LLAGRANGIAN SKELETA AND PLANE CURVE SINGULARITIES
ROGER CASALS
Abstract.
We construct closed arboreal Lagrangian skeleta associated to links of isolatedplane curve singularities. This yields closed Lagrangian skeleta for Weinstein pairs ( C , Λ)and Weinstein 4-manifolds W (Λ) associated to max-tb Legendrian representatives of alge-braic links Λ ⊆ ( S , ξ st ). We provide computations of Legendrian and Weinstein invariants,and discuss the contact topological nature of the Fomin-Pylyavskyy-Shustin-Thurston clus-ter algebra associated to a singularity. Finally, we present a conjectural ADE-classificationfor Lagrangian fillings of certain Legendrian links and list some related problems. Introduction
The object of this note is to study a relation between the theory of isolated plane curvesingularities , as developed by V.I. Arnol’d and S. Gusein-Zade [8, 9, 10, 58], N. A’Campo[1, 2, 3, 4], J.W. Milnor [72] and others, and arboreal Lagrangian skeleta of Weinstein 4-manifolds. In particular, we construct closed Lagrangian skeleta for the infinite class ofWeinstein 4-manifolds obtained by attaching Weinstein 2-handles [27, 105] to the link of anarbitrary isolated plane curve singularity f : C −→ C . These closed Lagrangian skeletaallow for an explicit computation of the moduli of microlocal sheaves [57, 77, 94] and alsoexplain the symplectic topology origin of the Fomin-Pylyavskyy-Shustin-Thurston clusteralgebra [44] of an isolated singularity.1.1. Main Results.
The advent of Lagrangian skeleta and sheaf invariants have underscoredthe relevance of Legendrian knots in the study of symplectic 4-manifolds [21, 27, 47, 94,95]. The theory of arboreal singularities, as developed by D. Nadler [75, 76], provides alocal-to-global method for the computation of categories of microlocal sheaves [77]. Theseinvariants, in turn, yield results in terms of Fukaya categories [47, 48]. The existence ofarboreal Lagrangian skeleta has been crystallized by L. Starkston [97] in the context ofWeinstein 4-manifolds, where this article takes place.Given a Weinstein 4-manifold (
W, λ st ), it is presently a challenge to describe an associated ar-boreal Lagrangian skeleta L ⊆ W . In particular, there is no general method for finding closed arboreal Lagrangian skeleta , or deciding whether these exist. This manuscript explores thisquestion by introducing a new type of closed arboreal Lagrangian skeleta for Legendrianlinks Λ ⊆ ( S , ξ st ) which are maximal-tb representatives of the link of an isolated plane curvesingularity f ∈ C [ x, y ]. The discussion in this note unravels thanks to a geometric fact: Theorem 1.1.
Let f ∈ C [ x, y ] be an isolated plane curve singularity and Λ f ⊆ ( S , ξ st ) its associated Legendrian link. The Weinstein pair ( C , Λ f ) admits the closed arboreal La-grangian skeleton L ( ˜ f ) = M ˜ f ∪ ϑ , obtained by attaching the Lagrangian D -thimbles ϑ of ˜ f to the Milnor fiber M ˜ f , for any real Morsification ˜ f ∈ R [ x, y ] . (cid:3) The two objects Λ f and L ( ˜ f ) in the statement of Theorem 1.1 require an explanation, whichwill be given. We rigorously define the notion of a Legendrian link Λ f ⊆ ( S , ξ st ) associated The reader is referred to [52] for a beautiful and gentle introduction to the subject. That is, a compact arboreal Lagrangian skeleta L ⊆ ( W λ ) such that ∂ L = 0. a r X i v : . [ m a t h . S G ] S e p o the germ f ∈ C [ x, y ] of an isolated curve singularity in Section 2. Note that the smoothlink of the singularity f ∈ C [ x, y ], as defined by J. Milnor [72], and canonically associated to f , is naturally a transverse link T f ⊆ ( S , ξ st ) [37, 51, 55]. The Legendrian link Λ f ⊆ ( S , ξ st )will be a maximal-tb Legendrian approximation of T f . The notation ( C , Λ f ) refers to theWeinstein pair ( C , R (Λ f )), where R (Λ f ) ⊆ ( S , ξ st ) is a small (Weinstein) annular ribbonfor the Legendrian link Λ f .The Lagrangian skeleton L ( ˜ f ) is also defined in Section 2. Note that the Milnor fibration of f ∈ C [ x, y ] is a symplectic fibration on ( C , ω st ), whose symplectic fibers bound the transverselink T f ⊆ ( S , ξ st ). Nevertheless, the Lagrangian skeleton L ( ˜ f ) is built from the underlying topological Milnor fiber and the vanishing cycles associated to a real Morsification. Indeed, L ( ˜ f ) is obtained by attaching the Lagrangian thimbles of the morsification ˜ f to the (topo-logical) Milnor fiber, which is Lagrangian in L ( ˜ f ). Theorem 1.1 is a relative statement, beingabout a Weinstein pair ( C , Λ f ) and not just about a Weinstein manifold. Hence, it is usefulin the absolute context, as follows.Consider a Legendrian knot Λ ⊆ ( S , ξ st ) in the standard contact 3-sphere and the Weinstein4-manifold W (Λ) = D ∪ Λ T ∗ D obtained by performing a 2-handle attachment along Λ. Afront projection for Λ (almost) provides an arboreal skeleton for the Weinstein 4-manifold W (Λ) [97]. Nevertheless, the computation of microlocal sheaf invariants from this model isfar from immediate, nor exhibits the cluster nature of the moduli space of Lagrangian fillings.The symplectic topology of a Weinstein manifold is much more visible, and invariants morereadily computed, from a closed arboreal Lagrangian skeleton, i.e. an arboreal Lagrangianskeleton which is compact and without boundary. In particular, Theorem 1.1 provides sucha closed Lagrangian skeleton associated to a real Morsification: Corollary 1.2.
