Lagrangian fillings for Legendrian links of finite type
aa r X i v : . [ m a t h . S G ] J a n LAGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE
BYUNG HEE AN, YOUNGJIN BAE, AND EUNJEONG LEE
Abstract.
We prove that there are at least seeds many exact embedded Lagrangian fillingsfor Legendrian links of type
ADE . We also provide seeds many Lagrangian fillings with certainsymmetries for type
BCFG . Our main tools are N -graphs and the combinatorics of seed patternsof finite type. Contents
1. Introduction 11.1. Backgrounds 11.2. The results 2Acknowledgement 52. Legendrians and N -graphs 52.1. Geometric setup 52.2. N -graphs and Legendrian weaves 62.3. Legendrian isotopies and moves on N -graphs 83. Cluster algebras 113.1. Basics on cluster algebras 113.2. Cluster algebras of finite type 143.3. Folding 153.4. Combinatorics of exchange graphs 184. N -graphs and seeds 214.1. One-cycles in Legendrian weaves 224.2. N -graphs and flag moduli space 244.3. N -graphs and seeds 254.4. Legendrian mutations in N -graphs 275. Lagrangian fillings for of Legendrians type ADE N -graphs of type ADE
BCFG N -graphs of type A n − and D N -graphs of type D n +1 and E N -graphs 44References 451. Introduction
Backgrounds.
Legendrian knots are central object in the study of contact 3-dimensionalcontact manifolds. Classification of Legendrian knots are important as its own right, and also playa prominent role in constructing 4-dimensional Weinstein manifold.Classical Legendrian knot invariants are Thurston–Bennequin number and rotation number [20]which distinguish the pair of Legendrian knots with the same knot type. There are non-classical
Mathematics Subject Classification.
Primary: 53D10, 13F60. Secondary: 57R17.
Key words and phrases.
Legendrian link, Lagrangian filling, Cluster algebra. invariants including the Legendrian contact algebra via the method of Floer theory [12, 10], andthe space of constructible sheaves using microlocal analysis [21, 29]. These non-classical invariantsdistinguish the Chekanov pair, a pair of Legendrian knots of type m having the same classicalinvariants.Recently, the study of exact Lagrangian fillings for Legendrian links has been extremely plen-tiful. In the context of Legendrian contact algebra, an exact Lagrangian filling gives an aug-mentation through the functorial view point [11]. There are several level of equivalence betweenaugmentations and the constructible sheaves for Legendrian links from counting to categoricalequivalence [24]. By using these idea of augmentations and constructible sheaves, people con-struct infinitely many fillings for certain Legendrian links [7, 19, 8]. Here is the summarized listof methods of constructing Lagrangian fillings for Legendrian links:(1) Decomposable Lagrangian fillings via pinching sequences [11].(2) Alternating Legendrians and its conjugate Lagrangian fillings [28].(3) Legendrian weaves via N -graphs [30, 8].(4) Donaldson–Thomas transformation on augmentation varieties [27, 18, 19].The cluster structure introduced by [14] plays a crucial role in the above constructions andapplications. More precisely, the space of augmentations, or equally the moduli of constructiblesheaves adapted to Legendrian links, admits a structure of cluster algebra [28]. Note that a seedof cluster algebra consists of a quiver whose vertices are decorated with cluster variables. Aninvolutory operation at each vertex, called mutation , generates all seeds of the cluster pattern.The main point is to identify the mutation in the cluster pattern and an operation in the spaceof Lagrangian fillings. This geometric operation is deeply related to the Lagrangian surgery [26]and the wall-crossing phenomenon [2].Indeed, a Legendrian torus link of type (2 , n ) admits Catalan number many Lagrangian fillingsup to exact Lagrangian isotopy [25, 28, 30]. Interestingly enough, the Catalan number is thenumber of seeds in a cluster pattern of Dynkin type A n − . There are also Legendrian linkscorresponding to Dynkin type D n , E , E , and E [19]. A conjecture in [6, Conjecture 5.1] saysthat the number of distinct exact embedded Lagrangian fillings (up to exact Lagrangian isotopy)for Legendrian links of type ADE is exactly the same as the number of seeds of the correspondingcluster algebras.Furthermore, it is also conjectured in [6, Conjecture 5.4] that for Legendrian links of type A n − , D n +1 , E and D , Lagrangian fillings having certain Z / Z or Z / Z -symmetry form thecluster patterns of type B n , C n , F and G , which are Dynkin diagrams obtained by folding intro-duced in [16].1.2. The results.
Our main result is that there are seeds many Lagrangian fillings for Legendrianlinks of finite type. We deal with N -graphs in [8] to construct the Lagrangian fillings. An N -graph G on D gives a Legendrian surface Λ( G ) in J D while the boundary ∂ G on S induces aLegendrian link λ ( ∂ G ). Then projection of Λ( G ) along the Reeb direction becomes a Lagrangianfilling of λ ( ∂ G ).As mentioned above, we interpret an N -graph as a seed in the corresponding cluster pattern.A one-cycle in the Legendrian surface Λ( G ) corresponds to a vertex of the quiver, and a signedintersection between one-cycles gives an arrow between corresponding vertices. From constructiblesheaves adapted to Λ( G ), one can assign a monodromy to each one-cycle which becomes the clustervariable at each vertex.There is an operation so called a Legendrian mutation µ γ on an N -graph G along one-cycle [ γ ] ∈ H (Λ( G )) which is the counterpart of the mutation on the cluster pattern, see Proposition 4.19.The delicate and challenging part is that we do not know whether Legendrian mutations are alwayspossible or not. Simply put, this is because the mutation in cluster side is algebraic, whereas theLegendrian mutation is rather geometric.The main idea of our construction is to consider the following bichromatic (blue and red) graph G ( a, b, c ), i.e. N -graph with N = 3, bounding a Legendrian link λ ( a, b, c ), which is the closure of AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 3 the braid β ( a, b, c ) as follows: λ ( a, b, c ) = Cl( β ( a, b, c )) ⊂ J S , β ( a, b, c ) = σ σ a +11 σ σ b +11 σ σ c +11 . Then β ( a, b, c ) corresponds to the rainbow closure of the braid β ( a, b, c ) = σ σ a σ b − σ c . a b − cβ ( a, b, c ) r = | {z } r Figure 1.
The rainbow closure of the braid β ( a, b, c ) in R One-cycles B ( a, b, c ) of the Legendrian surface Λ( G ( a, b, c )) are given by the yellow- and green-shaded edges as depicted in Figure 2. See § G ( a, b, c ) , B ( a, b, c )) = a + b + c + Figure 2.
The tripod N -graph G ( a, b, c ) and the good tuple B ( a, b, c ) of cyclesThere are several good properties of G ( a, b, c ) as follows:(1) The geometric- and algebraic intersection numbers of the one-cycles in B ( a, b, c ) coincide.(2) The corresponding quiver Q ( a, b, c ) is bipartite, see § ADE . More precisely, the underlying graphs of Q (1 , b, c )for b + c − n , Q ( n − , , Q (2 , , n −
3) are the same as Dynkin diagrams oftype A n , D n , and E n , respectively.Let us consider the finite type N -graph G ( a, b, c ), that is, a + b + c >
1. Denote the correspond-ing rank n -root system by Φ = Φ( a, b, c ), where n = a + b + c −
2. Let us consider an exchangegraph E (Φ) of the corresponding cluster pattern whose vertices are the seeds and whose edgesconnect the vertices are given by a single mutation. Note from [9] that the exchange graph E (Φ)can be realized as vertices and edges of a polytope P (Φ) ⊂ R n called a generalized associahedron .The combinatorics of the exchange graph E (Φ) is the key ingredient in investigating the Leg-endrian mutability. All facets of the polytope P (Φ) can be recovered from a sequence of muta-tions obtained by a Coxeter element together with a subset of facets of P (Φ) corresponding to P (Φ([ n ] \ { i } )), see [16] or Proposition 3.15. We call this specific sequence of mutations a Coxetermutation . In order to interpret a Coxeter mutation in terms of N -graphs, let us consider a parti-tion B + , B − of the one-cycles B , consisting of yellow- and green-shaded edges, respectively. Then BYUNG HEE AN, YOUNGJIN BAE, AND EUNJEONG LEE the N -graph realization of the Coxeter mutation is called the Legendrian Coxeter mutation andgiven by the sequence of Legendrian mutations: µ G = Y γ ∈ B − µ γ · Y γ ∈ B + µ γ . Then the resulting N -graph µ G ( G ( a, b, c ) , B ( a, b, c )) becomes the N -graph shown in Figure 3 upto a sequence of Move (II) in Figure 7. Figure 3.
Legendrian Coxeter mutation on ( G ( a, b, c ) , B ( a, b, c ))Removing the gray-shaded annulus region, the only difference between ( G ( a, b, c ) , B ( a, b, c )) and µ G ( G ( a, b, c ) , B ( a, b, c )) is the reversing of the color. Note that the intersection pattern betweenone-cycles and the Legendrian mutability are preserved under the action of the Legendrian Coxetermutation µ G . Moreover, the operation µ Q also acts on the face poset of the generalized associa-hedron of the root system Φ. By the induction argument on the rank of root system, we concludethat there in no (geometric) obstruction to realize each seed via the N -graph, especially for finitetype case. This guarantees that there are at least seeds many Lagrangian fillings for λ ( a, b, c ).For the infinite type, i.e. a + b + c ≤
1, the operation µ Q is of infinite order and so is µ G ,hence Legendrian weaves Λ( µ r G ( G ( a, b, c ) , B ( a, b, c )))produce infinitely many distinct Lagrangian fillings. Indeed, the quiver Q ( a, b, c ) is also bipartiteand the one can perform the Legendrian Coxeter mutation µ G on the N -graph G ( a, b, c ) by stackingthe gray-shaded annulus like as before. Therefore, there is no obstruction to realize seeds obtainedby mutations µ r G via the N -graphs. Since the order of the Legendrian Coxeter mutation is infinite(see Lemma 3.20), we obtain infinitely many N -graphs and hence infinitely many exact embeddedLagrangian fillings for the Legendrian link λ ( a, b, c ) with a + b + c ≤ Theorem 1.1 (Theorem 5.12) . For each a, b, c ≥ , the Legendrian knot or link λ ( a, b, c ) hasdistinct infinitely many Lagrangian fillings if a + 1 b + 1 c ≤ , or equivalently, the tripod Q ( a, b, c ) is of infinite type. Theorem 1.2 (Theorem 5.13) . There are at least seeds many distinct exact embedded Lagrangianfillings for Legendrian links of type
ADE . There are several way of constructing exact embedded Lagrangian fillings as mentioned above.Especially in D case, there are 34 distinct Lagrangian fillings constructed by the method of thealternating Legendrians in [3, 28], while the above N -graphs give seeds many 50 Lagrangian fillings.The remaining finite type Dynkin diagrams, which are non-simply laced, are of type BCFG ,obtain by the folding procedure from type
ADE , see § AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 5 seeds and mutations in B n , C n , F , and G cluster patterns can be regarded as certain subsets ofseeds and sequences of mutations in A n − , D n − , E , and D , respectively. Those specified seedsof type ADE admit N -graphs with certain symmetries given by an action of a finite group G , andwe call such seeds and N -graphs G -admissible . If a seed (or an N -graph) is again G -admissibleafter performing a sequence of mutations indexed by vertices in the same G -orbit, then we call it globally foldable with respect to G .The following four N -graphs are examples of type BCFG . Indeed, they are G (1 , , G (3 , , G (2 , , G (2 , , A B > D C < E F < D G < The colored regions represent how the group G acts on the N -graphs and the induced Lagrangianfillings. The first three N -graphs are globally foldable with respect to Z / Z by folding orange- andviolet-colored regions in an orientation preserving way. Similarly, the Z / Z -rotational symmetryof the last one implies that it is globally foldable with respect to Z / Z by folding three coloredregions. Theorem 1.3 (Theorem 6.11) . The following holds:(1) The Legendrian link λ ( A n − ) has (cid:0) nn (cid:1) Z / Z -admissible N -graphs which admits the clusterpattern of type B n .(2) The Legendrian link λ ( D n +1 ) has (cid:0) nn (cid:1) Z / Z -admissible N -graphs which admits the clusterpattern of type C n .(3) The Legendrian link λ ( E ) has Z / Z -admissible N -graphs which admits the clusterpattern of type F .(4) The Legendrian link λ ( D ) has Z / Z -admissible N -graphs which admits the cluster pat-tern of type G . Acknowledgement.
B. An was supported by Kyungpook National University Research Fund,2020. Y. Bae was supported by the National Research Foundation of Korea (NRF) grant fundedby the Korea government (MSIT) (No. 2020R1A2C1A0100320). E. Lee was supported by IBS-R003-D1. 2.
Legendrians and N -graphs We recall from [8] the notion of N -graphs and their combinatorial moves which encode theLegendrian isotopy data of corresponding Legendrian surfaces. As an application, we review how N -graphs can be use to find and to distinguish Lagrangian fillings for Legendrian links.2.1. Geometric setup.
