t for ( β, γ ) and consider the word ( i, t ) for ( s i β, γ ). Let Q t be the quiver associated to t . Let Q i, t be the quiver associated to ( i, t ) with the leftmost vertex c frozen.Recall that φ ∗ i ( A c ) = ∆ i ( x ) = ∆ i ( R i ( d )) = d. We define d to be the cluster variable A ′ c for the new frozen vertex c .For any other string (vertex) a associated to ( i, t ) as the left picture below, there is a corre-sponding string (vertex) a associated to t as the right right below. h th level ax k U + U − y j h th level ax k d λ ∨ k U + U − y j Let δ a := − h λ ∨ k , ω h i ∈ Z ; then φ ∗ i ( A a ) = ∆ h ( y j x k ) = ∆ h ( y j x k h k ) d − h λ ∨ k ,ω h i = A ′ a A ′ δ a c . In addition we define δ c := −h λ ∨ , ω i i = 1. Let I denote the vertices of Q i, t and let ǫ ij be theexchange matrix encoded by Q i, t . The set I ′ = I − { c } consists of vertices of Q t and I uf consistsof unfrozen vertices of Q t . We claim that for any a ∈ I uf , we have X b ∈ I ǫ ab δ b = 0 . (4.22)47o see this, recall that there is projection map p : Conf γβ ( C ) × G m −→ Conf γβ ( C ) π −→ Conf γβ ( B )As in [SW19, § γβ ( B ) is equipped with cluster Poission variables { X ′ a } a ∈ I uf such that p ∗ (cid:0) X ′ a (cid:1) = Y b ∈ I ′ A ′ ǫ ab b . (4.23)Consider the composition p ′ := Conf γs i β ( C ) π −→ Conf γs i β ( B ) −→ Conf γβ ( B )Here the second map is rational, obtained by forgetting the flag B − . Note that B − only changesthe decorations on the other flags. Therefore we have p = p ′ ◦ φ i . Therefore for a ∈ I uf we have p ∗ (cid:0) X ′ a (cid:1) = φ ∗ i ◦ p ′∗ (cid:0) X ′ a (cid:1) = φ ∗ i Y b ∈ I A ǫ ab b ! = A ′ ǫ ac c Y b ∈ I ′ A ′ ǫ ab δ b c A ′ ǫ ab b = A ′ P b ∈ I ǫ ab δ b c · Y b ∈ I ′ A ′ ǫ ab b . Comparing it with (4.23), we arrive at the identity (4.22).Note that identity (4.22) satisfies the assumptions stated in Proposition A.22. Therefore weknow that there is a unique way to extend the quiver of Conf γβ ( C ) so that φ i becomes a quasi-clusterisomorphism onto its image. Example 4.24.
We continue from Example 4.16. Consider the map φ : Conf γβ ( C ) × G m → Conf γs β ( C ). Let d = A ′ c be the coordinate for the G m factor. Then in the preimage, A A A d − α ∨ . A d − α ∨ − α ∨ . A d − α ∨ . A d α ∨ . A d α ∨ + α ∨ . A s s s s s s Such change of decorations gives rise the the pull-backs φ ∗ ( A c ) = A ′ c and φ ∗ ( A a ) = A ′ a d δ a = A ′ a A ′ δ a c for a = c . We record the numbers δ a at vertex a of the quiver Q as follows.1 1 − − −
11 1 − δ a we conclude that the cluster structure we put on Conf γβ ( C ) × G m is givenby the following quiver, where the right most vertex is the extra frozen vertex c .1 − − • • (cid:3) • • (cid:3) (cid:3) Convention 4.25.
For the rest of the paper, all mentions of double BS cells are for G = SL n andare base changed to F (i.e., as a variety over F ).48 Admissible Fillings and Cluster Seeds
This section aims to determine the cluster seed associated with each admissible filling, including thecluster chart, the cluster variables, and the quiver. We will prove that the cluster chart coincideswith the image of the functorial embedding described in Proposition 3.38. Hence, the cluster seedsof the augmentation variety can be used to distinguish non-Hamiltonian isotopic admissible fillings.The map from admissible fillings to the cluster seeds can be localized to each building blockcobordism. That is, every building block produces a cluster morphism, and their composition givesrise to an open embedding of a cluster chart into the augmentation variety. We shall also derive anexplicit construction for mutation sequences for the cluster seeds associated with admissible fillings.
Fix a word i = ( i , . . . , i l ) for a positive braid β ∈ Br + n . For k = 1 , . . . , l , let β i , ˆ k be the positive braiddefined by the word ( i , . . . , i k − , i k +1 , . . . , i l ). Recall the following open embedding (Propositions3.32 and 3.34) τ i ,k : Aug (cid:16) Λ β i , ˆ k (cid:17) × F × ∼ = → Aug (cid:16) Λ i , ˆ k (cid:17) → Aug (Λ β ) . Meanwhile, there is an open embedding ψ i ,k : Conf eβ i , ˆ k ( C ) × F × l −→ Conf s ik − ...s i s ik +1 ...s il ( C ) × F × φ ik −→ Conf s ik − ...s i s ik s ik +1 ...s il ( C ) l ′ −→ Conf eβ ( C ) , (5.1)where φ i k is constructed in Section 4.3, and the isomorphisms l, l ′ are sequences of left reflections.The goal of this section is to prove the following result. Proposition 5.2.
Let i be a word for β ∈ Br + n . The following diagram commutes Aug (cid:16) Λ β i , ˆ k (cid:17) × F × p γ × id ∼ = / / (cid:127) _ τ i ,k (cid:15) Conf eβ i , ˆ k ( C ) × F × d (cid:127) _ ψ i ,k (cid:15) (cid:15) Aug (Λ β ) γ ∼ = / / Conf eβ ( C ) where γ is the natural isomorphism defined in Theorem 4.10. Note that ψ i ,k is factored into 3 maps in (5.1). It suffices to prove the commutativity ofAug (cid:16) Λ β i , ˆ k (cid:17) × F × p ∼ = l ◦ ( γ c × id) / / (cid:127) _ τ (cid:15) Conf s ik − ...s i s ik +1 ...s il ( C ) × F × d (cid:127) _ φ (cid:15) (cid:15) Aug (Λ β ) ∼ = l ′− ◦ γ / / Conf s ik − ...s i s ik ...s il ( C ) (5.3)As shown in the proof of Proposition 3.34, the map τ is factored into two maps as τ : Aug (cid:16) Λ β i , ˆ k (cid:17) × F × ∼ = → Aug (cid:16) Λ i , ˆ k (cid:17) ֒ → Aug (Λ β ) . The subscripts p and d of the F × factors are coordinate symbols; the field F is fixed throughout the paper to bean algebraically closed field of characteristic 2. p ± to the right and recording themin the factor F × p . The second map is induced by a decorated exact Lagrangian cobordism S k , whichcan be computed explicitly using a matrix-scanning program (see (3.28) and (3.26)).Let us show that φ admits a similar factorization. First consider the standard representative ofa point in the image of φ . U − U − R i k − ( q k − ) · · · U − R i ( q ) . . . R i k − ( q k − ) U + R i k ( q k ) U + · · · R i k ( q k ) . . . R i l ( q l ) U + s i k − s i k − s i s i k s i k +1 s i l (5.4)From the proof of Proposition 4.19 we have d := q k = 0. We shall use the following identity R i k ( d ) = e − i k (cid:0) d − (cid:1) d α ∨ ik e i k (cid:0) d − (cid:1) . Let us delete U + in the lower left of (5.4), and act by (cid:0) e − i k (cid:0) d − (cid:1)(cid:1) − globally on the rest of (5.4).It gives rise to the Conf s ik − ...s i s ik +1 ...s il ( C ) factor of the preimage of (5.4) as follows. U − U − R i k − ( q k − ) e − i k (cid:0) d − (cid:1) · · · U − R i ( q ) . . . R i k − ( q k − ) e − i k (cid:0) d − (cid:1) d α ∨ ik e i k (cid:0) d − (cid:1) B + · · · d α ∨ ik e i k (cid:0) d − (cid:1) R i k +1 ( q k +1 ) . . . R i l ( q l ) B + s i k − s i k − s i s i k +1 s i l The following procedure transforms the above configuration to a standard representative:(1) move the unipotent factor e − i k (cid:0) d − (cid:1) inside each decorated flag in the top row all the way tothe left so that it can be absorbed into U − ;(2) move the unipotent factor e i k (cid:0) d − (cid:1) inside each decorated flag in the bottom row all the wayto the right so that it can be absorbed into B + ;(3) move the torus factor d α ∨ ik all the way to the right so that it can be absorbed into B + ;(4) replace every B + in the bottom row with U + to obtain a standard representative for theconfiguration.Let us focus on (1) first. For q ∈ F and l ∈ U − , there is a unique parameter q ′ := q + l i +1 ,i suchthat R i ( q ) l (cid:0) R i ( q ′ ) (cid:1) − ∈ U − (5.5)Set l ( k − = e − i k ( d − ). Using (5.5) recursively, we obtain a sequence of matrices l ( s ) ∈ U − andparameters q ′ s for s < k such that q ′ s = q s + l ( s ) i s +1 ,i s , l ( s − := R i s ( q s ) l ( s ) (cid:0) R i s (cid:0) q ′ s (cid:1)(cid:1) − . By definition, we have U − R i s ( q s ) . . . R i k − ( q k − ) e − i k ( d − ) = U − R i s ( q ′ s ) . . . R i k − ( q ′ k − )50ow we turn to (2). Let u ( k +1) = e i k (cid:0) d − (cid:1) . Similarly, we construct a sequence of matrices u ( s ) ∈ U for s > k inductively by setting q ′ s := q s + u ( s ) i s ,i s +1 , u ( s +1) := (cid:0) R i s (cid:0) q ′ s (cid:1)(cid:1) − u ( s ) R i s ( q s ) . We have e i k ( d − ) R i k +1 ( q k +1 ) . . . R i s ( q s ) B + = R i k +1 ( q ′ k +1 ) . . . R i s ( q ′ s ) B + . We show that (1) and (2) correspond to the matrix-scanning program for the decorated exactLagrangian cobordism S k . Recall from Theorem 4.10 that we have an isomorphism γ : Aug (Λ β ) → Conf eβ ( C ) which identifies the Reeb coordinates on Aug (Λ β ) and the corresponding affine coordi-nates on Conf eβ ( C ). Lemma 5.6.