Let f ∈ C [ x, y ] be an isolated curve singularity and Λ f its associated Legen-drian link. The 4-dimensional Weinstein manifold W (Λ f ) = D ∪ Λ f ( T ∗ D ∪ π (Λ f ) . . . ∪ T ∗ D )) admits the closed arboreal Lagrangian skeleton L ( ˜ f ) ∪ ∂ ( D ∪ π (Λ f ) . . . ∪ D ) , obtained by attach-ing the Lagrangian D -thimbles of ˜ f to the compactified Milnor fiber M f = M f ∪ ∂ ( D ∪ π ( ∂M f ) . . . ∪ D ) , for any real Morsification ˜ f ∈ R [ x, y ] . (cid:3) Let us see how Theorem 1.1 and Corollary 1.2 can be applied for two simple singularities,corresponding to the D and the E Dynkin diagrams. As we will see, part of the strength ofthese results is the explicit nature of the resulting Lagrangian skeleta and the direct bridgethey establish between the theory of singularities and symplectic topology.
Figure 1.
The D -Legendrian link Λ f ⊆ ( S , ξ st ) (Left) and a closed La-grangian arboreal skeleton for the Weinstein 4-manifold W (Λ f ) (Right), ob-tained by attaching 5 Lagrangian 2-disks to the cotangent bundle ( T ∗ Σ , λ st ). Example 1.3. ( i ) First, consider the germ of the D -singularity f ( x, y ) = xy + x , theLegendrian link associated to this singularity is depicted in Figure 1 (Left). The Weinstein -manifold W (Λ f ) = D ∪ Λ f ( T ∗ D ∪ T ∗ D ) admits the closed arboreal Lagrangian skeletondepicted in Figure 1 (Right). The D -Dynkin diagram is readily seen in the intersectionquiver of the boundaries of the Lagrangian 2-disks added to the (smooth compactification) ofthe genus 2 Milnor fiber. Figure 2.
Closed Lagrangian arboreal skeleton associated to the E -simplesingularity f ( x, y ) = x + y , according to Corollary 1.2.( ii ) Second, consider the germ of the singularity f ( x, y ) = x + y , the link of the singularityis the maximal-tb positive torus knot Λ f ∼ = Λ(3 , ⊆ ( S , ξ st ) . The Weinstein 4-manifold W (Λ f ) = D ∪ Λ f T ∗ D admits the closed arboreal Lagrangian skeleton depicted in Figure 2.This Lagrangian skeleton is built by attaching six Lagrangian 2-disks to the cotangent bundle ( T ∗ Σ , λ st ) of a genus 3 surface. These 2-disks are attached along the six curves in Figure2, whose intersection pattern is ( mutation equivalent to ) the E Dynkin diagram. (cid:3)
From now onward, we abbreviate “closed arboreal Lagrangian skeleton” to
Cal -skeleton. Let(
W, λ ) be a Weinstein 4-manifold, e.g. described by a Legendrian handlebody, a Lefschetzfibration or analytic equations in C n . There are two basic nested questions: Does it admita Cal-skeleton ? If so, how do you find one ? For instance, consider a max-tb Legendrianrepresentative Λ ⊆ ( ∂ D , λ st ) of any smooth knot, does W (Λ) admit a Cal-skeleton ? It mightbe that not all these Weinstein 4-manifolds W (Λ) admit such a skeleton: it is certainly notthe case if the Legendrian knot Λ were stabilized, hence the max-tb hypothesis. In general,the lack of exact Lagrangians in W (Λ) would provide an obstruction. Remark 1.4.