Let us start with the standard contact structure on R whose contactstructure is given by ξ = ker( dz − ydx ). Consider the symplectization ( R , d ( e s ( dz − ydx ))) andits contactization ( R , α = e s ( dz − ydx ) − dt ).Now consider a contact 3-dimensional space( R ℓ , ξ ) := { ( x, y, z, s, t ) ∈ R | ( s, t ) = ( ℓ, } BYUNG HEE AN, YOUNGJIN BAE, AND EUNJEONG LEE for each symplectization level s = ℓ . Take Legendrians λ ⊂ ( R , ξ ) and λ ⊂ ( R , ξ ) andconsider a Legendrian surface Λ ⊂ ( R , ξ = ker α ) ∩ { ≤ s ≤ } whose boundary is λ ∪ λ , i.e.,Λ ∩ ( R , ξ ) = λ , Λ ∩ ( R , ξ ) = λ . Let π : R → R be the projection along the contactization coordinate t , then π (Λ) becomes anexact Lagrangian (possibly immersed) cobordism from π ( λ ) to π ( λ ). Note that ∂ t is the Reebvector field of ( R , α ) and the above immersed points on π (Λ) correspond to Reeb chords in Λ.Relating the construction of the current article, any Legendrian link in ( R , ξ ) can be seen as asatellite link of the standard Legendrian unknot λ unknot ⊂ ( R , ξ ). Note that a neighborhood of λ unknot is contactomorphic to ( J S , ker( dz − p θ dθ )). Denote the corresponding contact embeddingby ι : J S ֒ → R .Let us denote the corresponding satellite links of λ , λ by λ β i = Cl( β i ) ⊂ ( J ( S × { i } ) , ker( dz − p θ dθ )) , i = 1 , . Here β i is a positive braid word for the satellite link of λ i , and Cl( β ) denotes the closure of abraid word β . Extending the contact embedding ι : J S ֒ → R , by abuse of notation, we have ι : J ( S × [1 , ֒ → R ∩ { ≤ s ≤ } . Denote the corresponding Legendrian surface byΛ ⊂ ( J ( S × [1 , , ker( dz − p θ dθ − p σ dσ )) , where σ is the coordinate for the interval [1 ,
2] which corresponds to the e s -coordinate. By a strictcontactomorphism, we can regardΛ ⊂ ( J S × R s × R t , ker( e s ( dz − p θ dθ ) − dt )) . Then its Lagrangian projection π ◦ ι (Λ) gives an exact Lagrangian cobordism from λ to λ , where π is the projection along the t -coordinate. Especially when λ = ∅ , the boundary J ( S × { } ) of J ( S × [1 , J D . Under the Lagrangian projection, this correspondsto a exact symplectic filling ( D , ω st ) of ( S , ξ st ) = ( R , ξ ) ∪ {∞} . We end this section by statingthe relation between Legendrian- and Lagrangian fillings. Lemma 2.1.
As in the above setup, let λ β ⊂ J S be a Legendrian link, and let ι ( λ β ) be aninduced Legendrian link in ( S , ξ st ) . Let Λ , Λ ′ ⊂ J D be two Legendrian surfaces, without Reebchords, bounding λ β . If the corresponding exact Lagrangian fillings π ◦ ι (Λ) , π ◦ ι (Λ ′ ) ⊂ ( D , ω st ) of ι ( λ β ) are exact Lagrangian isotopic relative to the boundary, then Λ , Λ ′ are Legendrian isotopicrelative to the boundary. N -graphs and Legendrian weaves.Definition 2.2. [8, Definition 2.2] An N -graph G on a smooth surface S is an ( N − G , . . . , G N − ) satisfying the following conditions:(1) Each graph G i is embedded, trivalent, possibly empty and non necessarily connected.(2) Any consecutive pair of graphs ( G i , G i +1 ), 1 ≤ i ≤ N −
2, intersects only at hexagonalpoints depicted as in Figure 4.(3) Any pair of graphs ( G i , G j ) with 1 ≤ i, j ≤ N − | i − j | > Figure 4.
A hexagonal point
AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 7
Remark . For the result of the current article, we mainly consider the case N = 3 and S = D .In other words, we focus bicolored graphs with monochromatic trivalent vertices and bichromatichexagonal points as in Figure 4.For any N -graph G on a surface S , we associate a Legendrian surface Λ( G ) ⊂ J S . Basically,we construct the Legendrian surface by weaving the wavefronts in S × R constructed from a localchart of S .Let G ⊂ S be an N -graph. A finite cover { U i } i ∈ I is called G -compatible if(1) each U i is diffeomorphic to the open disk ˚ D ,(2) U i ∩ G is connected, and(3) U i ∩ G contains at most one vertex.For each U i , we associate a wavefront Γ( U i ) ⊂ U i × R ⊂ S × R . Note that there are only fourtypes of local charts for any N -graph G as follows:Type 1. A chart without any graph component whose corresponding wavefront becomes [ i =1 ,...,N ˚ D × { i } ⊂ ˚ D × R . Type 2. A chart with single edge. The corresponding wavefront is the union of the A -germ alongthe two sheets ˚ D ×{ i } and ˚ D ×{ i +1 } , and trivial disks D ×{ i } , i ∈ { , . . . , N }\{ i, i +1 } .The local model of A comes from the origin of the singular surfaceΓ( A ) = { ( x, y, z ) ∈ R | x − z = 0 } See Figure 5(a).Type 3. A chart with a monochromatic trivalent vertex whose wavefront is the union of the D − -germ, see [1, § D × { i } , i ∈ { , . . . , N } \ { i, i + 1 } . The local modelfor Legendrian singularity of type D − is given by the image at the origin of δ − : R → R : ( x, y ) (cid:18) x − y , xy,
23 ( x − xy ) (cid:19) . See Figure 5(c).Type 4. A chart with a bichromatic hexagonal point. The induced wavefront is the union of the A -germ along the three sheets ˚ D × {∗} , ∗ = i, i + 1 , i + 2, and the trivial disks D × { i } , i ∈ { , . . . , N } \ { i, i + 1 , i + 2 } . The local model of A is given by the origin of the singularsurface { ( x, y, z ) ∈ R | ( x − z )( y − z ) = 0 } . See Figure 5(b). A (a) The germ of A A (b) The germ of A D − (c) The germ of D − Figure 5.
Three-types of wavefronts of Legendrian singularities.
Definition 2.4. [8, Definition 2.7] Let G be an N -graph on a surface S . The Legendrian weave Λ( G ) ⊂ J S is an embedded Legendrian surface whose wavefront Γ( G ) ⊂ S × R is constructed byweaving the wavefronts { Γ( U i ) } i ∈ I from a G -compatible cover { U i } i ∈ I with respect to the gluingdata given by G . Remark . Note that Λ( G ) is well-defined up to the choice of cover and up to planar isotopies.Let { ϕ t } t ∈ [0 , be a compactly supported isotopy of S . Then this induces a Legendrian isotopy ofLegendrian surface Λ( ϕ t ( G )) ⊂ J S relative to the boundary. BYUNG HEE AN, YOUNGJIN BAE, AND EUNJEONG LEE N ... 1 N ... i + 1 i ... 1 N ... i + 1 i ... 1 N ... i + 2 i + 1 i ... 1 Figure 6.
Four-types of local charts for N -graphs.2.3. Legendrian isotopies and moves on N -graphs. The idea of N -graph is useful in thestudy of Legendrian surface, because the Legendrian isotopy of the Legendrian weave Λ( G ) can beencoded in combinatorial moves of N -graphs. Theorem 2.6. [8, Theorem 1.1]
Let G be a local N -graph. The combinatorial moves in Figure 7and Figure 8 are Legendrian isotopies for Λ( G ) . (I) (II)(III) (IV)(V) (VI)(VI’) G (1 ,...,N ) (S) G (1 ,...,N ) ... ( N, N + 1)( N − , N )(2 , , Figure 7.
Combinatorial moves for Legendrian isotopies of surface Λ( G ). Herethe pairs (blue, red) and (red, green) are consecutive. Other pairs are not.2.3.1. N -graphs on D . Let λ β ⊂ J S be a Legendrian link obtained from a Legendrian line in R by satelliting the Legendrian unknot. Here λ β is the closure of a positive N -strand braid β .The braid word β consist of alphabets σ , . . . , σ N − , and these give an ( N − S which can be regarded as a boundary data of N -graphs on D . By the setup in § π ◦ ι (Λ( G )) induces an exact (possibly immersed) Lagrangian filling in ( R , ω st ) of ι ( λ β ). Let usdenote the equivalence class of a N -graph G ⊂ D up to the moves (I) , . . . , (XII) by [ G ]. AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 9 (VII) (VIII)(IX) (X)(XI) (XII)
Figure 8.
Combinatorial moves for Legendrian isotopies of surface Λ( G ): Theseare moves involving A -swallowtail singularities, the orange vertex. Here theorange lines are locus of cusp singularities. Remark . For N -graphs G on D the stabilization, becomes Move (S) in Figure 9: G (1 ,...,N ) (S) G (1 ,...,N ) (1 , , N − , N )( N, N + 1) · · ·
Figure 9.
A stabilization of an N -graph on D . Definition 2.8. An N -graph G ⊂ D is called free if the induced Legendrian weave Λ( G ) ⊂ J D can be woven without interior Reeb chord. Example 2.9. [8, Example 7.3] Let G ⊂ D be a 2-graph such that D \ G is simply connectedrelative to the boundary ∂ D ∩ ( D \ G ). Then G is free if and only if G has no faces contained in˚ D . Note that each of such faces admits at least one Reeb chord, see Figure 10. Figure 10. N -graphs with Reeb chords To investigate the Reeb chords of Λ( G ) in J D , let us consider the wavefront Γ( G ) in D × R .Label the sheets of the wavefront Γ( G ) = N [ i =1 Γ i (2.1)by the z -coordinate from the bottom to top. Let f i : D → R be a function whose graph becomesΓ i , and let h ij : D → R be a difference function given by f j − f i for any i, j ∈ [ N ] with i < j .By the construction h − i i +1 (0) gives G i ⊂ G . The critical points of h ij on ˚ D \ G are the possiblecandidates for the Reeb chords. In other words, to guarantee that G is free, it suffices to showthat h ij has no critical point on ˚ D \ G . F F F F F F F F F F F F Figure 11. G (2 , , G , and G Now apply this idea to the 3-graph G ( a, b, c ) in the introduction. In order to construct h and h , consider the following graph complements ˚ D \ G i for i = 1 ,
2. Let us denote the closureof connected components of ˚ D \ G i by { F i ; k } k ∈ K i . Each F i ; k is a polygon and exactly one edgecomes from the boundary ∂ D . Figure 12.
The vertical gray lines present gradient flow lines of h | F k . Herethe bottom (black) line comes from ∂ D .We then consider functions h i i +1 , i = 1 , • h i i +1 is smooth and nonnegative. • h i i +1 ( x ) = 0 if and only if x ∈ G i . • h i i +1 has no critical point on ˚ D \ G i . • For any k ∈ K i , positive gradient flow lines of h i i +1 | F i ; k head for the edge from ∂ D , seeFigure 12.By the construction of h i i +1 and the definition of Reeb chord, there is no Reeb chord connectingΓ i and Γ i +1 for i = 1 ,
2. Now consider the the gradient flow lines of h + h to see the Reebchords from Γ to Γ . Without loss of generality, we may assume that k∇ h k < k∇ h k excepta small neighborhood of G . Then by the configuration of G ( a, b, c ) = G ∪ G the gradient flowlines of h + h never vanish except the hexagonal point. In conclusion, we can construct thewavefront Γ( G ( a, b, c )) without interior Reeb chords. Lemma 2.10.
The -graph G ( a, b, c ) is free. AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 11 N -graphs on A . Let A be the oriented annulus with two boundary components ∂ + A and ∂ − A , and let G be a N -graph on A . We say that G is of type ( λ + , λ − ) if G on ∂ + A and ∂ − A are given by Legendrian links λ + and λ − , respectively. We may regard the N -graph G of type( λ + , λ − ) as a cobordism between λ + and λ − .Suppose that two annular N -graphs G and G are of type ( λ , λ ) and ( λ , λ ). Then two N -graphs can be merged or piled in a natural way to obtain the annular N -graph, denoted by G · G of type ( λ , λ ). Let G ⊂ D be an N -graph with ∂ G = λ , then the padding operation G G is defined by gluing along the boundary λ . Note that if there is a rotational symmetry on λ ,then the operation G G is well-defined only up to that symmetry.(RIII) ... (R0) ... Figure 13.
Reidemeister moves in J S or R avoiding cusp singularities.Let us illustrate elementary annulus N -graph s coming from the Legendrian isotopies in J S .The following two Legendrian Reidemeister moves (RIII) and (R0) can be interpreted as N -graphs G (RIII) and G (R0) on the annulus A , respectively, as depicted in Figure 14. The Move (I) and (V)of N -graphs in Figure 7 imply that the inverses G − and G − can be obtained by reversing therole of the inner- and outer boundaries.Suppose that there are certain rotational symmetry on N -graphs. Let us consider a rotationalannulus N -graph which is trivial as an N -graph but rotated respecting the symmetry. A typicalexample comes from Legendrian torus link λ ( n, m ) of maximal Thurston-Bennequin number. Theright one in Figure 14 is a rotational annulus N -graph for λ (3 , N -graphsplay a crucial role in producing a sequence of distinct exact Lagrangian fillings of positive braidLegendrian links, see [23, 7, 19]. 3. Cluster algebras
Cluster algebras, introduced by Fomin and Zelevinsky [14], are commutative algebras withspecific generators, called cluster variables , defined recursively. In this section, we recall basicnotions in the theory of cluster algebras. For more details, we refer the reader to [14, 15].Throughout this section, we fix m, n ∈ Z > such that n ≤ m , and we let F be the rationalfunction field with m independent variables over C .3.1. Basics on cluster algebras.Definition 3.1 (cf. [14, 15]) . A seed Σ = ( x , B ) is a pair of • a tuple x = ( x , . . . , x m ) of algebraically independent generators of F , that is, F = C ( x , . . . , x m ); • an n × m integer matrix B = ( b i,j ) i,j such that the principal part B pr := ( b i,j ) ≤ i,j ≤ n isskew-symmetrizable, that is, there exist positive integers d , . . . , d n such thatdiag( d , . . . , d n ) · B pr is a skew-symmetric matrix.We call elements x , . . . , x m cluster variables and call B exchange matrix . Moreover, we call x , . . . , x n unfrozen (or, mutable ) variables and x n +1 , . . . , x m frozen variables.To define cluster algebras, we introduce mutations on seeds, exchange matrices, and quivers asfollows. G (RIII) · G = GG (R0) · G = G Figure 14.