Define b ′ s := Φ ∗ S k ( b s ) for s = k as in (3.28) and (3.26) . Then for s = k , b ′ s = q ′ s Proof.
We only prove the case when 1 ≤ s < k . The proof for the case when s > k is similar.Let be the n × n identity matrix. Recall the truncation matrices M ± associated to an n × n matrix M . Recall Z i ( b ) in (3.3). By a calculation similar to (5.5), we get Z i ( b ) (cid:0) + M − (cid:1) Z i ( b − M i +1 ,i ) − = + (cid:16) Z i ( b ) M − Z i ( b − M i +1 ,i ) − (cid:17) − . (5.7)Recall the matrix L ( s ) used in (3.28). We claim that l ( s ) = + L ( s ) (5.8)The lemma follows directly from (5.8) since b ′ s = b s + L ( s ) i s +1 ,i s = b s + l ( s ) i s +1 ,i s = q s + l ( s ) i s +1 ,i s = q ′ s . Now we prove (5.8) by an induction on s . For the base case s = k −
1, we see that l ( k − = e − i k (cid:0) d − (cid:1) = e − i k (cid:0) p − (cid:1) = + L ( k − . Inductively, suppose l ( s ) = + L ( s ) . Note that R i s ( q s ) = Z i s ( b s ); therefore l ( s − = R i s ( q s ) l ( s ) (cid:0) R i s (cid:0) q ′ s (cid:1)(cid:1) − = Z i s ( b s ) (cid:16) + L ( s ) (cid:17) Z i s (cid:0) b ′ s (cid:1) − = + L ( s − , where the last step is due to (5.7) and the definition of L ( s ) .The step (3) moves the torus factor d α ∨ ik through the factors R i k +1 (cid:0) q ′ k +1 (cid:1) . . . R i l ( q ′ l ). Since d α ∨ ik = D i k ( d ) = D i k ( p ) and R i s (cid:0) q ′ s (cid:1) = Z i s (cid:0) b ′ s (cid:1) , it follows that moving d α ∨ ik through R i k +1 (cid:0) q ′ k +1 (cid:1) . . . R i l ( q ′ l ) is the same as moving D i k ( p ) through Z i k +1 (cid:0) b ′ k +1 (cid:1) . . . Z i l ( b ′ l ), which is precisely what one needs to do to move the pair of marked points p ± all the way to the right to get the isomorphism Aug (cid:16) Λ β i , ˆ k (cid:17) × F × p ∼ = Aug (cid:16) Λ i , ˆ k (cid:17) (Proposition3.32). This observation proves the following proposition.51 roposition 5.9. Suppose D i k ( p ) Z i k +1 (cid:0) b ′ k +1 (cid:1) . . . Z i l (cid:0) b ′ l (cid:1) = Z i k +1 (cid:0) b ′′ k +1 (cid:1) . . . Z i l (cid:0) b ′′ l (cid:1) D ′ as stated in the proof of Proposition 3.32, and suppose d α ∨ ik R i k +1 (cid:0) q ′ k +1 (cid:1) . . . R i l (cid:0) q ′ l (cid:1) = R i k +1 (cid:0) q ′′ k +1 (cid:1) . . . R i l (cid:0) q ′′ l (cid:1) D ′′ for some invertible diagonal matrix D ′′ and new parameters q ′′ s which are certain rescaling of q ′ s .Then for s > k , b ′′ s = q ′′ s . Proof of Theorem 5.2.
It suffices to prove the commutativity of the diagram (5.3). The pull-backmap φ ∗ has been separated into steps (1) - (4). According to the above computation, the preimageof the standard representative is U − U − R i k − (cid:0) q ′ k − (cid:1) · · · U − R i ( q ′ ) . . . R i k − (cid:0) q ′ k − (cid:1) U + R i k +1 (cid:0) q ′′ k +1 (cid:1) U + · · · R i k +1 (cid:0) q ′′ k +1 (cid:1) . . . R i l ( q ′′ l ) U + s i k − s i k − s i s i k +1 s i k +2 s i l × F × d = q k Note that (cid:0) q ′ , . . . , q ′ k − , q ′′ k +1 , . . . , q ′′ l , d (cid:1) are the affine coordinates of the ambient space F l − × F × ⊃ Conf s ik − ...s i s ik +1 ...s l ( C ) × F × d .Under the top map, the coordinate functions (cid:0) q ′ , . . . , q ′ k − , q ′′ k +1 , . . . , q ′′ l , d (cid:1) are pulled back to (cid:0) b ′ , . . . , b ′ k − , b ′′ k +1 , . . . , b ′′ l , p (cid:1) determined by the following equations: b ′ s = q ′ s if s < k , b ′′ s = q ′′ s if s > k , p = d Now by Lemmas 5.6 and 5.9, we see that (cid:0) b ′ , . . . , b ′ k − , b ′′ k +1 , . . . , b ′′ l , p (cid:1) coincide with the pull-backsthrough the bottom and left maps in Diagram (5.3). Theorem 5.2 is now proved. Let β ∈ Br + n be a positive braid. For 1 ≤ i < n we define a cyclic rotation to be the Legendrianisotopy ρ i : Λ s i β → Λ βs i . Such a Legendrian isotopy can be realized as a composition of LegendrianReidemeister II moves. For example, one can use the following two RII moves to move a crossing s i from the bottom strands to the top on the left. There is also an analogous pair of RII moves onthe right to move a crossing s i from top to bottom. i RII ←→ RII ←→ i The goal of this section is to prove the following result.52 roposition 5.10.
The following diagram commutes:
Aug (Λ s i β ) γ ∼ = / / Φ ρi ∼ = (cid:15) (cid:15) Conf es i β ( C ) r i ◦ i r ∼ = (cid:15) (cid:15) Aug (Λ βs i ) γ ∼ = / / Conf eβs i ( C ) where γ is the natural isomorphism defined in Theorem 4.10. We start by proving several Lemmas.
Lemma 5.11.
Let β be a positive braid and let i = ( i , . . . , i l ) be a word for β . For the positivebraid closure Λ s i β , we define M (1) := Z i ( b ) Z i ( b ) Z i ( b ) . . . Z i l ( b l +1 ) and recursively define M ( k +1) ij := M ( k ) ij + M ( k ) ik t k M ( k ) kj as in Proposition 3.4. Then under the dga isomorphism induced by the isotopy ρ i , the degree 0 Reebchords b ′ k of Λ βs i are mapped as follows: Φ ∗ ρ i (cid:0) b ′ k (cid:1) = b k +1 ∀ ≤ k ≤ l, and Φ ∗ ρ i (cid:0) b ′ l +1 (cid:1) = t i M ( i ) i,i +1 . Proof.