Elementary annulus operations on N -graphs on D , and a rotationalannulus N -graph.(1) (Mutation on seeds) For a seed Σ = ( x , B ) and an integer 1 ≤ k ≤ n , the mutation µ k (Σ) = ( x ′ , B ′ ) is defined as follows: x ′ i = x i if i = k,x − k Y b k,j > x b k,j j + Y b k,j < x − b k,j j otherwise .b ′ i,j = − b i,j if i = k or j = k,b i,j + | b i,k | b k,j + b i,k | b k,j | . (2) (Mutation on exchange matrices) We define µ k ( B ) = ( b ′ i,j ), and say that B ′ = ( b ′ i,j ) is themutation of B at k .(3) (Mutation on quivers) We call a finite directed multigraph Q a quiver if it does not haveoriented cycles of length at most 2. The adjacency matrix B ( Q ) of a quiver is alwaysskew-symmetric. Moreover, µ k ( B ( Q )) is again the adjacency matrix of a quiver Q ′ . Wedefine µ k ( Q ) to be the quiver satisfying B ( µ k ( Q )) = µ k ( B ( Q )) , and say that µ k ( Q ) is the mutation of Q at k .We say a quiver Q ′ is mutation equivalent to another quiver Q if there exists a sequence ofmutations µ j , . . . , µ j ℓ which connects Q ′ and Q , that is, Q ′ = ( µ j ℓ · · · µ j )( Q ) . Also, we say a quiver Q is acyclic if there is no directed cycle. AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 13 · · · · · · · · · · · ·
33 33 3 T T Figure 15.
The n -regular trees for n = 2 and n = 3. Remark . It is proved in [5, Corollary 4] that if two acyclic quivers are mutation equivalent,then there exists a sequence of mutations from one to other such that intermediate quivers are allacyclic. Indeed, two mutation equivalent acyclic quivers have the same underlying (undirected)graph.An immediate check shows that µ k (Σ) is again a seed, and a mutation is an involution, that is,its square is the identity. Also, note that the mutation on seeds does not change frozen variables x n +1 , . . . , x m . Let T n denote the n -regular tree whose edges are labeled by 1 , . . . , n . Except for n = 1, there are infinitely many vertices on the tree T n . For example, we present regular trees T and T in Figure 15. A cluster pattern (or seed pattern ) is an assignment T n → { seeds in F } , t Σ t = ( x t , B t )such that if t t ′ k in T n , then µ k (Σ t ) = Σ t ′ . Let { Σ t = ( x t , B t ) } t ∈ T n be a cluster patternwith x t = ( x t , . . . , x m ; t ). Since the mutation does not change frozen variables, we may let x n +1 = x n +1; t , . . . , x m = x m ; t . Definition 3.3 (cf. [15]) . Let { Σ t = ( x t , B t ) } t ∈ T n be a cluster pattern with x t = ( x t , . . . , x m ; t ).The cluster algebra A ( { Σ t } t ∈ T n ) is defined to be the C [ x n +1 , . . . , x m ]-subalgebra of F generatedby all the cluster variables S t ∈ T n { x t , . . . , x n ; t } .If we fix a vertex t ∈ T n , then a cluster pattern { Σ t } t ∈ T n is constructed from the seed Σ t .In this case, we call Σ t an initial seed . Because of this reason, we simply denote by A (Σ t ) thecluster algebra given by the cluster pattern constructed from the initial seed Σ t . Example 3.4.
Let n = m = 2. Suppose that an initial seed is given byΣ t = (cid:18) ( x , x ) , (cid:18) − (cid:19)(cid:19) . We present a part of the cluster pattern obtained by the initial seed Σ t . (cid:18) ( x , x ) , (cid:18) −
11 0 (cid:19)(cid:19) Σ t = (cid:18) ( x , x ) , (cid:18) − (cid:19)(cid:19)(cid:18) ( x x , x ) , (cid:18) − (cid:19)(cid:19) (cid:18)(cid:16) x x , x (cid:17) , (cid:18) −
11 0 (cid:19)(cid:19)(cid:18)(cid:16) x x , x + x x x (cid:17) , (cid:18) −
11 0 (cid:19)(cid:19) (cid:18)(cid:16) x x , x + x x x (cid:17) , (cid:18) − (cid:19)(cid:19) up torelabelling µ µ µ µ µ Remark . There is another mutation operation called the cluster X -mutation . Let { Σ t =( x t , B t ) } t ∈ T n be a cluster pattern with x t = ( x t , . . . , x m ; t ). For t ∈ T n and i ∈ [ n ], we set y t = ( y t , . . . , y n ; t ) by y i ; t = Y j ∈ [ m ] x b ( t ) i,j j ; t where B t = ( b ( t ) i,j ). Then the assignment t ( y t , B t ) is called a cluster Y -pattern and for t t ′ k in T n , we have y i ; t ′ = ( y i ; t y max { b ( t ) i,k , } k ; t (1 + y k ; t ) − b ( t ) i,k if i = k,y − k ; t otherwise;see [17, Proposition 3.9]. For t t ′ k in T n , the operation sends ( y t , B t ) to ( y t ′ , B t ′ ) is calledthe cluster X -mutation (or, X -cluster mutation). For exchange matrices and quivers, the cluster X -mutation is defined the same as before.3.2. Cluster algebras of finite type.
The number of cluster variables in Example 3.4 is finiteeven though the number of vertices in the graph T is infinite. We call such cluster algebras offinite type . More precisely, we recall the following definition. Definition 3.6 ([15]) . A cluster algebra is said to be of finite type if it has finitely many clustervariables.It has been realized that classifying finite type cluster algebras is related to studying exchangematrices. The
Cartan counterpart C ( B pr t ) = ( c i,j ) of the principal part B pr t of an exchange matrixis defined by c i,j = ( i = j, −| b i,j | otherwise . Since B pr t is skew-symmetrizable, its Cartan counterpart C ( B pr t ) is symmetrizable. Note that twomutation equivalent acyclic quivers produce the same Cartan counterpart (cf. Remark 3.2). Thefollowing theorem presents a classification of cluster algebras of finite type. Theorem 3.7 ([15]) . Let { Σ t = ( x t , B t ) } t ∈ T n be a cluster pattern with an initial seed Σ t =( x t , B t ) . Let A ( B t ) be the corresponding cluster algebra. Then we have the following.(1) The cluster algebra A ( B t ) is of finite type if and only if C ( B pr t ) is a Cartan matrix offinite type.(2) If the cluster algebra A ( B t ) is of finite type, then there is a bijective correspondencebetween the set of positive roots for C ( B pr t ) and the set of noninitial cluster variables.More precisely, for the set Π = { α , . . . , α n } of simple roots, a positive root P ni =1 d i α i isassociated to a cluster variable of the form f ( x t ) x d t · · · x d n n ; t , f ( x t ) ∈ C [ x t , . . . , x m ; t ] . Here, x t , . . . , x m ; t are cluster variables in the initial seed x t . Accordingly, there is abijective correspondence between the set of cluster variables and the set Φ ≥− of almostpositive roots Φ ≥− := Φ + ∪ ( − Π) , where Φ it the root system whose Cartan matrixis C ( B pr t ) . We provide a list of finite type root systems and their Dynkin diagram in Table 1. In whatfollows, we fix an ordering on the simple roots as in Table 1; our conventions agree with that in thestandard textbook of Humphreys [22]. In Table 2, we provide enumeration on the number of clustervariables and clusters in each cluster algebra of finite (irreducible) type (cf. [13, Figure 5.17]).
Definition 3.8.
For a quiver Q , we say that Q is of type X if it is mutation equivalent toa quiver Q ′ whose underlying unoriented graph on mutable vertices is the Dynkin diagram oftype X . Equivalently, Q is of type X if it is mutation equivalent to a quiver Q ′ whose Cartancounterpart C ( B pr ( Q ′ )) of the principal part of the adjacency matrix is the Cartan matrix of type X . Example 3.9.
Continuing Example 3.4, the Cartan counterpart of the principal part B pr t is givenby C ( B pr t ) = (cid:18) − − (cid:19) , AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 15
Φ Dynkin diagram A n ( n ≥ n − n B n ( n ≥ n − n − n > C n ( n ≥ n − n − n < D n ( n ≥ n − n − n − n E E E F > G < Table 1.
Dynkin diagrams of finite typewhich is the Cartan matrix of Lie type A . Accordingly, by Theorem 3.7, the cluster algebra A (Σ t )is of finite type. Indeed, there are five cluster variables and we present the bijective correspondencebetween them and the set of almost positive roots as described in Theorem 3.7(2). x x x x x x x + x x x − α − α α α α + α Here, α and α are simple roots of the Lie algebra of type A .3.3. Folding.
Under certain conditions, one can fold seed patterns to produce new ones. Thisprocedure is used to study cluster algebras of type
BCFG from those of simply-laced type
ADE .As before, we fix m, n ∈ Z > such that n ≤ m .Φ A n B n C n D n E E E F G n + 2 (cid:18) n + 2 n + 1 (cid:19) (cid:18) nn (cid:19) (cid:18) nn (cid:19) n − n (cid:18) n − n − (cid:19)
833 4160 25080 105 8 n ( n + 3)2 n ( n + 1) n ( n + 1) n
42 70 128 28 8
Table 2.
Enumeration of seeds and cluster variables
Let Q be a labeled quiver having m vertices labeled 1 , . . . , m . Let G be a finite group acting onthe set [ m ]. The notation i ∼ i ′ will mean that i and i ′ lie in the same G -orbit. To study foldingof cluster algebras, we prepare some terminologies. Definition 3.10 (cf. [13, § . Let Q be a labeled quiver having m vertices and G a finite groupacting on the set [ m ].(1) The quiver Q (or the corresponding m × n exchanged matrix B = B ( Q )) is G -admissible if(a) for any i ∼ i ′ , index i is mutable if and only if so is i ′ ;(b) for any indices i and j , and any g ∈ G , we have b i,j = b g ( i ) ,g ( j ) ;(c) for mutable indices i ∼ i ′ , we have b i,i ′ = 0;(d) for any i ∼ i ′ , and any mutable j , we have b i,j b i ′ ,j ≥ G -admissible quiver Q , we call a G -orbit mutable (respectively, frozen ) if it consistsof mutable (respectively, frozen) vertices.For a G -admissible quiver Q , we define the matrix B G = B ( Q ) G = ( b GI,J ) whose rows (respec-tively, columns) are labeled by the G -orbits (respectively, mutable G -orbits) by b GI,J = X i ∈ I b i,j where j is an arbitrary index in J . We then say B G is obtained from B (or from the quiver Q ) by folding with respect to the given G -action. Example 3.11.
Let Q be a quiver of type D given as follows. B ( Q ) = − − − The finite group G = Z / Z acts on [4] by sending 1
2. Here, we decoratevertices of the quiver Q with green and yellow colors for presenting sources and sinks, respectively.One may check that the quiver Q is G -admissible. By setting I = { } and I = { , , } , weobtain b GI ,I = X i ∈ I b i, = b , = 1 ,b GI ,I = X i ∈ I b i, = b , + b , + b , = − . Accordingly, we obtain the matrix B G = (cid:18) − (cid:19) whose Cartan counterpart is the Cartan matrixof type G .For a G -admissible quiver Q and a mutable G -orbit I , we consider a composition of mutationsgiven by µ I = Y i ∈ I µ i which is well-defined because of the definition of admissible quivers. If µ I ( Q ) is again G -admissible,then we have that ( µ I ( B )) G = µ I ( B G ) . We notice that the quiver µ I ( Q ) is not G -admissible in general. Therefore, we present the followingdefinition. Definition 3.12.
Let G be a group acting on the vertex set of a quiver Q . We say that Q is globally foldable with respect to G if Q is G -admissible and moreover for any sequence of mutable G -orbits I , . . . , I ℓ , the quiver ( µ I ℓ . . . µ I )( Q ) is G -admissible. AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 17
For a globally foldable quiver, we can fold all the seeds in the corresponding seed pattern.Let m G denote the number of orbits of the action of G on [ m ]. Let F G be the field of rationalfunctions in m G independent variables. Let ψ : F → F G be a surjective homomorphism. A seedΣ = ( x , B ( Q )) is called ( G, ψ ) -admissible if • Q is a G -admissible quiver; • for any i ∼ i ′ , we have ψ ( x i ) = ψ ( x i ′ ).In this situation, we define a new “folded” seed Σ G = ( x G , B G ) in F G whose exchange matrixis given as before and cluster variables x G = ( x I ) are indexed by the G -orbits and given by x I = ψ ( x i ). Proposition 3.13 (cf. [13, Corollary 4.4.11]) . Let Q be a quiver which is globally foldable withrespect to a group G acting on the set of its vertices. Let Σ = ( x , B ( Q )) be a seed in the field F of rational functions freely generated by a cluster x = ( x , . . . , x m ) . Define ψ : F → F G so that Σ is a ( G, ψ ) -admissible seed. Then, for any mutable G -orbits I , . . . , I ℓ , the seed ( µ I ℓ . . . µ I )(Σ) is ( G, ψ ) -admissible, and moreover the folded seeds (( µ I ℓ . . . µ I )(Σ)) G form a seed pattern in F G with the initial seed Σ G = ( x G , ( B ( Q )) G ) . Example 3.14.
The quiver in Example 3.11 is globally foldable, and moreover the correspondingseed pattern is of type G . In fact, seed patterns of type BCFG are obtained by folding quivers oftype
ADE in general. In Figure 16, we present the corresponding quivers of type
ADE . We decoratevertices of quivers with yellow and green colors for presenting source and sink, respectively. Asone may see, we have to put arrows on the Dynkin diagram alternatingly. For each case, the finitegroup action that makes each quiver globally foldable is given as follows.(1) A n − B n : The finite group G = Z / Z acts on the set [2 n −
1] of vertices of the quiverof type A n − by i n − i for i ∈ [2 n − . There are n orbits: I i = { i, n − i } for i ∈ [ n ].(2) D n +1 C n : The finite group G = Z / Z acts on the set [ n + 1] of vertices of the quiver oftype D n +1 by i i for i ∈ [ n − ,n n + 1 , n + 1 n. There are n orbits: I i = { i } for i ∈ [ n − I n = { n, n + 1 } .(3) E F : The finite group G = Z / Z acts on the set [6] of vertices of the quiver of type E by i i for i = 2 , , , , , . There are 4 orbits: I = { , } , I = { , } , I = { } , and I = { } .(4) D G : The finite group G = Z / Z acts on the set [4] of vertices of the quiver of type D by 2 , , , . There are 2 orbits: I = { } and I = { , , } .The alternating colorings on quivers of type ADE provide that on quivers of type
BCFG as displayedin the right column of Figure 16. A n − B n D n +1 C n E F D G n − nn + 12 n − n − n − n − n > n − n − nn + 1 1 2 n − n − n < > < Figure 16.
Foldings in Dynkin diagrams of finite type (for seed patterns)3.4.
Combinatorics of exchange graphs.
The exchange graph of a cluster pattern is the n -regular (finite or infinite) connected graph whose vertices are the seeds of the cluster pattern andwhose edges connect the seeds related by a single mutation. For example, the exchange graphin Example 3.4 is a cycle graph with 5 vertices. In this section, we recall the combinatorics ofexchange graphs which will be used later. For more details, we refer the reader to [15, 16, 17].As we already have seen in Theorem 3.7, cluster algebras of finite type are classified by Cartanmatrices of finite type. Moreover, for a cluster algebra of finite type, the exchange graph dependsonly on the exchange matrix (see [15]). Because of this reason, we denote by E (Φ) the exchangegraph of a cluster pattern corresponding to the root system Φ.To study the combinatorics of exchange graphs of cluster algebras, we prepare some terminolo-gies. We call a graph over [ m ] bipartite if there is a function ε : [ m ] → { + , −} , called a coloring ,such that for all i and j in [ m ], b i,j = 0 = ⇒ ( ε ( i ) = + ,ε ( j ) = − . Here, b i,j is the adjacency matrix of the graph. For example, every tree is bipartite, but cyclegraphs with an odd number of vertices are not bipartite. Dynkin diagrams of finite type arebipartite since they are trees.Let Φ be a rank n root system of finite type with the set of simple roots Π = { α i | i ∈ [ n ] } andthe set of positive roots Φ + . For every subset J ⊂ [ n ], let Φ( J ) denote the root subsystem of Φspanned by the set of simple roots { α i | i ∈ J } . Note that Φ( J ) may not be irreducible even ifΦ is. Let W be the Weyl group of Φ which is generated by the simple reflections s i = s α i . Sincethe Dynkin diagram of Φ is a bipartite graph, let I + and I − be two parts of the set [ n ]; they aredetermined uniquely up to renaming. Recall that a Coxeter element is the product of all simplereflections. The order h of a Coxeter element in W is called the Coxeter number of Φ. We presentthe known formula of Coxeter numbers h in Table 3 (see [4, Appendix]).Φ A n B n C n D n E E E F G h n + 1 2 n n n − Table 3.
Coxeter numbers
AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 19
Let ∆(Φ) be a simplicial complex whose ground set is Φ ≥− and maximal simplices are called clusters . The dual graph of ∆(Φ) is known to be the exchange graph E (Φ). Recall from [9, 16]that there is a polytopal realization of the simplicial complex ∆(Φ), that is, there is a simpleconvex polytope P (Φ) ⊂ R n such that the dual complex of P (Φ) agrees with ∆(Φ). The polytope P (Φ) is called the generalized associahedron . We denote by F β the facet of the polytope P (Φ)corresponding to a root β ∈ Φ ≥− . Here, a facet of a polytope of dimension n is a face of dimension n − µ Q = µ − µ + of a sequence of mutations where µ ε = Y i ∈ I ε µ i for ε ∈ { + , −} . We call µ Q a Coxeter mutation . It is known from [16, Proposition 3.2] that both µ + and µ − acton the face poset of the polytope P (Φ). Moreover, we have the following properties. Proposition 3.15 (cf. [16, Propositions 2.5, 3.2, and 3.7]) . The following holds.(1) Both µ + and µ − act on the face poset of the polytope P (Φ) .(2) Suppose that h = 2 e is even. The map ( r, i ) µ r Q ( F − α i ) induces a bijection { , , . . . , e } × I → F ( P (Φ)) where F ( P (Φ)) is the set of facets, which are codimension one faces, of the polytope P (Φ) .(3) The face poset of a facet µ r Q ( F − α i ) in P (Φ) is the same as that of the generalized associ-ahedron P (Φ([ n ] \ { i } )) of dimension n − .(4) The facets corresponding to negative simple roots − α , . . . , − α n intersect at a vertex. As a direct consequence of Proposition 3.15, we have the following lemma which will be usedlater.
Lemma 3.16.
Let Σ be a seed in a cluster pattern of finite type with even Coxeter number h = 2 e . Suppose that Σ ∈ µ r Q ( F − α i ) for r ∈ { , , . . . , e } and α i ∈ Π . Then, there exists asequence j , . . . , j ℓ ∈ [ n ] \ { i } which gives a sequence µ j , . . . , µ j ℓ of mutations from µ r Q (Σ t ) to Σ inside a facet µ r Q ( F − α i ) , that is, µ r Q (Σ t ) , µ j ( µ r Q (Σ t )) , ( µ j µ j )( µ r Q (Σ t )) , . . . , ( µ j ℓ · · · µ j )( µ r Q (Σ t )) ∈ µ r Q ( F − α i ) and Σ = ( µ j ℓ · · · µ j )( µ r Q (Σ t )) . Proof.
Since Σ t ∈ F − α i , we have µ r Q (Σ t ) ∈ µ r Q ( F − α i ). Accordingly, both seeds µ r Q (Σ t ) andΣ are contained in the same facet µ r Q ( F − α i ), so there exists a sequence µ j , . . . , µ j ℓ of mutationsfrom µ r Q (Σ t ) to Σ inside µ r Q ( F − α i ) as desired. (cid:3) Example 3.17.
Consider the root system Φ of type A . In this case, the Coxeter number is 4,which is even (cf. Table 3). In Table 4, we present how µ Q acts on the set of facets. Here, weuse the convention that I + = { , } and I − = { } . The corresponding generalized associahedronis presented in Figure 17. We label each facet the corresponding almost positive root. The back-side facets are associated with the set of negative simple roots. As one may see that the faceposets of µ r Q ( F − α i ) are the same as that of the generalized associahedron P (Φ([ n ] \ { i } )). Indeed,the facets µ r Q ( F − α ) and µ r Q ( F − α ) are pentagons, and the facets µ r Q ( F − α ) are squares. ForΣ = F − α ∩ F − α ∩ F − α , we decorate the vertices { µ r Q (Σ) | k = 0 , , } with green. r µ r Q ( F − α ) µ r Q ( F − α ) µ r Q ( F − α )0 F − α F − α F − α F α + α F α F α + α F α F α + α + α F α Table 4.
Computation µ r Q ( F − α i ) for type A
30 BYUNG HEE AN, YOUNGJIN BAE, AND EUNJEONG LEE α + α α + α α α α α + α + α Figure 17.
The type A generalized associahedron Example 3.18.
We consider the generalized associahedron of type D and present four facetscorresponding to the negative simple roots in Figure 18. The facet corresponding to − α iscombinatorially equivalent to P (Φ( { } )) × P (Φ( { } )) × P (Φ( { } )), which is a 3-cube presentedin the boundary. The intersection of these four facets is a vertex sits in the bottom colored ingreen. The Coxeter mutation µ Q acts on the face poset of the permutohedron, especially, fourgreen vertices are in the same orbit. Remark . As we have seen in Example 3.14, bipartite coloring on quivers of type
ADE inducethat on quivers of type
BCFG . Accordingly, if a seed pattern of simply-laced type X gives a seedpattern of type Y via the folding procedure, then the Coxeter mutation of type Y is the same asthat of type X . More precisely, for a globally foldable seed Σ with respect to G defining a clusteralgebra of type X and its Coxeter mutation µ X Q , we have µ Y Q (Σ G ) = ( µ X Q (Σ)) G . Here, µ Y Q is the Coxeter mutation on the seed pattern determined by Σ G .Moreover, Coxeter numbers of X and Y are the same. Indeed, h ( A n − ) = h ( B n ) = 2 n,h ( D n +1 ) = h ( C n ) = 2 n,h ( E ) = h ( F ) = 12 ,h ( D ) = h ( G ) = 6 . In the remaining part of this section, we recall [17] which considers the combinatorics on mu-tations in a more general setting. Let Q be a bipartite quiver and I + and I − be the bipartitedecomposition of the vertex set of Q . Consider the composition µ Q = µ − µ + of a sequence ofmutations where µ ε = Y i ∈ I ε µ i for ε ∈ { + , −} . We call µ Q a Coxeter mutation as before. We enclose this section by recalling the following resultwhich will be used later.
Lemma 3.20 ([17, Theorem 8.8]) . Let Σ t = ( x t , B t ) be an initial seed. Suppose that theexchange matrix B t is the adjacency matrix of a bipartite quiver Q . Then the set { µ r Q (Σ t ) } r ∈ Z ≥ of seeds is finite if and only if the Cartan counterpart C ( B pr t ) is a Cartan matrix of finite type. AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 21 (a) The generalized associahedron of type D .(b) F − α . (c) F − α . (d) F − α . Figure 18.
The generalized associahedron of type D and facets correspondingto some negative simple roots − α , − α , and − α . Moreover, for a quiver Q of finite type, the order the µ Q -action is given by ( h + 2) / if h iseven, or h + 2 otherwise. N -graphs and seeds Let us recall from [8] how to construct a seed from an N -graph G . Each one-cycle in Λ( G )corresponds to a vertex of the quiver, and a monodromy along that cycle gives a coordinate function at that vertex. The quiver is obtained from the intersection data among one-cycles.Moreover, there is an operation in N -graph, called Legendrian mutation , which is a counterpartof the mutation in the cluster structure. The Legendrian mutation is crucial in constructing anddistinguishing N -graphs. In turn, these will give seeds many Lagrangian fillings of Legendrianlinks.4.1. One-cycles in Legendrian weaves.
Let G ⊂ D be a free N -graph and Λ( G ) be the inducedLegendrian weave. We express one-cycles of Λ( G ) in terms of subgraphs of G . Definition 4.1.
A subgraph T ⊂ G is said to be admissible if it satisfies the following conditions: • every vertex of T is at most trivalent, • each univalent vertex in T is a trivalent vertex in G , • each bivalent vertex in T corresponding to a hexagonal point in G connects two oppositeedges in G , and • each trivalent vertex in T corresponding to a hexagonal point in G and connects threeedges in the same color.An admissible graph T ⊂ G is good if it is a (connected) tree and only univalent vertices of T are trivalent vertices in G .For each admissible subgraph T , we can define an oriented immersed loop ℓ ( T ) ⊂ D is defined bypaths whose local pictures look as depicted in Figure 19. Each arc ℓ j ( T ) ⊂ ℓ ( T ) cut by G is labelledas s j ∈ { , . . . , N } , which lifts to the s j -th sheet Γ s j via π D : Γ( G ) → D . By concatenating thelifts, we have an oriented embedded loop γ ( T ) in Λ( G ) and a one-cycle [ γ ] ∈ H (Λ( G ) , Z ) is calleda T -cycle if [ γ ] = [ γ ( T )]. i + i i + i + 1 i i + 1 i + i i + i + i i + i + i i + i + i i + (a) Near a trivalent vertex of G i + 2 i + 1 i + 2 i + 2 i + 2 i + 1 i + 2 i + i + i + i + i + 2 ii ii i i i + i + i + i + i + 1 i + 1 (b) Near a hexagonal point of G Figure 19.
Local configurations on cycles and corresponding arcs of G ⊂ D Example 4.2 ((Long) I -cycles) . For an edge e of G connecting two trivalent vertices, let I ( e ) bethe subgraph of G consisting of a single edge e . Then I ( e ) is a good subgraph of G and the cycle[ γ ( I ( e ))] depicted in Figure 20(a) is called an I -cycle .In general, a linear chain of edges ( e , e , . . . , e n ) satisfying AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 23 • e i connects a trivalent vertex and a hexagonal point for i = 1 , n ; • e i and e i +1 meet at a hexagonal point in the opposite way, see Figure 20(b), for i =2 , . . . , n − I ( e , . . . , e n ), and the cycle [ γ ( I ( e , . . . , e n ))] is called a long I -cycle . SeeFigure 20(b). Example 4.3 ( Y -cycles) . Let e , e , e be monochromatic edges joining a hexagonal point h andtrivalent vertices v i for i = 1 , ,
3. Then the subgraph Y ( e , e , e ) consisting of three edges e , e and e is a good subgraph of G and it defines a cycle [ γ ( Y ( e , e , e ))] called an upper or lower Y -cycle according to the relative position of sheets that edges represent. See Figures 20(c) and20(d). e ii + 1 i i + 1 (a) An I -cycle γ ( I ( e )) e e ii + 2 i + 1 i + 2 i + 1 i + 1 i + 2 i + 2 (b) A long I -cycle γ ( I ( e , e )) i + i + i + i +1 i + i + i + i + i + (c) An upper Y -cycle γ ( Y ( e , e , e )) i i i i + 1 i + 1 i + 1 i + 1 i + 1 i + 1 i iii i i (d) A lower Y -cycle γ ( Y ( e , e , e )) Figure 20. (Long) I - and Y -cyclesOne of the benifit of cycles from admissible subgraphs is that one can keep track how cycles arechanged under the N -graph moves described in Figure 7, especially under Move (I) and Move (II).Note that Move (III) can be decomposed into a sequence of Move (I) and Move (II). Some of suchchanges are given in Figure 21. Then it is easy to check that any T -cycle coming from a goodsubgraph T can be transformed to an I -cycle. Remark . It is important to note that not every cycle can be represented by a subgraph. Forexample, the cycle on the left of the following picture can not be expressed by a subtree but itcan be after Move (I). γ = = γ ( T ) (I)4 BYUNG HEE AN, YOUNGJIN BAE, AND EUNJEONG LEE(I) (I)(I) (II)(II) (II)(II) (II) Figure 21.