We will prove with 2D projections instead of the symplectization. They are equivalent forReidemeister moves following Remark 3.18.By [Che02, ENS02], a Legendrian isotopy induces a quasi-isomorphism of the CE dga’s. Weshall separate the cyclic rotation into several Reidemeister moves and compute the induced mapon degree 0 Reeb chords. We will use bordered dga [Siv11] to compute their images under Φ ∗ ρ i .The cyclic rotation can be carried out in three steps: (i) using two Reidemeister II to move thecrossing s i from the bottom to the top of the braid closure on the right side of the link, depictedin (5.12); (ii) using a planar isotopy to move the crossing s i from the right all the way to the leftof the braid; (iii) using two Reidemeister II to move the crossing s i from the top to the bottom ofthe braid closure on the left side of the link. We write Φ ∗ ρ i = Φ ∗ ( iii ) ◦ Φ ∗ ( ii ) ◦ Φ ∗ ( i ) . The planar isotopydoes not change the dga, that is, Φ ∗ ( ii ) = id. We focus on steps (i) and (iii).Let us denote the Legendrian link obtained from step (i) as Λ s i β . Within step (i), we can localizethe front diagram so that only the two strands intersecting at the crossing s i are relevant.We choose the vertical borders so that the local strip contains only the two right cusps and thecrossing that are relavant to the story.1324 b ′ a ′ i a ′ i +1 x jk ( A , ∂ ) RII ←− ˜ b ˜ c ˜ d ˜ a i +1 ˜ a i x jk ( A , ∂ ) RII ←− c a i a i +1 x jk ( A , ∂ ) (5.12)Let ( A s , ∂ s ) , s = 1 , , A s is generated by Reeb chords in the Legendrian tangle53ogether with formal variables x jk with 1 ≤ j < k ≤
4; the grading and differentials on the formalvariables are defined by | x jk | := µ ( j ) − µ ( k ) − ∂x jk = X j In terms of the standard representative, the right reflection map maps U − U − y · · · U − y m U − e i ( q ) s i y m U + x U + · · · x l U + s j s j s j m s i s i s i s i l − U − y · · · U − y m U + x U + · · · x l U + x l R i ( q ′ ) U + s j s j s j m s i s i s i l s i where ∆ i denotes the i th principal minor of a matrix and q ′ := ∆ i ( zs i )∆ i ( z ) with z := R i ( q ) y m x l (see (4.6) for the definition of the matrix R i ( q ) ).Proof. Note that U − R i ( q ) y m x l U + . The product z := R i ( q ) y m x l is Gaussian decompos-able, i.e., z ∈ U − TU + . Let z = [ z ] − [ z ] [ z ] + be its Gaussian decomposition. Now act on the standardrepresentative in Conf γs i β ( C ) by [ z ] − [ z ] − − R i ( q ) y m , which yields the picture on the left below. . . . s i / / U − [ z ] ①①①①①①①①① U + . . . s i / / U − [ z ] s i U + s i / / s i U + Now it is not hard to see that the new flag B l +1 should be s i B + , and the A l = U + induces thedecoration s i U + on it, as shown in the picture on the right above. Then we forget the flag U − [ z ] to get the image of r i in Conf γβs i ( C ). To restore the standard representative, we need to acton the whole configuration again by ( R i ( q ) y m ) − [ z ] − [ z ] ; this action restores all decorated flags A , . . . , A m , A , . . . , A l to their original position, and the new decorated flag A l +1 becomes( R i ( q ) y m ) − [ z ] − [ z ] s i U + = x l [ z ] − s i U + By following the Gaussian elimination process, one can see that[ z ] + = ∆ ( zs )∆ ( z ) ∗ · · · ∗ ∗ ∆ ( zs )∆ ( z ) · · · ∗ ∗ · · · ∗ ∗ ... ... ... . . . ... ...0 0 0 · · · ∆ n − ( zs n − )∆ n − ( z ) · · · . Note that s i [ z ] + e i (cid:16) ∆ i ( zs i )∆ i ( z ) (cid:17) s i is still an upper triangular unipotent matrix. Therefore x l [ z ] − s i U + = x l e i (cid:18) ∆ i ( zs i )∆ i ( z ) (cid:19) s i U + = x l R i (cid:18) ∆ i ( zs i )∆ i ( z ) (cid:19) U + = x l R i (cid:0) q ′ (cid:1) U + . Corollary 5.14. Fix a word i for a positive braid β . Let ( q , . . . , q l +1 ) be the affine coordinateson Conf es i β ( C ) corresponding to the word ( i, i ) and let (cid:0) q ′ , . . . , q ′ l +1 (cid:1) be the affine coordinates on Conf eβs i ( C ) corresponding to the word ( i , i ) . Let z := R i ( q ) R i ( q ) . . . R i l ( q l +1 ) . Then the compo-sition r i ◦ i r : Conf es i β ( C ) → Conf eβs i ( C ) pulls back the affine coordinates as (cid:0) r i ◦ i r (cid:1) ∗ (cid:0) q ′ k (cid:1) = q k +1 for ≤ k ≤ l , (cid:0) r i ◦ i r (cid:1) ∗ (cid:0) q ′ l +1 (cid:1) = ∆ i ( zs i )∆ i ( z )56 roof of Proposition 5.10. Fix a word i = ( i , . . . , i l ) for β . Let us denote the coordinates onthe spaces in the bottom row with primed notation and denote the coordinates on the spacesin the top row with unprimed notation. Note that γ maps ( b , . . . , b l +1 ) ( q , . . . , q l +1 ) and (cid:0) b ′ , . . . , b ′ l +1 (cid:1) (cid:0) q ′ , . . . , q ′ l +1 (cid:1) . It then follows from the definition of γ in Theorem 4.10 that γ ∗ ( q k ) = b k and γ ∗ (cid:0) q ′ k (cid:1) = b ′ k . On the other hand, from the computation of Φ ρ i and r i ◦ i r so far we see that for 1 ≤ k ≤ l ,Φ ∗ ρ i ( b ′ k ) = b k +1 and (cid:0) r i ◦ i r (cid:1) ∗ ( q ′ k ) = q k +1 . Therefore we haveΦ ∗ ρ i ◦ γ ∗ (cid:0) q ′ k (cid:1) = Φ ∗ ρ i (cid:0) b ′ k (cid:1) = b k +1 = γ ∗ ( q k +1 ) = γ ∗ ◦ (cid:0) r i ◦ i r (cid:1) ∗ (cid:0) q ′ k (cid:1) . It remains to consider the pull-backs of the coordinate q ′ l +1 . By Corollary 5.14 we see that (cid:0) r i ◦ i r (cid:1) ∗ (cid:0) q ′ l +1 (cid:1) = ∆ i ( zs i )∆ i ( z ) for z = R i ( q ) R i ( q ) · · · R i l ( q l +1 )which is equal to M := Z i ( b ) Z i ( b ) · · · Z i l ( b l +1 )under the identification by γ . Therefore γ ∗ ◦ (cid:0) r i ◦ i r (cid:1) ∗ (cid:0) q ′ l +1 (cid:1) = ∆ i ( M s i )∆ i ( M ) = ∆ { ,...,i − ,i +1 }{ ,...,i } ( M )∆ i ( M ) , where ∆ JI denotes the determinant of the submatrix formed by the rows in set I and columns inset J . By Proposition 3.10 we further see that∆ i ( M ) = i Y k =1 t − k and ∆ { ,...,i − ,i +1 }{ ,...,i } ( M ) = M ( i ) i,i +1 i − Y k =1 t − k . Therefore by Lemma 5.11 we have γ ∗ ◦ (cid:0) r i ◦ i r (cid:1) ∗ (cid:0) q ′ l +1 (cid:1) = t i M ( i ) i,i +1 = Φ ∗ ρ i (cid:0) b ′ l +1 (cid:1) ∆ i ( c l +1 ) = Φ ∗ ρ i ◦ γ ∗ c ′ (cid:0) q ′ l +1 (cid:1) . Theorem 5.15. Let ( L, P ) be a decorated admissible filling of Λ β . Recall from Proposition 3.38that φ + ◦ Φ L is an open embedding of an algebraic torus into Aug (Λ β ) . The image of φ + ◦ Φ L is acluster chart on Aug (Λ β ) (with the cluster structure induced by the natural isomorphism γ definedin Theorem 4.10).Proof. Recall that admissible fillings are built from (1) pinchings, (2) braid moves, (3) cyclic rota-tions, and (4) minimum cobordisms. Below is a summary of how each of them can be interpretedin terms of cluster varieties.(1) Proposition 5.2 proves that any pinching defines a quasi-cluster isomorphism onto its image,which is defined by the localization (freezing) of a cluster variable.572) It is known that a braid move · · · s i s i +1 s i · · · ∼ · · · s i +1 s i s i +1 · · · gives rise to a single clustermutation on the double Bott-Samelson cell [SW19] and hence a braid move corresponds to asingle cluster mutation on the augmentation variety as well.(3) Proposition 5.10 proves that any cyclic rotation gives quasi-cluster isomorphism.(4) A minimum cobordism induces an isomorphism between algebraic tori, which are clustervarieties with totally frozen vertices.Combining the above results, we see that for any admissible filling L , φ + ◦ Φ L : Aug( ∅ , P ) → Aug (Λ β ) is a quasi-cluster isomorphism from a totally frozen cluster variety (i.e., an algebraictorus) onto its image. This implies that the image must be a cluster chart on Aug (Λ β ) as well. Corollary 5.16. If L and L ′ are two admissible fillings of Λ β that give rise to distinct clusterseeds, then L and L ′ are non-Hamiltonian isotopic.Proof. From the surjectivity of the map Aug (Λ β ) ∼ = Conf eβ ( C ) → Conf eβ ( B ) we know that thequiver associated to Aug (Λ β ) is full-ranked. Then by Theorem A.12 we can use cluster seeds todistinguish cluster charts. This implies the statement because if L and L ′ are Hamiltonian isotopic,then the image of φ + ◦ Φ L must coincide with φ + ◦ Φ L ′ .For the remaining of this section we discuss how to compute cluster seeds associated to admis-sible fillings. Initial Seed. Any braid word i = ( i , . . . , i l ) of β gives rise to an initial seed for Aug (Λ β ). Theidea is that the braid word i corresponds to a unique triangulation for the double Bott-Samelsoncell Conf eβ ( C ), namely subdividing a triangle into smaller triangles along the base according to i ;then one can pull seed back via the isomorphism γ : Aug (Λ β ) → Conf eβ ( C ) (Theorem 4.10) andget an initial seed on Aug (Λ β ). Note that since all these initial seeds are from the