Cycles under Move (I) and (II).On the other hand, there might be a one-cycle having two different subgraph presentations asfollows: ∼ ∼
Therefore, there is a bit subtle issue for picking up nice cycles in a consistent way.
Definition 4.5.
Let G ⊂ D be an N -graph, and Λ( G ) be an induced Legendrian surface in J D .A cycle [ γ ] ∈ H (Λ( G )) is good if [ γ ] = [ γ ( T )] for some good T ⊂ G .A tuple of linearly independent good cycles B = { [ γ i ] } i ∈ I in H (Λ( G )) is good if for any pair ofcycles [ γ i ] and [ γ j ], G can be transformed into G ′ via N -graph moves so that two cycles [ γ i ] and[ γ j ] become I -cycles in H (Λ( G ′ )). Remark . Suppose [ γ i ] and [ γ j ] be a pair of cycles in a good tuple B of H (Λ( G )). If G is free,then by Example 2.9, there is no bigon in G up to N -graph moves. So, under this assumption,two corresponding I -cycles in Definition 4.5 intersect at most one trivalent vertex. Definition 4.7.
Let ( G , B ) and ( G ′ , B ′ ) be pairs of an N -graph and good tuples of one-cycles.We say that ( G , B ) and ( G ′ , B ′ ) are equivalent if there is a sequence of moves between G and G ′ inducing moves as depicted in Figure 21 between representatives of cycles in B and B ′ . We denotethe equivalent class of ( G , B ) by [ G , B ]. Remark . For two equivalent pairs ( G , B ) and ( G ′ , B ′ ), all moves between N -graphs G and G ′ can be realized by isotopies betwen Legendrian weaves Λ( G ) and Λ( G ′ ) by Theorem 1.1 in [8], andthen the induced isomorphism H (Λ( G )) ∼ = H (Λ( G ′ )) identifies B with B ′ .4.2. N -graphs and flag moduli space. We recall from [8] a central algebraic invariant M ( G )of the Legendrian weave Λ( G ). The main idea is to consider moduli spaces of constructible sheaves AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 25 associated to Λ( G ). To introduce a legible model for such constructible sheaves, let us consider afull flag, i.e. a nested sequence of subspaces in C N ; F • ∈ { ( F i ) Ni =0 | dim F i = i, F j ⊂ F j +1 , ≤ j ≤ N − , F N = C N } . Definition 4.9. [8] Let G ⊂ D be an N -graph. Let { F i } i ∈ I be a set of closures of connectedcomponents of D \ G , call each closure a face . The framed flag moduli space f M ( G ) is a collectionof flags F Λ( G ) = {F • ( F i ) } i ∈ I in C N satisfying the following:Let F , F be a pair of faces sharing an edge in G i . Then the corresponding flags F • ( F ) , F • ( F )satisfy ( F j ( F ) = F j ( F ) , ≤ j ≤ N, j = i ; F i ( F ) = F i ( F ) . (4.1)Let us consider the general linear group GL N action on M ( G ) by acting on all flags at once.The flag moduli space of the N -graph G is defined by the quotient space (a stack, in general) M ( G ) := f M ( G ) / GL N . Let S h ( D × R ) be the category of constructible sheaves on D × R . Under the identification J D ∼ = T ∞ , − ( D × R ), an N -graph G ⊂ D gives a LegendrianΛ( G ) ⊂ J D ∼ = T ∞ , − ( D × R ) ⊂ T ∞ ( D × R ) . This can be used to define a Legendrian isotopy invariant S h G ) ( D × R ) of S h ( D × R ) consistingof constructible sheaves • whose singular support at infinity lies in Λ( G ) ⊂ T ∞ ( D × R ), • whose microlocal rank is one, and • which are zero near D × {−∞} .See [21, 29] for the detail. Theorem 4.10 ([8, Theorem 5.3]) . The flag moduli space M ( G ) is isomorphic to S h G ) ( D × R ) .Hence M ( G ) is a Legendrian isotopy invariant of Λ( G ) .Remark . Indeed, the actual theorem is about a connected surface, not only for D .Let λ = λ β be a Legendrian in J S , which gives us an ( N − X = ( X , . . . , X N − )in S which given by the alphabet σ , . . . , σ N − of the braid word β . Let { f j } j ∈ J be the set ofclosures of connected components of S \ X . The flags F λ = {F • ( f j ) } j ∈ J in C N satisfying exactlythe same conditions in (4.1) will be called simply by flags on λ . It is well known that the modulispace M ( X ) of such flags F λ up to GL N is isomorphic to S h λ ( S × R ) which is a Legendrianisotopy invariant, see [29, Theorem 1.1]. Definition 4.12.
Let G ⊂ D be an N -graph, and let F λ be flags adapted to λ ⊂ J ∂ D givenby ∂ G . An N -graph G is good , if the flags F λ uniquely determine flags F Λ( G ) in Definition 4.9.Note that G ( a, b, c ) in the introduction is good in an obvious way. If an N -graph G ⊂ D isgood and [ G ] = [ G ′ ], then G ′ is also good.4.3. N -graphs and seeds. Let G ⊂ D be an N -graph, and B = { [ γ i ] } i ∈ [ n ] ⊂ H (Λ( G )) be a goodtuple of cycles. For two cycles [ γ i ] and [ γ j ], let i ([ γ i ] , [ γ j ]) be the algebraic intersection number in H (Λ( G )) which can be computed explicitly as follows: without loss of generality, we may assumethat both [ γ i ] and [ γ j ] are I -cycles represented by γ ( I ( e )) and γ ( I ( e ′ )) for some edges e and e ′ in G , respectively. Suppose that e and e ′ intersect at the vertex in G . Then two representatives of γ i and γ j look locally as depicted in Figure 22 and their intersection is defined to be ± Definition 4.13.
For each a pair ( G , B ) of an N -graph and a good tuple of cycles, we define aquiver Q = Q ( G , B ) as follows:(1) the set of vertices is [ n ] where B = { [ γ i ] | i ∈ [ n ] } ⊂ H (Λ( G )), and e ′ e γ i γ j γ i γ j (+) (a) Positively intersecting I -cycles e e ′ γ i γ j γ i γ j ( − ) (b) Negatively intersecting I -cycles Figure 22. I -cycles with intersections.(2) the ( i, j )-entry b i,j for B ( Q ) = ( b i,j ) is the algebraic intersection number between [ γ i ] and[ γ j ] b i,j = i ([ γ i ] , [ γ j ]) . In order to assign a cluster variable to each one-cycle. Let us review the microlocal monodromyfunctor from [29] µ mon : S h • Λ → L oc • (Λ) . In our case, this functor sends microlocal rank-one sheaves F Λ( G ) ∈ S h G ) ( D × R ) , or equiv-alently, flags {F • ( F i ) } i ∈ I ∈ M ( G ) to rank-one local systems µ mon( F Λ( G ) ) on the Legendriansurface Λ( G ). Then (cluster) variables x for the triple ( G , B , F Λ( G ) ) are defined by x = (cid:0) µ mon( F Λ( G ) )([ γ ]) , . . . , µ mon( F Λ( G ) )([ γ n ]) (cid:1) . Let us denote the above assignment byΨ( G , B , F Λ( G ) ) = ( x (Λ( G ) , B , F Λ( G ) ) , Q (Λ( G ) , B )) . By the Legendrian isotopy invariance of S h G ) ( D × R ) in [21], and the functorial property ofthe microlocal monodromy functor µ mon [29], the assignment Ψ is well-defined up to isotopy ofΛ( G ). That is, if two triples (Λ( G ) , B , F Λ( G ) ) and (Λ( G ′ ) , B ′ , F Λ( G ′ ) ) are Legendrian isotopic, thenthey give us the same seed via Ψ.Especially when an N -graph G is good, see Definition 4.12, F Λ( G ) is determined by the flags F λ ∈ S h λ ( ∂ D × R ) at the boundary, where the Legendrian link λ is given by ∂ G . So we have Theorem 4.14. [8, § Let G ⊂ D be a good N -graph with a good tuple B of cycles in H (Λ( G )) , and with flags F λ on λ ⊂ J S at the boundary. Then the assignment Ψ to a seed in acluster structure Ψ( G , B , F λ ) = ( x (Λ( G ) , B , F λ ) , Q (Λ( G ) , B )) is well-defined up to Legendrian isotopy. As a corollary, the seed Ψ( G , B , F λ ) can be used to distinguish a pair of Legendrian surfacesand hence, by Lemma 2.1, a pair of Lagrangian fillings. Corollary 4.15.
As in the above setup, if two triples ( G , B , F λ ) , ( G ′ , B ′ , F λ ) with the same bound-ary condition define different seeds, then two induced Lagrangian fillings π ◦ ι (Λ( G )) , π ◦ ι (Λ( G ′ )) bounding ι ( λ ) are not exact Lagrangian isotopic to each other. The monodromy µ mon( F Λ( G ) ) along a loop [ γ ] ∈ H (Λ( G )) can be obtained by restricting theconstructible sheaf F Λ( G ) to a tubular neighborhood of γ . Let us investigate how the monodromycan be computed explicitly in terms of flags {F • ( F i ) } i ∈ I .Let us consider an I -cycle [ γ ] represented by a loop γ ( e ) for some monochromatic edge e as inFigure 23(a). Let us denote four flags corresponding to each region by F , F , F , F , respectively. AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 27
Suppose that e ⊂ G i , then by the construction of flag moduli space M ( G ), a two-dimensionalvector space V := F i +1 ( F ∗ ) / F i − ( F ∗ ) is independent of ∗ = 1 , , ,
4. Moreover, F i ( F ∗ ) / F i − ( F ∗ )defines a one-dimensional subspace v ∗ ⊂ V for ∗ = 1 , , ,
4, satisfying v = v = v = v = v . Then µ mon( F Λ( G ) ) along the one-cycle [ γ ( e )] is defined by the cross ratio µ mon( F Λ( G ) )([ γ ]) := h v , v , v , v i = v ∧ v v ∧ v · v ∧ v v ∧ v . Suppose that local flags { F j } j ∈ J near the upper Y -cycle [ γ U ] look like in Figure 23(b). Let G i and G i +1 be the N -subgraphs in red and blue, respectively. Then the 3-dimensional vector space V = F i +2 ( F ∗ ) / F i − ( F ∗ ) is independent of ∗ ∈ J . Now regard a, b, c and A, B, C are subspacesof V of dimension one and two, respectively. Then the microlocal monodromy along the Y -cycle[ γ U ] becomes µ mon( F Λ( G ) )([ γ U ]) := B ( a ) C ( b ) A ( c ) B ( c ) C ( a ) A ( b ) . Here B ( a ) can be seen as a paring between the vector a and the covector B .Now consider the lower Y -cycle [ γ L ] whose local flags given as in Figure 23(c). We already haveseen that the orientation convention of the loop in Figure 20 for the upper and lower Y -cycle isdifferent. Then microlocal monodromy along [ γ L ] follows the opposite orientation and becomes µ mon( F Λ( G ) )([ γ L ]) := C ( a ) B ( c ) A ( b ) C ( b ) B ( a ) A ( c ) . Here, B ( a ) is a pairing between the vector B and covector a which is the same as the above. γv v v v (a) I -cycle with flags. ( b, B ) ( a , A ) ( c , C ) ( a , a b ) ( c , b c ) ( a , a c ) ( c , a c ) ( b , a b ) ( b , b c ) γ U (b) Upper Y -cycle with flags. ( b, B ) ( a , A ) ( c , C ) ( A B , A ) ( B C , C ) ( A C , A ) ( A C , C ) ( A B , B ) ( B C , B ) γ L (c) Lower Y -cycle with flags. Figure 23. I - and Y -cycles with flags.4.4. Legendrian mutations in N -graphs. Let us define an operation called (
Legendrian ) muta-tion on N -graphs G which corresponds to a geometric operation on the induced Legendrian surfaceΛ( G ) that producing a smoothly isotopic but not necessarily Legendrian isotopic to Λ( G ), see [8,Definition 4.19]. Note that operation has an intimate relation with the wall-crossing phenomenon[2], Lagrangian surgery [26], and quiver (or cluster) mutations [14]. Definition 4.16. [8] Let G be a (local) N -graph and e ∈ G i ⊂ G be an edge between two trivalentvertices corresponding to an I -cycle [ γ ] = [ γ ( e )]. The mutation µ γ ( G ) of G along γ is obtained byapplying the local change depicted in the left of Figure 24.For the Y -cycle, the Legendrian mutation becomes as in the right of Figure 24. Note that themutation at Y -cycle can be decomposed into a sequence of Move (I) and Move (II) together witha mutation at I -cycle.Let us remind our main purpose of finding exact embedded Lagrangian fillings for a Legen-drian links. The following lemma guarantees that Legendrian mutation preserves the embeddingproperty of Lagrangian fillings. µ γ µ γ ′ γ γ ′ (a) A mutation along I -cycle. µ γ µ γ ′ γ γ ′ (b) A mutations along Y -cycle. Figure 24.
Legendrian mutations at I - and Y -cycles. Proposition 4.17. [8, Lemma 7.4]
Let G ⊂ D be a free N -graph. Then mutation µ ( G ) at any I -or Y -cycle is again free N -graph. Proposition 4.18.
Let G ⊂ D be a good N -graph. Then mutation µ γ ( G ) at I -cycle γ is againgood N -graph.Proof. The proof is straightforward from the notion of the good N -graph in Definition 4.12 andof the Legendrian mutation depicted in Figure 24(a). Note that the Legendrian mutation µ γ ( G )at Y -cycle γ is also good, since µ γ ( G ) is a composition of Moves (I) and (II), and a mutation at I -cycle. (cid:3) An important observation is the Legendrian mutation on ( G , B ) induces a cluster mutation onthe induced seed ( x (Λ( G ) , B , F λ ) , Q (Λ( G ) , B )). Proposition 4.19 ([8, § . Let G ⊂ D be a good N -graph and B be a good tuple of cycles in H (Λ( G )) . Let µ γ i ( G , B ) be a Legendrian mutation of ( G , B ) along a one-cycle γ i then the followingholds: for flags F λ on λ , Ψ( µ γ i ( G , B ) , F λ ) = µ i (Ψ( G , B , F λ )) . Here, µ i is the cluster X -mutation at the vertex i ( cf. Remark 3.5 ) . Lagrangian fillings for of Legendrians type
ADE
Tripods.
Let λ ⊂ J S be a Legendrian knot or link which bounds a Legendrian surface Λ( G )in J D for some free N -graph G . We fix a good tuple B of cycles in the sense of Definition 4.5,and fix flags F λ on λ . Then by Theorem 4.14, we obtain a seed Ψ( G , B , F λ ) which is a pair of aset of cluster variables x (Λ( G ) , B , F λ ) and a quiver Q (Λ( G ) , B ).We say that the pair ( G , B ) is of finite type or of infinite type if so is the cluster algebra definedby Q (Λ( G ) , B ). In particular, it is said to be of type ADE if the quiver Q (Λ( G ) , B ) is of type A n , D n , E , E or E (see Definition 3.8). Braid words β of Legendrians, N -graphs and good tuplesof cycles ( G , B ) of type ADE are depicted in Table 5.One can generalize these quivers of type
ADE as follows:
Definition 5.1 (Tripod quiver) . For a, b, c ≥
1, the tripod Q ( a, b, c ) of type ( a, b, c ) is a bipartitequiver such that(1) the set of vertices is [ n ] for n = a + b + c − A a , A b , and A c , and(3) the vertex where A a , A b , and A c are glued together is called the central vertex , labelled as1 and colored as +.We define an N -graph G ( a, b, c ) on D as the concatenation of three N -graphs G ( A a ) , G ( A b ) and G ( A c ) by making one Y -cycle γ and define a good tuple B ( a, b, c ) of cycles as the union of cyclesin three N -graphs. The N -graph obtained by switching colors from G ( a, b, c ) and the induced setof chosen cycles will be denoted by ¯ G ( a, b, c ) and ¯ B ( a, b, c ), respectively.The pictorial definitions of Q ( a, b, c ) , G ( a, b, c ) and B ( a, b, c ) are depicted in Figure 25. AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 29 β ( G , B ) Q σ n +31 γ n γ n − · · · γ γ γ n σ σ σ σ σ σ n − γ n − γ n γ n − γ n − n − n − n − nn − n − σ σ σ σ σ σ n − γ γ γ γ γ n −
43 2 5 n Table 5. N -graphs and their quivers of type ADE
It is obvious that if a, b or c is one, then it is the same as A n for n = a + b + c − a, b or c is two include quivers of type D n and E n .Notice that the boundary of the Legendrian weave Λ( G ( a, b, c )) denoted by λ ( a, b, c ) is expressedas the braid word λ ( a, b, c ) = Cl( β ( a, b, c )) , β ( a, b, c ) = σ σ a +11 σ σ b +11 σ σ c +11 . Then this braid is equivalent to the following: β ( a, b, c ) = σ σ a +11 σ σ b +11 σ σ c +11 = σ σ ( σ σ ) σ a σ b − σ c ( σ σ ) σ = ∆ σ σ a σ b − σ c ∆ . aa +1 a +2 a +3 a + b − a + ba + b +1 a + b +2 a + b + c − (a) The tripod Q ( a, b, c ) a + b + c + (b) ( G ( a, b, c ) , B ( a, b, c )) a + b + c + (c) (¯ G ( a, b, c ) , ¯ B ( a, b, c )) Figure 25.
The tripod N -graph G ( a, b, c ) and the chosen set B ( a, b, c ) of cyclesHence λ ( a, b, c ) ⊂ J S corresponds to the rainbow closure of the braid β ( a, b, c ) = σ σ a σ b − σ c . a b − cβ ( a, b, c ) r = | {z } r Remark . One can easily check the quiver Q brick ( a, b, c ) from the brick diagram of β ( a, b, c )described in [19] looks as follows: Q brick ( a, b, c ) = σ σ σ σ · · · σ σ σ σ σ · · · σ σ σ σ σ · · · σ σ a z }| { | {z } b − c z }| { · · · · · ·· · · Then this quiver Q brick ( a, b, c ) is obviously mutation equivalent to the bipartite quiver Q ( a, b, c ). AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 31
It is not hard to check that β (1 , b, c ) is a stabilization of β ( A n ) = σ n +31 for n = b + c − β ( A n ) = σ n +31 = σ b +11 σ c +11 (S) −→ σ σ σ σ b +11 σ σ c +11 = β (1 , b, c ) . Lemma 5.3.
The N -graph G (1 , b, c ) is a stabilization of G ( A n ) for n = b + c − .Proof. According to Remark 2.7 and Figure 9, a stabilization of G ( A n ) is given as the secondpicture in Figure 26. Then by adding an annular N -graph corresponding to a sequence of (RIII),we obtain the third, which produces the fourth by applying the following generalized push-throughmove. (II ∗ ) Now we add an annular N -graph consisting of (RIII)’s as above to obtain the fifth N -graph inFigure 26, which is the same as G (1 , b, c ) as desired up to Move (II) at the center. (cid:3) Definition 5.4.
Let ( G , B ) and ( G ′ , B ′ ) be pairs of N -graphs and good tuples of cycles. We saythat ( G , B ) is Legendrian mutation equivalent to ( G ′ , B ′ ) if there exists a sequence µ of Legendrianmutations which sends ( G , B ) to ( G ′ , B ′ ) up to equivalence. That is,[ µ ( G , B )] = [( G ′ , B ′ )] . In particular, ( G , B ) is said to be of type X or of type ( a, b, c ) if it is Legendrian mutationequivalent to ( G ( X ) , B ( X )) or ( G ( a, b, c ) , B ( a, b, c )), respectively.5.2. Coxeter mutation for tripods.
For a bipartite quiver Q , we have two sets of vertices I + and I − so that all edges are oriented from I + to I − . By definition, for a tripod Q ( a, b, c ), thecentral vertex 1 is lying in I + . Let µ + and µ − be sequences of mutations defined by compositions ofmutations corresponding to each and every vertex in I + and I − , respectively. A Coxeter mutation µ Q is the composition µ Q = µ − µ + = Y i ∈ I − µ i · Y i ∈ I + µ i . Remark . For any sequence µ of mutations, we will use the right-to-left convention. Namely,the rightmost mutation will be applied first on the quiver Q .Similarly, we define the Legendrian Coxeter mutation, which will be denoted by µ G , on abipartite N -graph G as follows: Definition 5.6 (Legendrian Coxeter mutation) . For a bipartite N -graph G with decomposed setsof cycles B = B + ∪ B − , we define the Legendrian Coxeter mutation µ G as the composition ofLegendrian mutations µ G = Y γ ∈ B − µ γ · Y γ ∈ B + µ γ . Lemma 5.7.
The effect of the Legendrian Coxeter mutation on ( G ( A n ) , B ( A n )) is the clockwise πn +3 -rotation.Proof. We may assume that the Coxeter element µ G can be represented by the sequence µ G = µ − µ + = ( µ γ µ γ µ γ · · · )( µ γ µ γ µ γ . . . ) . Then the action of µ G on G ( A n ) is as depicted in Figure 27, which is nothing but the clockwise πn +3 -rotation of the original N -graph ( G ( A n ) , B ( A n )) as claimed. (cid:3) Remark . The order of the Coxeter mutation is either ( n + 3) / n is odd or n + 3 otherwise.Since the Coxeter number h = n + 1 for A n , this verifies Lemma 3.20. G ( A n ) = = G (1 , b, c ) (S) L e g e n d r i a n i s o t o p y (II ∗ ) L e g e n d r i a n i s o t o p y (II ∗ )(II) Figure 26.
The N -graph G (1 , b, c ) is a stabilization of G ( A n ) for n = b + c − G , B , F λ ) be a triple of a good N -graph, a good tuple of cycles and flags F λ on λ . Supposethat the quiver Q ( G , B ) is bipartite and µ G ( G , B ) is well-defined. Then by Proposition 4.19, we AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 33 · · · ( G ( A n ) , B ( A n )) · · · µ + ( G ( A n ) , B ( A n )) ··· µ G ( G ( A n ) , B ( A n )) µ + µ − Figure 27.
Legendrian Coxeter mutation µ G on ( G ( A n ) , B ( A n ))have Ψ( µ G ( G , B ) , F λ ) = µ Q (Ψ( G , B , F λ )) . In particular, for quivers of type A n or tripods we have the following corollary. Corollary 5.9.
For each n ≥ and a, b, c ≥ , the Legendrian Coxeter mutation µ G on ( G ( A n ) , B ( A n )) or ( G ( a, b, c ) , B ( a, b, c )) corresponds to the Coxeter mutation µ Q on Q ( A n ) or Q ( a, b, c ) , respec-tively. By the mutation convention mentioned above, for each tripod G ( a, b, c ), we always take amutation at the central Y -cycle γ first. After the Legendrian mutation on ( G ( a, b, c ) , B ( a, b, c ))at γ , we have the N -graph on the left in Figure 28(a). Then there are three shaded regions thatwe can apply the generalized push-through moves so that we obtain the N -graph on the right inFigure 28(a). Notice that in each triangular shaded region, the N -subgraph looks like the N -graphof type A a − , A b − or A c − . Moreover, the mutations corresponding to the rest sequence is just acomposition of Coxeter mutations of type A a − , A b − and A c − , which are essentially the same asthe clock wise rotations. Therefore, the result of the Coxeter mutation will be given as depictedin Figure 28(b).Then one can observe that this is very similar to the original N -graph G ( a, b, c ). Indeed, theinside is identical to G ( a, b, c ) but the colors are switched, which is ¯ G ( a, b, c ) by definition. Thecomplement of ¯ G ( a, b, c ) in µ Q ( G ( a, b, c ) , B ( a, b, c )) is an annular N -graph. Definition 5.10 (Coxeter padding) . For each triple a, b, c , the annular N -graph depicted inFigure 29 is denoted by C ( a, b, c ) and called the Coxeter padding of type ( a, b, c ). We also denotethe Coxeter padding with color switched by ¯ C ( a, b, c ).Notice that two Coxeter paddings C ( a, b, c ) and ¯ C ( a, b, c ) can be glued without any ambiguityand so we can also pile up Coxeter paddings C ( a, b, c ) and ¯ C ( a, b, c ) alternatively as many timesas we want.We also define the concatenation of the Coxeter padding ¯ C ( a, b, c ) on the pair ( G ( a, b, c ) , B ( a, b, c ))as the pair ( G ′ , B ′ ) such that(1) the N -graph G ′ is obtained by gluing ¯ C ( a, b, c ) on G ( a, b, c ), and(2) the tuple B ′ of cycles is the set of I - and Y -cycles identified with B ( a, b, c ) in a canonicalway. Proposition 5.11.
The Legendrian Coxeter mutation on ( G ( a, b, c ) , B ( a, b, c )) or (¯ G ( a, b, c ) , ¯ B ( a, b, c )) is given as the concatenation µ G ( G ( a, b, c ) , B ( a, b, c )) = C ( a, b, c )(¯ G ( a, b, c ) , ¯ B ( a, b, c )) ,µ G (¯ G ( a, b, c ) , B ( a, b, c )) = ¯ C ( a, b, c )( G ( a, b, c ) , B ( a, b, c )) . Proof.
This follows directly from the above observation. (cid:3) ∗ ) (a) After the mutation at the central vertex(b) After Legendrian Coxeter mutation Figure 28.
Legendrian Coxeter mutation for ( G ( a, b, c ) , B ( a, b, c ))It is important that this proposition holds only when we take the Legendrian Coxeter muta-tion on the very standard N -graph with the tuple of cycles ( G ( a, b, c ) , B ( a, b, c )). Otherwise, theLegendrian Coxeter mutation will not be expressed as simple as above. Theorem 5.12.
For a, b, c ≥ with a + b + c ≤ , The Legendrian knot or link λ ( a, b, c ) in J S admits infinitely many distinct exact embedded Lagrangian fillings.Proof. By Proposition 5.11, the effect of the Legendrian Coxeter mutation on ( G ( a, b, c ) , B ( a, b, c ))is just to attach the Coxeter padding on (¯ G ( a, b, c ) , ¯ B ( a, b, c )). In particular, as mentioned earlier,for each r ≥
0, the iterated Legendrian Coxeter mutation µ r G ( G ( a, b, c ) , B ( a, b, c ))is well-defined. Each of these N -graphs define a Legendrian weave Λ( µ r G ( G ( a, b, c ) , B ( a, b, c ))),whose Lagrangian projection is a Lagrangian filling L r ( a, b, c ) := ( π ◦ ι )(Λ( µ r G ( G ( a, b, c ) , B ( a, b, c )))as desired. Therefore it suffices to prove that Lagrangians L r ( a, b, c ) for r ≥ a + b + c ≤ a + b + c ≤
1, or equivalently, Q ( a, b, c ) is of infinite type. Then the orderof the Coxeter mutation is infinite by Lemma 3.20 and so is the order of the Legendrian Coxeter AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 35 (a) C ( a, b, c ) (b) ¯ C ( a, b, c ) Figure 29.
Coxeter paddings C ( a, b, c ) and ¯ C ( a, b, c )mutation by Corollary 5.9. In particular, for fixed flags F λ on λ , the set (cid:8) Ψ( µ r G ( G ( a, b, c ) , B ( a, b, c )) , F λ ) | r ≥ (cid:9) is the set of infinitely many pairwise distinct seeds in the cluster pattern for Q ( a, b, c ). Hence byCorollary 4.15, we have pairwise distinct Lagrangian fillings L r ( a, b, c ). (cid:3) N -graphs of type ADE . In this section, we will prove one of the main theorem.