same clusterstructure on Conf eβ ( C ), the resulting cluster structure on Aug (Λ β ) is independent from the choiceof braid word i . •• • • · · · • s i s i s i s i l Recall from Section 4.2 that one can turn a triangulation into a quiver via a string diagram.We simplify this combinatorial procedure for positive braid links Λ β to the following. • Draw a projection of the braid β to R according to the word i with strands orientated from leftto right. Associate an unfrozen vertex to each bounded component of the diagram. Associatea frozen vertex to each open region between strands at the either end of the diagram. (Thereare no vertices for the top/bottom unbounded regions.) • For each crossing, draw the following arrow pattern (colored in blue) between neighboringvertices (dashed arrows are of 1 / • ••• Sum up the arrows between each pair of vertices. Delete the frozen vertices (along with theirincident arrows) that are associated to the open regions not bounded on the left.Note that each vertex of the obtained quiver lies to the right of a unique crossing. Hence we canenumerate the quiver vertices by 1 , , . . . , l as well. Moreover, by following the recipe in Section 4.2it is not hard to see that the initial cluster K coordinates are given by A k = ∆ i k ( R i ( ± b ) R i ( ± b ) . . . R i k ( ± b k )) , where ∆ i k denotes the i k th minor of an n × n matrix (3.9) and the matrices R i are defined in (4.6). Proposition 5.17. The n − number of frozen cluster variables are A m = m Y i =1 t − i , ≤ m < n. Proof. Fix a word i = ( i , . . . , i l ) for the positive braid β . According to Section 4.2, the frozenvariables on Conf eβ ( C ) are the principal minors of the matrix product R i ( q ) . . . R i l ( q l ). Since R i k ( q k ) = Z i k ( b k ), it follows that the frozen cluster variables are principal minors of the matrixproduct Z i ( b ) . . . Z i l ( b l ). The proposition then follows from the fact that the principal minors of Z i ( b ) · · · Z i l ( b l ) are Q mi =1 t − i (Proposition 3.10). Pinching. Fix a braid word i = ( i , . . . , i l ) for β and consider a saddle cobordism (pinching) S k .We learned that i determines an initial seed α on Conf es i ...s il ( C ). By Propositions 4.21 and 5.2 weknow that the pinching gives rise to a quasi-cluster isomorphism τ i ,k from Aug (cid:16) Λ β i , ˆ k (cid:17) × F × onto theimg τ i ,k . In particular, within this image, there is a cluster seed α ′ that is an extension of the initialseed on Aug (cid:16) Λ β i , ˆ k (cid:17) defined by the braid word ( i , . . . , i k − , i k +1 , . . . , i l ). Moreover, Proposition5.2 shows that this cluster seed α ′ can be obtained by a mutation sequence that corresponds to thesequences of left reflections l and l ′ in (5.1).Let us first describe how to realize the left reflection sequence l . The goal is to get to atriangulation of the following form. B B · · · A m B B B · · · B l s i k − s i k − s i s i k s i k +1 s i k +2 s i l i k th c (5.18)Such a triangulation can be achieved via the following recursive process for 1 ≤ j < k :(1) turn the left most triangle s i j upside down to s i j ;592) move this new upside-down triangle to the immediate right of the triangle s i k using diag-onal flipping;Note that only step (2) involves mutation, and the mutation for each 1 ≤ j < k is given by V j := µ ( ijtj ) ◦ · · · ◦ µ ( ij ) ◦ µ ( ij ) (5.19)where t j := { l | j < l ≤ k, i l = i j } .After the left reflection sequence l , we need to localize (freeze) the left most cluster variable A c associated to the string c in (5.18). This deletes the triangle s i k from the triangulation. B B · · · A m B B · · · B l s i k − s i k − s i s i k +1 s i k +2 s i l Then we apply the left reflection sequence l ′ . The induced move on the triangulation is the followingrecursive process for k > j ≥ s i j upside down to s i j ;(2) move this new triangle to the right through all triangles of the orientation .Again, only step (2) involves mutations, which can be described explicitly for each k > j ≥ W j := ( V − j if i j = i k ,µ ( ij ) ◦ µ ( ij ) ◦ . . . µ ( ijtj ) if i j = i k . Combining the left reflection sequences l and l ′ , we see that the total mutation sequence for thepinching S k is W ◦ W ◦ · · · ◦ W k − ◦ V k − ◦ V k − ◦ · · · V . (5.20) Remark 5.21. Note that between the mutation sequences W k − and V k − there is a step thatisolates the quiver vertex (cid:0) i k (cid:1) from the braid picture (Propositions 4.21 and 5.2). Therefore thequiver vertex will no longer be used in subsequent mutations for the admissible filling. Proposition 5.22. Let i = ( i , . . . , i l ) be a word of β . Let S k : β ′ → β be a saddle cobordism thatpinches the crossing i k . Let Q and Q ′ be the initial quiver associated with β and β ′ respectively.Then ( Q ′ ) uf is a full subquiver of a quiver that is mutation equivalent to Q uf . roof. Let Q ′′ be the quiver obtained from Q via the mutation sequence (5.20). Then ( Q ′ ) uf is thefull subquiver of ( Q ′′ ) uf complementary to the vertex (cid:0) i k (cid:1) by Propositions 4.21 and 5.2. Proposition 5.23. For any word i = ( i , . . . , i l ) of β , the initial seed on Aug (Λ β ) defined by i isassociated with the admissible filling obtained by pinching the crossings in i one-by-one from left toright.Proof. It follows from the fact that no left reflection is needed for the pinching sequence given bythe identity permutation. Corollary 5.24. All degree 0 Reeb chords in Λ β are cluster variables and H ( A (Λ β ) , F ) c is acluster algebra.Proof. For any degree 0 Reeb chord b k , there is an admissible filling L which pinches b k as its firstmove. Then by construction we know that Φ ∗ L ( b k ) = p k is precisely the cluster variable that definesimg( φ + ◦ Φ S k ) (Propositions 4.21 and 5.2). Therefore all b k are cluster variables.To show that H ( A (Λ β ) , F ) c is a cluster algebra, note that by Theorem 4.10 we know that H ( A (Λ β ) , F ) c = O (Aug (Λ β )) is already an upper cluster algebra. On the other hand, we knowthat H ( A (Λ β ) , F ) c is generated by the degree 0 Reeb chords b i and the formal variables t ± i ,and from the proof of Proposition 5.17 we know that all frozen variables are polynomials in thedegree 0 Reeb chords b i as well. Therefore we can conclude that H ( A (Λ β ) , F ) c is indeed a clusteralgebra. Braid Move. A braid move changes a braid word locally by a Legendrian Reidemeister III (R3),namely · · · s i s i +1 s i · · · ∼ · · · s i +1 s i s i +1 · · · . By a result of Ekholm and K´alm´an [EK08, Section 6],the dga homomorphism induced by R3 coincides with the R3 formula of Chekanov [Che02, § φ ∗ in(4.12). Therefore we can use the natural isomorphism γ from Theorem 4.10 to compute the effectof a braid move on the cluster seeds by computing its effect on Conf eβ ( C ), which is a single mutationat the quiver vertex bounded by the three crossings involved in the braid move [SW19]. i + 2 i + 1 i c µ c ←→ i + 2 i + 1 i c Cyclic Rotation. Recall from Proposition 5.10 that a cyclic rotation ρ i : Λ s i β → Λ βs i can berealized as a composition of reflections r i ◦ i r . Since the cluster seed is computed via pull-back, themutation sequence should correspond to (1) moving a triangle from bottom to top on the right, (2)pushing it forward through all the other triangles, and then (3) moving it back from top to bottomon the left. •• •• s i β (1) •• •• s i β (2) •• •• s i β (3) • • •• s i β Note that only step (2) involves cluster mutations, and the sequence of mutations is R := µ ( i ) ◦ · · · ◦ µ ( ini ) , n i is the number of unfrozen quiver vertices on the i th level. Minimum Cobordism. Like we discussed in the proof of Theorem 5.15, minimum cobordismsgive rise to isomorphisms between algebraic tori and hence no cluster mutation is involved. Proposition 5.25. If ( L, P ) : Λ β ′ → Λ β is a decorated admissible cobordism, then(1) φ + ◦ Φ L : Aug (cid:0) Λ β ′ , P (cid:1) → Aug (Λ β ) is an open embedding.(2) Q uf β ′ is a full subquiver of a quiver that is mutation equivalent to Q uf β .Proof. (1) It suffices to note that a saddle cobordism (pinch) gives rise to an open embedding(Propositions 4.21 and 5.2), whereas cyclic rotations and braid moves give rise to an isomorphism.