Theorem 5.13.
Let λ be a Legendrian knot or link which is either λ ( A n ) or λ ( a, b, c ) of type ADE . Then it admits exact embedded Lagrangian fillings as many as seeds in its seed pattern ofthe same type.
Indeed, this theorem follows from the generalized questions.
Question . For given N -graph G with a chosen set B of cycles, can we take a Legendrianmutation as many times as we want? Or equivalently, after applying a mutation µ k on ( G , B ), isthe tuple µ k ( B ) still good in µ k ( G )?This question has been raised previously in [8, Remark 7.13]. One of the main reason makingthe question nontrivial is that the potential difference of geometric and algebraic intersectionsbetween two cycles. More concretely, two cycles γ and γ as shown in Figure 30, can never beisotoped off to each other but their signed intersections following the rule in Figure 22 vanishes.Hence in the corresponding quiver to the first local N -graph, there are no arrows between thecorresponding vertices 1 and 2. However, after a sequence of Move (II), we can deform γ into γ ( e ) for an edge e as depicted in the third picture of Figure 30. The mutation µ γ ( e ) transforms γ to γ ′ , which is not good and so it is not clear how to define a mutation µ γ ′ .Instead of attacking this question directly, we will prove the following: Proposition 5.15.
Let λ be as above and F λ be flags on λ . Suppose that Σ is a seed in the seedpattern of the same type with the initial seed Σ t = ( Ψ( G ( A n ) , B ( A n ) , F λ ) λ = λ ( A n );Ψ( G ( a, b, c ) , B ( a, b, c ) , F λ ) λ = λ ( a, b, c ) . Then λ admits either an N -graph ( G , B ) on D such that G is either a -graph if λ = λ ( A n ) or a -graph if λ = λ ( a, b, c ) , and Σ = Ψ( G , B , F λ ) . Under the aid of this proposition, one can prove Theorem 5.13. γ γ γ γ ′ γ ′ (II) (II) ◦ (II) µ γ Figure 30.
Non-disjoint cycles γ and γ with signed intersection number zero,and a mutation µ e . Proof of Theorem 5.13.
Let λ be given as above. Then by Proposition 5.15, we have pairs of N -graphs and good tuples of cycles which have a one-to-one correspondence Ψ with seeds in theseed pattern of Q ( a, b, c ). Hence any pair of the Lagrangian fillings coming from these N -graphsis never exact Lagrangian isotopic by Corollary 4.15. This completes the proof. (cid:3) We will use the following observations: let P (Φ) be the generalized associahedron for the rootsystem Φ of type ADE (cf. Theorem 3.7 and § ≥− .(3) For the initial seed Σ t , we may assume that the facets of codimension one including Σ t correspond to negative simple roots. Namely, there are exactly n -facets F = { F − α i | α i ∈ Π } . (4) The orbits of F under the action of the Legendrian Coxeter mutation µ Q exhaust all facets. Proof of Proposition 5.15.
For a Legendrian link λ of type ADE , we fix flags F λ on λ . Let usdefine the initial 2-graph or 3-graph with the chosen tuple of cycles ( G t , B t ) as( G t , B t ) = ( ( G ( A n ) , B ( A n )) λ = λ ( A n );( G ( a, b, c ) , B ( a, b, c )) λ = λ ( a, b, c ) , which defines the initial seed Σ t via ΨΣ t = Ψ( G t , B t , F λ ) = ( x (Λ( G t ) , B t , F λ ) , Q (Λ( G t ) , B t )) . Suppose that Σ is a seed in the cluster pattern. Then we need to to prove that there exists an N -graph ( G , B ) such that Ψ( G , B , F λ ) = Σ.By Proposition 3.15 and Lemma 3.16, there exist an integer r and a sequence µ ′ of mutationssuch that Σ = µ ′ ( µ r Q (Σ t )) , where µ ′ joins µ r Q (Σ t ) and Σ inside a facet.If λ = λ ( A n ), then µ G acts on ( G t , B t ) as the (cid:16) πn +2 (cid:17) -rotation, which obviously commutes withLegendrian mutation µ ′ . Hence it suffices to show the well-definedness of µ ′ ( G t , B t ). AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 37
Otherwise, as seen earlier, the action of the Legendrian Coxeter mutation µ r G on ( G t , B t ) isobtained by the concatenation of sequences of C = C ( a, b, c ) and ¯ C = ¯ C ( a, b, c ) to either ( G t , B t )or (¯ G t , ¯ B t ). µ r G ( G t , B t ) = ( C ¯ C · · · ¯ C ( G t , B t ) r is even , C ¯ C · · · C (¯ G t , ¯ B t ) r is odd . Let us regard the sequence µ ′ of mutations as the sequence of Legendrian mutations. Since theconcatenation of C or ¯ C do not touch any chosen cycle in B t , two operations—the concatenationof C or ¯ C , and the mutation µ ′ — commute. Therefore µ ′ ( µ r G ( G t , B t )) = ( C ¯ C · · · ¯ C ( µ ′ ( G t , B t )) r is even , C ¯ C · · · C ( µ ′ (¯ G t , ¯ B t )) r is odd , and the proposition follows if µ ′ ( G t , B t ) and µ ′ (¯ G t , ¯ B t ) are well-defined. Since G t and ¯ G t areessentially the same, it suffices to show the well-definedness of µ ′ ( G t , B t ) as before.Now we will prove the well-definedness of µ ′ ( G t , B t ) in both cases by using induction on n .Suppose that µ ′ is a sequence of mutations in a facet F β for some β ∈ Φ ≥− , and µ r Q ( F − α i ) = F β for some α i ∈ Π. Then the facet F β is combinatorially equivalent to the lower dimensionalgeneralized associahedron F β ∼ = P (Φ([ n ] \ { i } )) ∼ = P (Φ ) × · · · × P (Φ m ) . Here, Φ([ n ] \ { i } ) is not necessarily irreducible and we denote by Φ , . . . , Φ ℓ , ℓ ≤ n ] \{ i } ) = Φ ×· · ·× Φ ℓ . Moreover, in terms of quivers, if we denote the connectedcomponents of Q \ { i } by Q (1) , . . . , Q ( ℓ ) , then we may say that Φ j and Q ( j ) are of the same type.Therefore the sequence µ ′ of mutations can be decomposed into µ (1) , . . . , µ ( ℓ ) on Q (1) , . . . , Q ( ℓ ) ,respectively.Similarly, in N -graph G t , the i -th cycle [ γ i ] separates ( G t , B t ) into at most three parts { ( G , B ) , . . . , ( G ℓ , B ℓ ) } , as seen in Figure 31. This means that µ ( j ) ( G j , B j ) is well-defined for all 1 ≤ j ≤ ℓ = ⇒ µ ′ ( G t , B t ) is well-defined . Indeed, if λ = λ ( A n ), then we have the following two cases:(1) if γ i corresponds to a bivalent vertex, then for some 1 ≤ r, s with r + s + 1 = n , we havetwo 2-subgraphs { ( G ( A r ) , B ( A r )) , ( G ( A s ) , B ( A s )) } ;(2) if γ i corresponds to a leaf, then we have the 2-subgraph { ( G ( A n − ) , B ( A n − ) } . Otherwise, if λ = λ ( a, b, c ), then we have the following three cases:(1) if γ i corresponds to the central vertex, then we have three 3-subgraphs { ( G (3) ( A a − ) , B (3) ( A a − )) , ( G (3) ( A b − ) , B (3) ( A b − )) , ( G (3) ( A c − ) , B (3) ( A c − )) } , (2) if γ i corresponds to a bivalent vertex, then for some 1 ≤ r, s with r + s + 1 = a , up topermuting indices a, b, c , we have two 3-subgraphs { ( G (3) ( A s ) , B (3) ( A s )) , ( G ( r, b, c ) , B ( r, b, c )) } , (3) otherwise, if γ i corresponds to a leaf, then up to permuting indices a, b, c , we have the3-subgraph { ( G ( a − , b, c ) , B ( a − , b, c )) } . Some of separations are depicted in Figure 31. Here, G (3) ( A s ) is the 3-graph which looks like the2-graph G ( A s ). Indeed, there are no edges in red and so the well-definedness of each mutation on G (3) ( A s ) is the same as G ( A s ). Therefore we can safely replace G (3) ( A s ) with G ( A s ) for the proof.However, for each 1 ≤ j ≤ ℓ , the N -subgraph G j is either G ( a ′ , b ′ , c ′ ) with n ′ = a ′ + b ′ + c ′ − G ( A n ′ ), where n ′ < n . Therefore the proposition follows from the induction on n once we establishthe initial step, which is when n = 1, that is, either( G (1 , , , B (1 , , G ( A ) , B ( A )) . Since there are no obstructions for mutations on these N -graphs, we are done for the initialcondition for the induction. (cid:3) → ( G ( ) ( A a − ) , B ( ) ( A a − )) ( G ( ) ( A b − ) , B ( ) ( A b − )) ( G ( ) ( A c − ) , B ( ) ( A c − ) ) → ( G ( r, b, c ) , B ( r, b, c )) ( G (3) ( A s ) , B (3) ( A s )) Figure 31.
Separations of ( G ( a, b, c ) , B ( a, b, c )) at γ i Remark . In the above proof, it is not claimed that two mutations µ ′ and µ G commute. Indeed,if we first mutate ( G t , B t ) via µ ′ , then the result may not look like either ( G t , B t ) or (¯ G t , ¯ B t )and hence µ G will not work as expected.6. Lagrangian fillings admitting cluster structures of type
BCFG
In this section, we will construct cluster structures of type
BCFG on certain N -graphs by usingthe folding of N -graphs. Throughout this section, let us assume that a triple ( X , Y , G ) is one of( A n − , B n , Z / Z ) , ( D n +1 , C n , Z / Z ) , ( E , F , Z / Z ) , ( D , G , Z / Z )and that the group G is generated by τ . Remark . For Q ( D n +1 ) = Q ( n − , , , Q ( E ) = Q (2 , ,
3) and Q ( D ) = Q (2 , , AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 39 ( G , B ) ( G , B ) ττ (a) Z / Z -action on ( G , B ) of type A n − ( G , B ) ( G , B ) ( G , B ) τ ττ (b) Z / Z -action on ( G , B ) of type D Figure 32.
Rotation actions on N -graphs of type A n − and D Rotations and N -graphs of type A n − and D . Let ( G , B ) be a pair of a 2- or 3-graphand a good tuple of cycles of type X = A n − or D , respectively. We define a new pair ( τ ( G ) , τ ( B ))such that τ ( G ) and τ ( B ) are obtained by the (2 π/N )-rotation on ( G , B ).We say that ( G , B ) is G -admissible if(1) the N -graph G has the (2 π/N )-rotation symmetry so that τ ( G ) = G ,(2) the tuples of cycles B and τ ( B ) are identical up to relabelling as follows: if X = A n − , γ i ↔ γ n − i , and if X = D , γ → γ , γ → γ , γ → γ . In particular, τ preserves γ n if X = A n − and γ if X = D . Figure 33 shows examples andnon-examples of G -admissible N -graphs. (a) G -admissible N -graphs (b) non- G -admissible N -graphs Figure 33. G -admissible or non- G -admissible N -graphs for Legendrians of type A n − and D Remark . Notice that the Coxeter padding C for G ( A n − ) is empty and C (2 , ,
2) has obviouslythe Z / Z -rotational symmetry. For each G -admissible ( G , B ) of type A n − or D , so is the following C · · · ¯ CC ( G , B ) or ¯ C · · · ¯ CC ( G , B ) . Lemma 6.3.
Let Q be a quiver of type A n − . Suppose that Q is invariant under the action τ ( i ) = 2 n − i for all i ∈ [2 n − . Then there is no oriented cycle of the form j → i → τ ( j ) → τ ( i ) → j for any i, j = n .Proof. It is well known that any minimal cycle in Q is of length 3. Therefore, if such an orientedcycle exists, then there must be an edge i − τ ( i ) or j − τ ( j ) in Q . Hence b i,τ ( i ) = 0 or b j,τ ( j ) = 0for B = ( b k,ℓ ) = B ( Q ).This is impossible because Q is Z / Z -admissible and so b i,τ ( i ) = b τ ( i ) ,τ ( τ ( i )) = b τ ( i ) ,i = − b i,τ ( i ) = ⇒ b i,τ ( i ) = 0 . Therefore we are done. (cid:3)
Proposition 6.4.
Let ( G , B ) be of type A n − as above. If ( G , B ) is Z / Z -admissible, then so isthe quiver Q (Λ( G ) , B ) .Proof. For Q = Q (Λ( G ) , B ), since the generator τ ∈ Z / Z acts on B as τ ( γ i ) = γ n − i , we have the Z / Z -action on the set [2 n −
1] of vertices of Q as τ ( i ) = 2 n − i and therefore for each i ∈ [2 n − i ∼ n − i. We will check the conditions (a), (b), (c), and (d) for admissibility according to Definition 3.10.(a) Since all vertices in Q are mutable, the condition (a) is obviously satisfied.(b) Let B = ( b i,j ) = B ( Q ). Then for each i, j ∈ [2 n − b i,j is given by the alge-braic intersection number ( γ i , γ j ), which is the same as ( γ n − i , γ n − j ) since G has the π -rotationsymmetry. Hence b i,j = b τ ( i ) ,τ ( j ) . (c) On the other hand, for each i ∈ [2 n − i ], we have b i,τ ( i ) = ( γ i , γ τ ( i ) ) = ( γ τ ( i ) , γ τ ( τ ( i )) ) = ( γ τ ( i ) , γ i ) = − b i,τ ( i ) , which implies that b i,τ ( i ) = 0 . (d) Finally, we need to prove that for each i, j , b i,j b τ ( i ) ,j ≥ . If j = n , then since τ ( n ) = n , we have b i,n b τ ( i ) ,n = b i,n b τ ( i ) ,τ ( n ) = b i,n b i,n ≥ . Similarly, if i = n , then b n,j b τ ( n ) ,j = b n,j b n,j ≥ . Suppose that for some i, j = n , b i,j b τ ( i ) ,j < . By changing the roles of i and τ ( i ) if necessary, we may assume that b i,j < < b τ ( i ) ,j . Then wealso have b τ ( i ) ,τ ( j ) < < b i,τ ( j ) , which implies that there is an oriented cycle in Q j → i → τ ( j ) → τ ( i ) → j. However, this contradicts to Lemma 6.3 and therefore Q satisfies all conditions in Definition 3.10. (cid:3) Similarly, we have the following proposition as well.
AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 41
Proposition 6.5.
Let ( G , B ) be of type D . If ( G , B ) is Z / Z -admissible, then so is the quiver Q ( G , B ) .Proof. Let Q = Q ( G , B ). Then by definition of the Z / Z -action, we have2 ∼ ∼ . (a) and (b) This is obvious as before.(c) Let B = ( b i,j ) = B ( Q ). Suppose that b , = 0. Then by (b), b , = b , = b , = 0and so Q has a directed cycle either2 → → → → → → . Then according to b , , the underlying graph of the quiver Q is either the complete graph K ora disconnected graph, but both are impossible. Therefore b , = b , = b , = 0 . (d) The only entries we need to check are b ,j ’s, which are all equal by (b). Therefore b ,j b ,j ′ ≥ . (cid:3) Partial rotations and N -graphs of type D n +1 and E . Recall from Table 5 that theLegendrians λ ( D n +1 ) = λ ( n − , ,
2) and λ ( E ) = λ (2 , ,
3) whose braid representatives are β ( D n +1 ) = σ σ n σ σ σ σ , β ( E ) = σ σ σ σ σ σ . Let us identify D with the unit disk in C and define the ray R θ in D as R θ = { ( r, θ ) ∈ D ⊂ C | ≤ r ≤ } . In each case, we assume that three σ ’s are on the end points of three rays R , R π/ and R π/ ,which are points { , e π/ , e π/ } ⊂ S ⊂ C . We also assume that two same blocks of σ ∗ in each λ are contained in the angle [2 π/ , π/
3] and [4 π/ , π ].Let ( G , B ) be a 3-graph of type X = D n +1 or E . We consider the intersection ( G , B ) ∩ R θ between ( G , B ) with the ray R θ , which consists of colored points or intervals possibly togetherwith labels γ j . Definition 6.6 (Ray symmetry) . We say that the pair ( G , B ) is ray-symmetric if the intersections( G , B ) ∩ R θ for θ = 0 , π/ , π/ R , ( G , B ) ∩ R ) ∼ = ( R π/ , ( G , B ) ∩ R π/ ) ∼ = ( R π/ , ( G , B ) ∩ R π/ ) . (6.1) R R π/ R π/ ( R , ( G , B ) ∩ R ) =( R π/ , ( G , B ) ∩ R π/ ) =( R π/ , ( G , B ) ∩ R π/ ) = ∼ = ∼ = Figure 34.
Ray-symmetricity
Then we define a Z / Z -action on a ray-symmetric ( G , B ) as follows:(1) cut D into three sectors D , D and D along the rays R θ for θ = 0 , π/ π/ G , B ) gives us three 3-subgraphs { ( G , B ) , ( G , B ) , ( G , B ) } , G i = G ∩ D i , (2) change two subgraphs contained in sectors whose angles are in between [2 π/ , π/
3] and[4 π/ , π ] by rotating certain angles.(3) The result will be denoted by ( τ ( G ) , τ ( B )).Notice that each subgraph G i may not satisfy the condition of N -graphs but the final resultwill be an well-defined 3-graph since G is ray-symmetric. However, if G is not ray-symmetric, thenthe Z / Z -action is never well-defined. We call this action the partial rotation and see Figure 35for the pictorial definition. ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) ( G , B ) τ cut τ cutpartial rot.glue partial rot. glue Figure 35.
Partial rotationWe say that ( G , B ) is Z / Z -admissible if it is invariant under the partial rotation up to relabelingof cycles as follows:(1) if X = D n +1 , then γ n ↔ γ n +1 , (2) if X = E , then γ ↔ γ , γ ↔ γ . Remark . Similar to Remark 6.2, the Coxeter padding C = C ( n − , ,
2) for G ( D n +1 ) or C (2 , ,
3) for G ( E ) has also the Z / Z -symmetry under the partial rotation. Therefore for any Z / Z -admissible ( G , B ) under the partial rotation of type D n +1 or E , so is the following C · · · ¯ CC ( G , B ) or ¯ C · · · ¯ CC ( G , B ) . Lemma 6.8.
Let Q be a quiver of type E . Suppose that Q is invariant under the action τ (3) = 5 , τ (4) = 6 . AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 43
Then there is no oriented cycle, which is either → → → → or → → → → . Proof.
We will use the essentially same argument as the proof of Lemma 6.3.If Q = µ ( Q ( E )), then by Proposition 3.15 and Lemma 3.16, there exist an integer r and asequence µ ′ of mutations such that Q = µ ′ ( µ r Q ( Q ( E ))) = µ ′ ( Q ( E )) . Moreover, µ ′ misses at least one mutation µ i .(1) If i = 1, then Q ( E ) \ { } consists of three quivers { } ⊂ Q ( A ) , { , } ⊂ Q ( A ) , { , } ⊂ Q ( A ) . In particular, there are no direct edges between two sets of vertices { , } and { , } .(2) If i = 2, then µ ′ can be regarded as a sequence of mutations of type A . By Lemma 6.3,we are done.(3) If i = 3, then we may assume that µ ′ also misses µ due to the symmetry. Hence we haveseparated quivers { , } ⊂ Q ( A ) , { } ⊂ Q ( A ) , { } ⊂ Q ( A ) . Then after the mutation, the vertices 4 and 6 can be joined only with 3 and 5, respectively,and so we never have an edge between 3 and 6 or between 4 to 5.(4) If i = 4, then as above, we may assume that µ ′ misses µ as well and we may regard µ ′ asa sequence of mutations on Q ( D ) { , , , } ⊂ Q ( D ) . Moreover, µ ′ consists of mutations corresponding to Z / Z -orbits, which are µ , µ and µ , = ( µ µ ). Here the group Z / Z folds D onto C , and up to Coxeter mutations, thereare only three facets F C − α ∼ = P (Φ( C )) , F C − α ∼ = P (Φ( A )) × P (Φ( A )) , F C − α ∼ = P (Φ( A )) , Hence all possible quivers are obtained by one of the following ways: µ µ , µ · · · | {z } k ( Q ( D )) , ≤ k ≤ µ µ , µ · · · | {z } k ( Q ( D )) , ≤ k ≤ µ µ µ · · · | {z } k ( Q ( D )) , ≤ k ≤ µ µ , µ µ , µ µ , )( Q ( D )) and ( µ µ µ µ µ )( Q ( D ))are obtained from Q ( D )) by permuting vertices 3 ↔ ↔
2, respectively. Finally,one can directly check that we have no such cycles by the exhaustive search in this fulllist. (cid:3)
Proposition 6.9.
Let ( G , B ) be of type X = D n +1 or E . If ( G , B ) is Z / Z -admissible, then so isthe quiver Q ( G , B ) .Proof. (a) and (b): This is obvious as before.(c) Let B = ( b i,j ) = B ( Q ). Then by (b), b i,τ ( i ) = b τ ( i ) ,τ ( τ ( i )) = b τ ( i ) ,i = − b i,τ ( i ) = ⇒ b i,τ ( i ) = 0 . (d) If X = D n +1 , then we only need to show b i,n b i,n +1 ≥ for i < n . This is obvious since b i,n +1 = b τ ( i ) ,τ ( n +1) = b i,n . If X = E , then all we need to show are inequalities b i,j b i,j +2 ≥ , b , b , ≥ i = 1 , j = 3 , b i,j +2 = b τ ( i ) ,τ ( j +2) = b i,j . Suppose that b , b , <
0. Then since b , = b , and b , = b , , the Q has a loop either3 → → → → → → → → . (cid:3) Q ( A n − ) as a tripod Q (1 , n, n ) . As observed in Lemma 5.3, one can think Q (1 , n, n ) and G (1 , n, n ) for A n − instead of Q ( A n − ) and G ( A n − ).The major difference is now we have to use 3-graphs and partial rotations instead of 2-graphsand π -rotations. Then it can be easily checked that the above two notions are identical and so the Z / Z -admissibility for 3-graphs of type A n − is also well-defined. Moreover, the 3-graph analogueunder the partial rotation of Proposition 6.4 will be true.6.3. Global foldability of N -graphs. Let ( G , B ) be of type X . We say that ( G , B ) is globallyfoldable with respect to G if ( G , B ) is G -admissible and for any sequence of mutable G -orbits I , . . . , I ℓ , there exists a G -admissible ( G ′ , B ′ ) such that Q (Λ( G ′ ) , B ′ ) = ( µ I ℓ · · · µ I )( Q (Λ( G ) , B )) . Theorem 6.10.
The N -graph with a good tuple of cycles ( G ( X ) , B ( X )) is globally foldable withrespect to G .Proof. Let us define the initial quiver Q t by Q t = Q (Λ( G ( X )) , B ( X )) . For a sequence of mutable G -orbits I , . . . , I ℓ , we have an integer r and µ ′ Y by Proposition 3.15and Lemma 3.16 such that in the cluster pattern of type Y , two sequences of mutations (cid:0) ( µ X I ℓ · · · µ X I )( Q t ) (cid:1) G = µ ′ Y (cid:0) ( µ Y Q ) r (cid:0) Q Gt (cid:1)(cid:1) will produce the same seed.On the other hand, as seen in Remark 3.19, the Coxeter mutation µ X Q will correspond to theCoxeter mutation µ Y Q via the folding. Moreover, µ ′ Y comes from a sequence µ ′ X of mutations viathe folding such that µ ′ X is the composition of mutations at G -orbits and happens inside somefacet F β ⊂ P (Φ( X )). Hence we have µ ′ Y (cid:0) ( µ Y Q ) r (cid:0) Q Gt (cid:1)(cid:1) = µ ′ Y (cid:0) (( µ X Q ) r ( Q t )) G (cid:1) = (cid:0) µ ′ X (( µ X Q ) r ( Q t )) (cid:1) G . By Proposition 5.15, there exists a pair ( G ′ , B ′ ) satisfying that Q (Λ( G ′ ) , B ′ ) = Q ( µ ′ X (( µ X Q ) r ( Q t ))) = ( µ X I ℓ · · · µ X I )( Q t ) . Finally, we need to show that ( G ′ , B ′ ) can be assumed to be G -admissible. As in the proof ofProposition 5.15, ( G ′ , B ′ ) is obtained by taking a µ ′ X on either ( G ( X ) , B ( X )) or (¯ G ( X ) , B ( X )) andattaching Coxeter paddings. As observed in Remarks 6.2 and 6.7, the Coxeter paddings themselvesare already G -admissible and the process attaching them preserve the G -admissibility in each case.Therefore we only need to show the G -admissibility of µ ′ X ( G ( X ) , B ( X )).We will use the essentially same strategy as the proof of Proposition 5.15. Since µ ′ X missessome µ γ i , it misses all µ γ i ′ for i ∼ i ′ . Then one can split ( G ( X ) , B ( X )) in a G -admissible way. Thatis, the set of N -subgraphs { ( G , B ) , . . . , ( G ℓ , B ℓ ) } AGRANGIAN FILLINGS FOR LEGENDRIAN LINKS OF FINITE TYPE 45 is closed under the G -action. In this case, G may permute N -subgraphs as well. Now we split µ ′ X into { µ ′ , . . . , µ ′ ℓ } such that each µ ′ i is a sequence of mutations of Q ( G i , B i ). Then µ ′ X ( G ( X ) , B ( X ))is G -admissible if so is µ ′ i ( G i , B i ) for each 1 ≤ i ≤ ℓ .Since each ( G i , B i ) is strictly simpler than ( G ( X ) , B ( X )) in terms of the number of vertices andis again of type ADE , the rest of the proof follows from induction and we omit the detail. (cid:3)
As a direct consequence, we will prove the following theorem:
Theorem 6.11.
The following holds:(1) The Legendrian link λ ( A n − ) has (cid:0) nn (cid:1) Z / Z -admissible N -graphs which admits the clusterpattern of type B n .(2) The Legendrian link λ ( D n +1 ) has (cid:0) nn (cid:1) Z / Z -admissible N -graphs which admits the clusterpattern of type C n .(3) The Legendrian link λ ( E ) has Z / Z -admissible N -graphs which admits the clusterpattern of type F .(4) The Legendrian link λ ( D ) has Z / Z -admissible N -graphs which admits the cluster pat-tern of type G .Proof. By Theorem 6.10, it is already known that for each X , the quiver of type X is globallyfoldable with respect to G . By Propositions 6.4, 6.5 and 6.9, the quiver Q ( G ( X ) , B ( X )) is alsoglobally foldable with respect to G .Let Σ t = Ψ( G ( X ) , B ( X ) , F λ ) = ( x (Λ( G ( X )) , B ( X ) , F λ ) , Q (Λ( G ( X )) , B ( X ))) be the initial seed.Without loss of generality, we may denote cluster variables in x by x = ( x , . . . , x rk( X ) )and we define a field homomorphism ψ : F = C ( x , . . . , x rk( X ) ) → F G = C ( x I , . . . , x I rk( Y ) ) by ψ ( x i ) = x I for any i in a G -orbit I . Then by construction, the initial seed Σ t is ( G, ψ )-admissible. See § Y as desired, and we aredone. (cid:3) References [1] V. I. Arnold.
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Email address : [email protected] Department of Mathematics Education, Kyungpook National University, Republic of Korea
Email address : [email protected] Department of Mathematics, Incheon National University, Republic of Korea
Email address : [email protected]@ibs.ac.kr