(2) Proposition 5.22 shows that the unfrozen quiver after a pinch is a full subquiver. Thestatement then follows since cyclic rotations and braid moves can both be realized as quiver muta-tions. Let us revisit Example 1.6, which concerns the closure of β = s s s s s s s s ∈ Br +4 . Considerthe following step-by-step admissible filling L of Λ β :(1) pinch the 6th, the 7th, and the 1st crossings in that order, yielding the positive braid linkΛ s s s s s ;(2) apply a braid move to the middle three crossings, yielding Λ s s s s s ;(3) cyclically rotate the last crossing to the front, yielding Λ s s s s s ;(4) pinch the remaining crossings from right to left, yielding Λ e , a disjoint union of 4 unknots;(5) filling up each unknot with a minimum cobordism.Recall the initial seed from Example 1.6:1 23 45 67 8 (5.26) A = b , A = b , A = 1 + b b , A = b , A = b + b b + b b b ,A = b + b b , A = 1 + b b + b b b + b b + b b b b , A = b + b b b + b b . Let us now follow the recipes we have developed in the previous section to compute the mutationsequence we need in order to reach the cluster seed associated with the admissible filling L .We start by pinching the 6th crossing: note that the 6th crossing is 3, and the mutation sequenceis a composition of the following mutations V = µ ◦ µ , V = µ , V = µ , V = ∅ , V = ∅ ,W = µ ◦ µ , W = ∅ , W = µ , W = ∅ , W = ∅ . µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ = µ ◦ µ ◦ µ ◦ µ ◦ µ . Note that the quiver vertex 2 is “temporary frozen” due to the pinch (Propositions 4.21 and 5.2)and will not appear in later mutations any more. Therefore from now on, the quiver vertex 6 is theonly vertex on the 3rd level and it is frozen.Next we pinch the 7th crossing, which is 1. By going over a computation similar to the above,we see that the only non-trivial mutations are V = µ ◦ µ ◦ µ , V = µ ◦ µ , V = µ , W = µ ◦ µ ,and W = µ . The combined mutation sequence for this pinch is µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ . Note that the quiver vertex 1 is “temporary frozen” as well and will not be used any more. Fromnow on, the quiver vertices on the 1st level are (cid:0) (cid:1) = 3 , (cid:0) (cid:1) = 5, and (cid:0) (cid:1) = 7.Next we pinch the 1st crossing, which is 1. No mutation is needed for this pinching since no leftreflection sequence is needed. However, from now on the quiver vertex 3 will no longer be availablefor mutation, and the quiver vertices on the 1st level is updated as (cid:0) (cid:1) = 5 and (cid:0) (cid:1) = 7.Next we perform a braid move among the middle three crossings. Since the braid move goesfrom (1 , , 1) to (2 , , µ . Note that after this step, the quiver vertex 5 now drops down to the 2nd level. There are now threequiver vertices on the 2nd level, namely (cid:0) (cid:1) = 5, (cid:0) (cid:1) = 4, and (cid:0) (cid:1) = 8 (which is frozen).Next we need to do a cyclic rotation to move the last crossing 2 to the front of the braid. Thiscyclic rotation requires us to mutate all non-frozen vertices on the 2nd level from right to left.Therefore the mutation sequence for this step is µ ◦ µ . Let us leave it as an exercise for the readers to verify that the mutation sequence for pinchingΛ (2 , , , , from right to left needs the following mutation sequences (step-by-step):( ∅ ) ◦ ( ∅ ) ◦ ( µ ) ◦ ( µ ◦ µ ) ◦ ( µ ◦ µ ◦ µ ◦ µ ) = µ ◦ µ ◦ µ . Therefore the total mutation sequence for the admissible filling L is( µ ◦ µ ◦ µ ) ◦ ( µ ◦ µ ) ◦ ( µ ) ◦ ( µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ) ◦ ( µ ◦ µ ◦ µ ◦ µ ◦ µ )= µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ ◦ µ . If we apply this mutation sequence to the initial seed (5.26), we get1 23 45 67 863 ′ = b , A ′ = b , A ′ = b , A ′ = b , A ′ = 1 + b b , A ′ = b + b b ,A ′ = 1 + b b + b b + b b b + b b b b , A ′ = b + b b + b b b . (5.27)Let us verify that this is indeed the cluster seed corresponding to the admissible filling L byexpressing the cluster coordinates A ′ i as monomials in terms of the local coordinates p i on the imageof φ + ◦ Φ L . By using the matrix scanning technique, one can find that pinching the 6th, the 7th,and the 1st crossings give the following maps on the degree 0 Reeb chords: b p , b b + b p − , b b + p − , b b ,b b + p − , b p , b p , b b . Then the braid move changes b , b , and b by b b , b b + b b , b b . ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ ∗ ∗∗ ∗∗ ∗∗ ∗ ∗ ∗ ∗ b only, and according to Lemma 5.11, the change is given by(without shifting of indices): b t b p t p p + t b p − b p t p p . Note that b is the furthest Reeb chord on the left. ∗∗ ∗∗ ∗∗ t t ∗ ∗ ∗ ∗ b p , b p + p − p , b p , b p , b p + p p − p . Composing all the functorial dga homomorphisms together, we see that Φ ∗ L maps the degree 0Reeb chords as follows: b p , b p + p p p − , b p + p − , b p p ,b p + p − + p − p , b p , b p ,b t p p t p p + t p p − p p t p p + t p p t p p + t p p p − p p t p p + t p p − p − p p t p p . To get rid of the the factors t and t in the last expression, we use the t -minima conditions p p p t = 1 and p p − p p − p p − t = 1 , and get that Φ ∗ L ( b ) = p p − p + p p − p p − p − p + p p − p p − p p − p p − p − p . Substituting these assignments into the cluster variables A ′ i in (5.27) and making the formalvariables p i commute, we get A ′ = p , A ′ = p , A ′ = p , A ′ = p p , A ′ = p p ,A ′ = p p , A ′ = p p p , A ′ = p p p p p p p p . In the construction of Aug (Λ β ), we put one marked point near each resolved right cusp. Thisis more than the minimal requirement to have one marked point in each connected component[NR13]. In this section, we will construct a cluster structure on the reduced augmentation varietyand relate it to local systems on exact Lagrangian fillings when Λ β is a knot.Let η be the number of components in Λ β and let n , n , . . . , n η be the number of marked pointsin each component. Define R := { r ⊂ { , . . . , n } | | r | = η and { i, j } ∈ r = ⇒ t i and t j are in distinct link component } . Note that each element r ∈ R records a configuration with one marked point per component;therefore R is the set of all possible marked point reduction and hence | R | = n n · · · n η . For each r ∈ R , we define Aug r (Λ β ) to be the augmentation variety obtained by performingmarked point reduction on Λ β according to r . Geometrically Aug r (Λ β ) is the subvariety of Aug (Λ β )obtained by setting t i = 1 for all i / ∈ r .Let us investigate how setting marked points to be 1 affects cluster K structures. Recall fromProposition 5.17 that the frozen variable A k associated to the open region on the far right betweenthe k th and the ( k + 1)st levels is equal to Q ki =1 t − i . Setting a marked point t i = 1 causes one ofthe following on the frozen cluster variables. 65 If i = 1, then setting t = 1 makes A = 1; • If 1 < i < n , then setting t i = 1 makes A i = A i − ; • If i = n , then setting t n = 1 makes A n − = 1 (recall from Corollary 3.15 that t t . . . t n = 1).Note that (1) and (3) are equivalent to deleting the corresponding frozen vertex, whereas (2)is equivalent to merging the two corresponding frozen vertices into one, which is a quasi-clusterembedding by Proposition A.23. Therefore we conclude with the following theorem. Theorem 6.1. For each r ∈ R , let Aug r (Λ β ) ⊂ Aug (Λ β ) be the reduced augmentation varietyobtained by setting t i = 1 for all i / ∈ r . Then Aug r (Λ β ) inherits cluster K structures from Aug (Λ β ) . Example 6.2. Consider the (2 , l ) torus link Λ β , where β = s l ∈ Br +2 . The number of componentsdepend on the parity of l .When l is odd, Λ β is a knot. · · · b b b l ∗ ∗ t t The marked point reduction either sets t = 1 or t = 1; but either restriction leads to the other oneand hence both of the result in the same cluster K structure. Below is a more concrete examplewith l = 3: b b b t − When l is even, Λ β is a 2-component link. Since Λ β already starts with only one marked pointin each component, no reduction of marked points is needed and the cluster variety always carriesone frozen variable. Lemma 6.3. Suppose Λ β is a knot and suppose t i is the only marked point remaining after areduction r of marked points. Then ǫ ( t i ) = 1 for any augmentation ǫ ∈ Aug r (Λ β ) .Proof. Recall from Corollary 3.15 that on Aug (Λ β ) there is always Q nk =1 t k = 1.Let β ∈ Br + n and let ( L, P ) be a decorated admissible filling of Λ β . For a marked point reduction r ∈ R we define P r := P \ { t i | i / ∈ r } . Note that the complement of P r in L is still a disjoint union of topological disks. Proposition 6.4. Aug ( ∅ , P r ) ∼ = Triv ( L r , P r ) .Proof. It follows from the same argument as in the proofs of Propositions 3.35 and 3.36. Proposition 6.5. Suppose Λ β is a knot. Then Aug ( ∅ , P r ) ∼ = Loc F × ( L ) , the moduli space of F × -local systems on L as a topological surface. roof. By Lemma 6.3, the value of the marked curve connecting to the unique initial marked pointis fixed. Therefore the only remaining parameters of Aug ( ∅ , P r ) are p , . . . , p l , all of which arenon-zero.Notice that { p , . . . , p l } may not be algebraically independent due to the monodromy conditionsat t -minima. Let us now reduce it to a subset of independent free parameters. We observe thatthese l oriented marked curves and the n number of t -minima sitting at the bottom of the minimumcobordisms form a connected directed graph G embedded in L . Pick a spanning tree of G consistingof n − G ′ will have one vertex, and l − n + 1 orientedloops. Note that the parameters p i associated to the edges of this spanning tree can all be fixedusing the remaining l − n + 1 parameters due to the trivial monodromy conditions at the t -minima.321 b b b b b b β p p p p p p G ∗ p p p p G ′ We claim that the l − n + 1 loops in the embedded graph G ′ is a basis of H ( L, ∂L ). Note that sinceΛ β is assumed to be a knot, L is topologically a once punctured genus g surface. Therefore thenatural map H ( L ) → H ( L, ∂L ) is an isomorphism. On the other hand, χ ( L ) = − tb (Λ β ) = n − l by [Cha10], yielding 1 − g = n − l. Therefore H ( L ) ∼ = Z g = Z l − n +1 . Moreover, it is not hard to see that the l − n + 1 oriented loopsembedded in L form a basis of H ( L ). Therefore the l − n + 1 oriented loops in the embeddedgraph G ′ is a basis of H ( L, ∂L ).To finish the proof, it suffices to observe thatAug ( ∅ , P r ) ∼ = H (cid:0) G ′ (cid:1) ⊗ F × ∼ = H ( L, ∂L ) ⊗ F × ∼ = Hom (cid:0) H ( L ) , F × (cid:1) ∼ = Loc F × ( L ) . Note that in the third isomorphism we use the Poincar´e duality H ( L ) ∗ ∼ = H ( L, ∂L ). Proposition 6.6. The functorial morphism φ + ◦ Φ L : Aug ( ∅ , P r ) → Aug r (Λ β ) is an open embed-ding of an algebraic torus and the image is a cluster chart on Aug r (Λ β ) .Proof. It suffices to note that the cluster charts of Aug r (Λ β ) are cut out form the cluster charts ofAug (Λ β ) by setting t i = 1 for all i / ∈ r . A Appendix: Basics of Cluster theory We provide a brief review on cluster algebras and cluster varieties in the skewsymmetric setting.We will follow the notations and conventions used by Fock and Goncharov [FG09]. Definition A.1. A quiver is a triple Q = (cid:0) I, I uf , ǫ (cid:1) where(1) I is a finite set; 672) I uf ⊂ I ;(3) ǫ is an I × I skew-symmetric matrix with Q -entries such that ǫ ij ∈ Z unless i, j / ∈ I uf .Alternatively, one can think of Q as a directed graph with vertex set I and exchange matrix ǫ . Thesubset I uf is called the subset of unfrozen vertices . The unfrozen part of the quiver Q is defined tobe the full subquiver Q uf of Q generated by the unfrozen vertices.Properties of quivers are often defined via their unfrozen parts. Definition A.2. A quiver Q is said to be connected if the underlying graph of Q uf is connected.A quiver Q is said to be acyclic if there is no directed cycle inside Q uf . A quiver Q is said to have full-rank if the submatrix ǫ | I uf × I is of full-rank. Definition A.3. Let Q = (cid:0) I, I uf , ǫ (cid:1) be a quiver and let k ∈ I uf . The mutation in the direction k produces a new quiver µ k Q = (cid:0) I ′ , I ′ uf , ǫ ′ (cid:1) where I ′ = I , I ′ uf = I uf , and ǫ ′ ij = (cid:26) − ǫ ij if k ∈ { i, j } ,ǫ ij + [ ǫ ik ] + [ ǫ kj ] + − [ − ǫ ik ] + [ − ǫ kj ] + if k / ∈ { i, j } , where [ n ] + = max { , n } . Two quivers are mutation equivalent if they are related by a sequence ofmutations. Denote by | Q | the class of quivers that are mutation equivalent to Q . Lemma A.4. The properties of connectedness and full-rank in Definition A.2 are invariant undermutations. Therefore they are properties for mutation equivalence classes of quivers.Proof. We will leave the proof as an exercise to the readers. A.1 Cluster Ensembles Definition A.5. A cluster K variety A is an affine variety together with an atlas of charts called cluster charts (up to codimension 2), which satisfy the following properties:(1) There is a quiver Q = (cid:0) I, I uf , ǫ (cid:1) and a collection of coordinate functions A ; α := ( A i ; α ) i ∈ I associated to each cluster chart α such that α is a split algebraic torus with coordinatefunctions (cid:16) A ± i ; α (cid:17) i ∈ I . We tend to suppress the subscript ; α whenever it is clear from context.(2) For any k ∈ I uf of the quiver Q associated to a cluster chart α , there is another cluster chart α ′ whose quiver Q ′ = µ k Q and whose coordinate functions A ′ are related to A by A ′ i = A − k Q j A [ − ǫ kj ] + j (cid:16) Q j A ǫ kj j (cid:17) if i = k,A i if i = k. We say that α ′ is a mutation of α in the direction k and denote it as α ′ = µ k α .(3) Any two cluster charts in the atlas are related by a finite sequence of mutations.The collection A of coordinate functions for each cluster chart is called a cluster , and the coordinatefunctions A i are called cluster K coordinates or cluster variables . A pair ( A , Q ) consisting of acluster and its corresponding quiver is called a cluster seed . We denote cluster seeds by the samesymbol for their corresponding cluster charts. 68he cluster variables A i for i ∈ I \ I uf are said to be frozen and they are invariant undermutations. There is a canonical 2-form on A which can be expressed in each cluster chart asΩ = X i,j ǫ ij dA i A i ∧ dA j A j . Definition A.6. There is a dual version of cluster K variety called a cluster Poisson variety X whose cluster charts are split algebraic tori χ with coordinate functions (cid:0) X ± χ (cid:1) i ∈ I , and theytransform according to the formulas below under a mutation in direction k : X ′ i = ( X − k if i = k , X i X [ ǫ ik ] + k (1 + X k ) − ǫ ik if i = k .The coordinate functions X i are called cluster Poisson coordinates . There is a canonical Poissonstructure on X , whose bivector field can be expressed in each cluster chart asΠ = X i,j ǫ ij X i X j ∂∂X i ∧ ∂∂X j . Let A and X be a pair of cluster varieties associated to the same quiver Q . There is a naturalone-to-one correspondence between the cluster seeds of A and the cluster seeds of X . Abusingnotation, we refer to a pair of corresponding cluster seeds as a cluster seed for the pair ( A , X ) anddenote it by s . There is a canonical morphism p : A → X , which is expressed in the correspondingcluster charts as p ∗ ( X i ) = Y j A ǫ ij j . Remark A.7. Since ǫ ij may not be integers when i, j are both frozen, the map p : A → X is notnecessarily well-defined algebraically. However, the map p : A → X uf is always well-defined. (Here X uf is the unfrozen quotient of X defined in Section A.3.) Definition A.8. The triple ( A , X , p ) associated to a mutation equivalence class | Q | of quivers iscalled a cluster ensemble . The coordinate ring O ( A ) is called the upper cluster algebra associatedto | Q | . The subalgebra of O ( A ) generated by cluster variables and the inverses of the frozen clustervariables is called the (ordinary) cluster algebra associated to | Q | and we denote it as ord( A ).The cluster algebra ord( A ) was introduced by Fomin and Zelevinsky in [FZ02]. The uppercluster algebra O ( A ) was introduced by Berenstein, Fomin, and Zelevinsky in [BFZ05]. A.2 Separation Formula In practice, we often fix a seed and refer it as the initial seed of the cluster variety/algebra. Anyseed can be chosen as an initial seed.Let s = ( α , χ ) be an initial seed with initial cluster coordinates ( A i ;0 ) i ∈ I and ( X i ;0 ) i ∈ I .The cluster coordinates of every cluster seed s = ( α, χ ) can be written in terms of the initialcluster coordinates via a rational coordinate transformation. That is, there exists matrices c ij ; s and69 ij ; s and a collection of polynomials ( F i ; s ) i ∈ I in terms of the initial unfrozen cluster Poissoncoordinates such that A i ; s = Y j A g ij ; s j ;0 F i ; s | X k ;0 = Q l A ǫkl ;0 l ;0 , (A.9) X i ; s = Y j X c ij ; s j ;0 Y k F ǫ ik ; s k ; s ! (A.10)These two formulas are also known as the separation formulas of cluster coordinates and they areintroduced by Fomin and Zelevinsky [FZ07, Proposition 3.13, Corollary 6.3]. The matrices c ij ; s and g ij ; s are called the c - and the g -matrices of s respectively, and the polynomials F j ; s are calledthe F -polynomials of s .We define a partial order on monomials Q X a i i such that Q X a i i ≥ Q X b i i if a i ≥ b i for all i .We say that Q X a i i is the highest term of a polynomial f if it is bigger (i.e. ≥ ) than any othermonomial in f . Below is a summary of properties of F -polynomials. Theorem A.11 ([LS15, GHKK18]) . Every F -polynomial f satisfies the following properties:(1) all coefficients of f are positive integers;(2) the constant term of f is 1;(3) the highest term of f is of coefficient 1. Here (2) and (3) are equivalent due to [FZ07, Proposition 5.3]. Theorem A.12. Let Q be a quiver of full rank and let A Q be its associated cluster K varietydefined over any algebraically closed field (of any characteristic). The cluster charts of distinctcluster seeds of A Q do not coincide.Proof. Let α and α ′ be two cluster charts of distinct cluster seeds and let ( A i ) i ∈ I and ( A ′ i ) i ∈ I betheir cluster coordinates respectively. By (A.9) we have A ′ i = Y j A g ij j F i | X k = Q l A ǫkll . Since α and α ′ are associated with distinct cluster seeds, there is at least one wall between theircluster chambers in the scattering diagram associated to Q [GHKK18]. Recall that F -polynomialsare generating functions of broken lines in scattering diagrams. Hence, there is at least one F -polynomial F k that is non-trivial as a polynomial in Z [ X i ] i ∈ I uf . By Theorem A.11, both thehighest term and the constant term of F k have coefficients 1. Therefore F k has at least two termsafter passing to Z p [ X i ] i ∈ I uf for any prime number p .The substitution X k = Q l A ǫ kl l gives rise to a homomorphism p ∗ from Z p [ X i ] i ∈ I uf to the Laurentpolynomial ring Z p [ A j ] j ∈ I . The quiver Q has full rank. Therefore p ∗ is injective and F k | X l = p ∗ ( X l ) is non-constant. In other words, at least one coordinate A ′ k of α ′ is not a monomial in termsof the coordinates of α . This implies that α and α ′ cannot coincide as torus charts because anybiregular isomorphism between algebraic tori over an algebraically closed field must have monomialcoordinate transformation. 70 .3 (Quasi)-Cluster Morphisms Definition A.13. Suppose σ : I ′ → I is an injective map such that(1) σ | I ′ uf : I ′ uf → I uf is a bijection,(2) ǫ ′ ij = ǫ σ ( i ) σ ( j ) for all i, j ∈ I ′ .Then σ induces a morphism of algebraic tori σ : α ′ → α and σ : χ → χ ′ , which can be extended tomorphisms of cluster varieties σ : A ′ → A and σ : X → X ′ , called cluster morphisms . If the abovemap σ : I ′ → I is bijective, then the induced cluster morphisms are called cluster isomorphism . Example A.14. Consider the inclusion of the unfrozen part Q uf = (cid:0) I uf , I uf , ǫ | I uf × I uf (cid:1) into Q . Thisinclusion induces cluster morphisms A uf → A and X → X uf . Definition A.15. A cluster automorphism is a cluster isomorphism from a cluster variety back toitself. Cluster automorphisms form a group G called the cluster modular group .If we fix an initial seed, then the pull-back of an initial cluster coordinate via a cluster auto-morphism is again a cluster coordinate (of another cluster seed), and we can express such pull-backas a rational expression in terms of the initial cluster coordinates again. Such an expression takesthe same form as that given by the separation formula (A.9) or (A.10). Therefore it makes senseto speak about the c -matrix, g -matrix, and F -polynomials of a cluster automorphism with respectto a given initial seed.We do not need to distinguish the cluster modular groups for A and X because of the following. Proposition A.16. A cluster automorphism σ is the identity map on A if and only if it is theidentity map on X .Proof. The separation formula (A.9) (resp. (A.10)) implies that σ is the identity map on A (resp. X ) if and only if its g -matrix G (resp. c -matrix C ) is the identity matrix with respect to some(equivalently any) initial seed. The proposition then follows from the tropical duality theorem[NZ12, Theorem 1.2], which says that C − = G t . Remark A.17. Define N := Z I = L i ∈ I Z e i for a quiver Q = (cid:0) I, I uf , ǫ (cid:1) and let N uf be thesublattice spanned by { e i } i ∈ I uf . The exchange matrix ǫ equips N with a skew-symmetric form {· , ·} : N × N → Q such that { e i , e j } = ǫ ij . Let M be the dual lattice of N . One should think of N as the character lattice of cluster chart χ and think of M as the character lattice of the cluster chart α corresponding to χ . For n ∈ N and m ∈ M we denote the corresponding character functionsas X n and A m . In particular, X e i are precisely the cluster Poisson coordinates X i , and the map p : A → X is induced by the linear map p ∗ : N → M, n 7→ { n, ·} . We will use this set-up to definequasi-clutser morphisms. More detailed discussions can be found in [Fra16, GS19, SW19]. Definition A.18. Let N and N ′ be the lattices associated to Q and Q ′ as described in the aboveremark. Suppose σ : N ′ → N is an injective linear map such that(1) σ | N ′ uf is an isomoprhism onto N uf and for any i ∈ I ′ uf , σ ( e ′ i ) = e j for some j ∈ I uf ,(2) σ preserves the skew-symmetric forms. 71hen σ induces a morphism of algebraic tori σ : χ → χ ′ which extends to a morphism σ : X → X ′ .On the dual side, σ induces a linear map σ : M → M ′ , which induces a morphism of algebraictori σ : α ′ → α and extends to a morphism σ : A ′ → A . We call morphisms σ : X → X ′ and σ : A ′ → A quasi-cluster morphisms . A quasi-cluster isomorphism is a quasi-cluster morphismthat is a biregular isomorphism whose inverse is also a quasi-cluster morphism. A quasi-clusterautomorphism is a quasi-cluster isomorphism from a cluster variety back to itself. Quasi-clusterautomorphisms form a group QG called the quasi-cluster modular group .It is not hard to see that the cluster modular group G is a subgroup of the quasi-cluster modulargroup QG , and there is a natural map QG → G uf where G uf denotes the cluster modular group forthe unfrozen part. Remark A.19. (Quasi-)cluster automorphisms are also known as (quasi-)cluster transformations .It is not hard to see that the restriction of quasi-cluster morphisms to cluster charts commutewith cluster mutations. Consequently we have the following theorem. Theorem A.20. Let V and W be two cluster varieties of the same type (either K or Poisson).If σ : V → W is a quasi-cluster morphism, then there is a one-to-one correspondence between theircluster charts, and σ restricts to a morphism between algebraic tori on the corresponding clustercharts. Below we construct two particular quasi-cluster morphisms that are crucially used in this paper. Changing a Frozen Vertex. Recall the lattice N = L i ∈ I Z e i associated with a quiver Q = (cid:0) I, I uf , ǫ (cid:1) in Remark A.17. Let k be a frozen vertex of Q . Let ( δ j ) j ∈ I is an | I | -tuple of integers.We consider a lattice N ′ = L i ∈ I Z e ′ i and define a linear map σ : N ′ → N such that σ (cid:0) e ′ i (cid:1) := (cid:26) e i if i = k, P j ∈ I δ j e j if i = k. The exchange matrix ǫ equips N with a skew-symmetric form {· , ·} , whose pull-back through σ induces a skew-symmetric form {· , ·} ′ on N ′ . Let ǫ ′ be an I × I matrix such that ǫ ′ ij = (cid:8) e ′ i , e ′ j (cid:9) ′ := (cid:8) σ ( e ′ i ) , σ ( e ′ j ) (cid:9) . Let A ′ and X ′ be the cluster varieties associated with the quiver Q ′ = (cid:0) I, I uf , ǫ ′ (cid:1) . Note that σ satisfies the conditions (1) and (2) of Definition A.18. Therefore it define quasi-cluster morphisms σ : A ′ → A and σ : X → X ′ . Let α (resp. α ′ ) be the K cluster chart associated with the quiver Q (resp. Q ′ ). Let χ (resp. χ ′ )be the Poisson cluster chart associated with the quiver Q (resp. Q ′ ). Then the pull-back maps of σ can be written in terms of these cluster charts as σ ∗ ( A i ) = (cid:26) A ′ i A ′ δ i k if i = k,A ′ k if i = k. and σ ∗ (cid:0) X ′ i (cid:1) = ( X i if i = k, Q j X δ j j if i = k. (A.21) Proposition A.22. With the above set-up, the followings are true. 1) If δ k = 1 , then the two quasi-cluster morphisms σ are quasi-cluster isomorphisms.(2) If P j ǫ ij δ j = 0 for i ∈ I uf , then there is no arrow between the vertex k and the unfrozen partof Q ′ .Proof. (1) is obvious. For (2), it suffices to note that for i ∈ I uf , (cid:8) e ′ i , e ′ k (cid:9) ′ = (cid:8) σ (cid:0) e ′ i (cid:1) , σ (cid:0) e ′ k (cid:1)(cid:9) = (cid:26) e i , X j δ j e j (cid:27) = X j δ j ǫ ij = 0 , which implies that there is no arrow between the vertex k and the unfrozen part of Q ′ . Merging Frozen Vertices. Let t and t be two frozen vertices in a quiver Q = (cid:0) I, I uf , ǫ (cid:1) .Define the quiver Q ′ = (cid:0) I ′ , I ′ uf , ǫ ′ (cid:1) , where I ′ := ( I \ { t , t } ) ⊔ { t } , I ′ uf := I uf , and ǫ ′ ij := ǫ ij if i, j = t,ǫ t j + ǫ t j if i = t,ǫ it + ǫ it if j = t. We say that Q ′ is obtained from Q by merging two frozen vertices t and t into a single frozenvertex t . Let N and N ′ be the lattices associated with the quivers Q and Q ′ as in Remark A.17.There is an injective linear map σ : N ′ → Ne ′ i e i for i = t,e ′ t e t + e t . Note that σ satisfies the conditions in Definition A.18. Therefore it defines quasi-cluster morphisms σ : A ′ → A and σ : X → X ′ . The next theorem is direct consequence of the construction of σ . Proposition A.23. The quasi-cluster morphism σ : A ′ → A embeds A ′ as a subvariety of A determined by the locus { A i = A j } . References [Aur07] Denis Auroux. Mirror symmetry and T -duality in the complement of an anticanonicaldivisor. J. G¨okova Geom. Topol. GGT , 1:51–91, 2007. arXiv:0706.3207 .[BFZ05] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky. Cluster algebras. III.Upper bounds and double Bruhat cells. Duke Math. J. , 126(1):1–52, 2005. arXiv:math/0305434 , doi:10.1215/S0012-7094-04-12611-9 .[Cha10] Baptiste Chantraine. Lagrangian concordance of Legendrian knots. Algebr. Geom.Topol. , 10(1):63–85, 2010. arXiv:math/0611848 , doi:10.2140/agt.2010.10.63 .73Che02] Yuri Chekanov. Differential algebra of Legendrian links. Invent. Math. , 150(3):441–483,2002. arXiv:math/9709233 , doi:10.1007/s002220200212 .[CZ20] Roger Casals and Eric Zaslow. Legendrian weaves: N-graph calculus, flag moduli andapplications, 2020. arXiv:2007.04943 .[EENS13] Tobias Ekholm, John Etnyre, Lenhard Ng, and Michael Sullivan. Knotcontact homology. Geom. Topol. , 17(2):975–1112, 2013. arXiv:1109.1542 , doi:10.2140/gt.2013.17.975 .[EES05] Tobias Ekholm, John Etnyre, and Michael Sullivan. The contact homology of Leg-endrian submanifolds in R n +1 . J. Differential Geom. , 71(2):177–305, 2005. URL: http://projecteuclid.org/euclid.jdg/1143651770 , arXiv:math/0210124 .[EGH00] Yakov Eliashberg, Alexander Givental, and Helmut Hofer. Introduction to symplec-tic field theory. In Visions in Mathematics , Special Volume, Part II, pages 560–673. Birkh¨auser Basel, 2000. GAFA 2000 (Tel Aviv, 1999). arXiv:math/0010059 , doi:10.1007/978-3-0346-0425-3_4 .[EHK16] Tobias Ekholm, Ko Honda, and Tam´as K´alm´an. Legendrian knots and exactLagrangian cobordisms. J. Eur. Math. Soc. (JEMS) , 18(11):2627–2689, 2016. arXiv:1212.1519 , doi:10.4171/JEMS/650 .[EK08] Tobias Ekholm and Tam´as K´alm´an. Isotopies of Legendrian 1-knotsand Legendrian 2-tori. J. Symplectic Geom. , 6(4):407–460, 2008. URL: http://projecteuclid.org/euclid.jsg/1232029298 , arXiv:0710.4382 .[Ekh07] Tobias Ekholm. Morse flow trees and Legendrian contact homology in1-jet spaces. Geom. Topol. , 11:1083–1224, 2007. arXiv:math/0509386 , doi:10.2140/gt.2007.11.1083 .[EL17] Tobias Ekholm and Yanki Lekili. Duality between Lagrangian and Legendrian invari-ants, 2017. arXiv:1701.01284 .[EN18] John Etnyre and Lenhard Ng. Legendrian contact homology in R , 2018. arXiv:1811.10966 .[ENS02] John Etnyre, Lenhard Ng, and Joshua Sabloff. Invariants of Legendrian knotsand coherent orientations. J. Symplectic Geom. , 1(2):321–367, 2002. URL: http://projecteuclid.org/euclid.jsg/1092316653 , arXiv:math/0101145 .[EP96] Yakov. Eliashberg and Leonid Polterovich. Local Lagrangian 2-knots are trivial. Ann.of Math. (2) , 144(1):61–76, 1996. doi:10.2307/2118583 .[Etn05] John Etnyre. Legendrian and transversal knots. In Handbook of knot the-ory , pages 105–185. Elsevier B. V., Amsterdam, 2005. arXiv:math/0306256 , doi:10.1016/B978-044451452-3/50004-6 .[EV18] John Etnyre and Vera V´ertesi. Legendrian satellites. Int. Math. Res. Not. IMRN ,2018(23):7241–7304, 2018. arXiv:1608.05695 , doi:10.1093/imrn/rnx106 .74FG06] Vladimir Fock and Alexander Goncharov. Cluster X -varieties, amalgamation, andPoisson-Lie groups. In Algebraic geometry and number theory , volume 253 of Progr.Math. , pages 27–68. Birkh¨auser Boston, Boston, MA, 2006. arXiv:math/0508408 , doi:10.1007/978-0-8176-4532-8_2 .[FG09] Vladimir Fock and Alexander Goncharov. Cluster ensembles, quantizationand the dilogarithm. Ann. Sci. ´Ec. Norm. Sup´er. (4) , 42(6):865–930, 2009. arXiv:math/0311245 , doi:10.24033/asens.2112 .[FR11] Dmitry Fuchs and Dan Rutherford. Generating families and Legendrian contact homol-ogy in the standard contact space. J. Topol. , 4(1):190–226, 2011. arXiv:0807.4277 , doi:10.1112/jtopol/jtq033 .[Fra16] Chris Fraser. Quasi-homomorphisms of cluster algebras. Adv. in Appl. Math. , 81:40–77,2016. arXiv:1509.05385 , doi:10.1016/j.aam.2016.06.005 .[Fuc03] Dmitry Fuchs. Chekanov-Eliashberg invariant of Legendrian knots:existence of augmentations. J. Geom. Phys. , 47(1):43–65, 2003. doi:10.1016/S0393-0440(01)00013-4 .[FZ02] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. I. Founda-tions. J. Amer. Math. Soc. , 15(2):497–529, 2002. arXiv:math/0104151 , doi:10.1090/S0894-0347-01-00385-X .[FZ07] Sergey Fomin and Andrei Zelevinsky. Cluster algebras. IV. Coefficients. Compos. Math. ,143(1):112–164, 2007. arXiv:math/0602259 , doi:10.1112/S0010437X06002521 .[Gei08] Hansj¨org Geiges. An introduction to contact topology , volume 109 of CambridgeStudies in Advanced Mathematics . Cambridge University Press, Cambridge, 2008. doi:10.1017/CBO9780511611438 .[GHKK18] Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich. Canonical basesfor cluster algebras. J. Amer. Math. Soc. , 31(2):497–608, 2018. arXiv:1411.1394 , doi:10.1090/jams/890 .[GPS18] Sheel Ganatra, John Pardon, and Vivek Shende. Sectorial descent for wrapped Fukayacategories, 2018. arXiv:1809.03427 .[GR91] I. M. Gelfand and V. S. Retakh. Determinants of matrices over noncommutative rings. Funktsional. Anal. i Prilozhen. , 25(2):13–25, 96, 1991. doi:10.1007/BF01079588 .[GS19] Alexander Goncharov and Linhui Shen. Quantum geometry of moduli spaces of localsystems and representation theory. Preprint, 2019. arXiv:1904.10491 .[K´al06] Tam´as K´alm´an. Braid-positive Legendrian links. Int. Math. Res. Not. , pages Art ID14874, 29, 2006. arXiv:math/0608457 , doi:10.1155/IMRN/2006/14874 .[Kum02] Shrawan Kumar. Kac-Moody groups, their flag varieties and representation theory ,volume 204 of Progress in Mathematics . Birkh¨auser Boston, Inc., Boston, MA, 2002. doi:10.1007/978-1-4612-0105-2 .75LS15] Kyungyong Lee and Ralf Schiffler. Positivity for cluster algebras. Ann. of Math. (2) ,182(1):73–125, 2015. arXiv:1306.2415 , doi:10.4007/annals.2015.182.1.2 .[Nad09] David Nadler. Microlocal branes are constructible sheaves. Selecta Math. (N.S.) ,15(4):563–619, 2009. arXiv:math/0612399 , doi:10.1007/s00029-009-0008-0 .[Ng01] Lenhard Ng. Maximal Thurston-Bennequin number of two-bridge links. Algebr. Geom.Topol. , 1:427–434, 2001. arXiv:math/0008242 , doi:10.2140/agt.2001.1.427 .[Ng03] Lenhard Ng. Computable Legendrian invariants. Topology , 42(1):55–82, 2003. arXiv:math/0011265 , doi:10.1016/S0040-9383(02)00010-1 .[NR13] Lenhard Ng and Daniel Rutherford. Satellites of Legendrian knots and representationsof the Chekanov-Eliashberg algebra. Algebr. Geom. Topol. , 13(5):3047–3097, 2013. arXiv:1206.2259 , doi:10.2140/agt.2013.13.3047 .[NRSSZ15] Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow. Augmen-tations are sheaves. Preprint, 2015. arXiv:1502.04939 .[NZ09] David Nadler and Eric Zaslow. Constructible sheaves and the Fukaya cat-egory. J. Amer. Math. Soc. , 22(1):233–286, 2009. arXiv:math/0604379 , doi:10.1090/S0894-0347-08-00612-7 .[NZ12] Tomoki Nakanishi and Andrei Zelevinsky. On tropical dualities in cluster alge-bras. In Algebraic groups and quantum groups , volume 565 of Contemp. Math. ,pages 217–226. Amer. Math. Soc., Providence, RI, 2012. arXiv:1101.3736 , doi:10.1090/conm/565/11159 .[Pan17] Yu Pan. Exact Lagrangian fillings of Legendrian (2 , n ) torus links. Pacific J. Math. ,289(2):417–441, 2017. arXiv:1607.03167 , doi:10.2140/pjm.2017.289.417 .[Pol91] Leonid Polterovich. The surgery of Lagrange submanifolds. Geom. Funct. Anal. ,1(2):198–210, 1991. doi:10.1007/BF01896378 .[PT20] James Pascaleff and Dmitry Tonkonog. The wall-crossing formula and La-grangian mutations. Adv. Math. , 361:106850, 67, 2020. arXiv:1711.03209 , doi:10.1016/j.aim.2019.106850 .[Sab05] Joshua Sabloff. Augmentations and rulings of Legendrian knots. Int. Math. Res. Not. ,2005(19):1157–1180, 2005. arXiv:math/0409032 , doi:10.1155/IMRN.2005.1157 .[Siv11] Steven Sivek. A bordered Chekanov-Eliashberg algebra. J. Topol. , 4(1):73–104, 2011. arXiv:1004.4929 , doi:10.1112/jtopol/jtq035 .[STWZ19] Vivek Shende, David Treumann, Harold Williams, and Eric Zaslow. Cluster varietiesfrom Legendrian knots. Duke Math. J. , 168(15):2801–2871, 2019. arXiv:1512.08942 , doi:10.1215/00127094-2019-0027 .[SW19] Linhui Shen and Daping Weng. Cluster structures on double Bott-Samelson cells.Preprint, 2019. arXiv:1904.07992 .76Syl19] Zachary Sylvan. On partially wrapped Fukaya categories. J. Topol. , 12(2):372–441,2019. arXiv:1604.02540 , doi:10.1112/topo.12088 .[Via14] Renato Vianna. On exotic Lagrangian tori in CP . Geom. Topol. , 18(4):2419–2476,2014. arXiv:1305.7512 , doi:10.2140/gt.2014.18.2419doi:10.2140/gt.2014.18.2419