For simplicity, we focus on oriented exact Lagrangians. Non-orientable Cal-skeleta should also be of interest. For instance, consider the max-tb Legendrian left -handedtrefoil knot Λ(3 ) ⊆ ( ∂ D , λ st ). Then W (Λ(3 )) admits a Cal-skeleton RP ∪ S D given byattaching a Lagrangian 2-disk to a Lagrangian RP , as shown in Figure 3. (cid:3) Figure 3.
Cal-skeleton RP ∪ S D associated to Λ(3 ) ⊆ ( ∂ D , λ st ). This seems appropriate, as D. Nadler (UC Berkeley), L. Starkston (UC Davis) and Y. Eliashberg (Stan-ford), the initial developers of arboreal Lagrangian skeleta, hold their positions in the state of California. ymplectic invariants of Weinstein 4-manifolds W include (partially) wrapped Fukaya cate-gories [12, 98] and categories of microlocal sheaves [77]. Microlocal sheaf invariants shouldbe particularly computable if a Cal-skeleton L ⊆ W is given, yet worked out examples arescarce in the literature. In Section 4, we use Theorem 1.1 to compute the moduli space ofsimple microlocal sheaves on some of the Cal-skeleta L from Corollary 1.2.Finally, Theorem 1.1 provides a context for the study of exact Lagrangian fillings of Leg-endrian links Λ f ⊆ ( S , ξ st ) associated to isolated plane curve singularities. Indeed, let L ( f ) = M f ∪ ϑ be a Cal-skeleton for the Weinstein pair ( C , Λ f ), as produced in Theorem1.1. The topological Milnor fiber M f may serve as a marked exact Lagrangian filling for theLegendrian link Λ f , and performing Lagrangian disk surgeries [93, 106] along the Lagrangianthimbles in ϑ is a method to construct additional exact Lagrangian fillings. In general, thisstrategy might be potentially obstructed, as the Lagrangian disks might acquire immersedboundaries when the Lagrangian surgeries are performed. That said, since Lagrangian diskssurgeries yield combinatorial mutations of a quiver, Theorem 1.1 might hint towards a struc-tural conjecture: we expect as many exact Lagrangian fillings Λ f as elements in the clustermutation class of the intersection quiver for the vanishing thimbles ϑ . Section 5 concludeswith a discussion on such conjectural matters. Acknowledgements : The author thanks A. Keating for many conversations on divides ofsingularities throughout the years. The author is supported by the NSF grant DMS-1841913,a BBVA Research Fellowship and the Alfred P. Sloan Foundation.2.
Lagrangian Skeleta for Isolated Singularities
In this section we introduce the necessary ingredients for Theorem 1.1 and prove it. We referthe reader to [9, 52, 73] for the basics of plane curve singularities and [36, 37, 51, 82] forbackground on 3-dimensional contact topology.2.1.
The Legendrian Link of an Isolated Singularity.
Let f ∈ C [ x, y ] be a bivariatecomplex polynomial with an isolated complex singularity at the origin ( x, y ) = (0 , ∈ C .The link of the singularity T f ⊆ ( S , ξ st ) is the intersection T f = V ( f ) ∩ S ε = { ( x, y ) ∈ C : f ( x, y ) = 0 } ∩ { ( x, y ) ∈ C : | x | + | y | = ε } , where ε ∈ R + is small enough. The intersection is transverse for ε ∈ R + small enough [30, 72],and thus T f is a smooth link. The link T f is in fact a transverse link for the contact structure ξ st = T S ∩ i ( T S ), as is the boundary of the (Milnor) fiber M f for the Milnor fibration[51, 55]. Equivalently, it is the transverse binding of the contact open book generated by f (cid:107) f (cid:107) : S \ T f −→ S . The link of a singularity was first introduced by W. Wirtinger and K. Brauner [19] andmasterfully studied by J. Milnor [72]. The book [30] comprehensively develops the smoothtopology of link of singularities and their connection to 3-manifold topology. The contacttopological nature of the associated open book was developed by E. Giroux [55].From a smooth perspective, the smooth isotopy class of T f is that of an iterated cable of theunknot [30]. Let K l,m be the oriented ( l, m )-cable of a smooth link K ⊆ S , i.e. an embeddedcurve in the boundary ∂ O p ( K ) of the solid torus O p ( K ) in the homology class l · [ λ ] + m · [ µ ],with λ the longitude and µ the meridian of O p ( K ). It is shown in [30, Chapter IV.7] that The correspondence [81, Theorem 1.3] and T. K´alm´an’s description [63] of augmentation varieties Aug(Λ)are also useful tools in this context. Potentially not Hamiltonian isotopic. See also W. Neumann’s article in E. K¨ahler’s volume [62]. n iterated cable K ( l ,m ) , ( l ,m ) ,..., ( l r ,m r ) ⊆ S is the link of an isolated singularity if and onlyif m i +1 > ( l i m i ) l i +1 , for 1 ≤ i ≤ r − Remark 2.1.
Given an isolated singularity f ( x, y ), there are algorithms for determining thesmooth type of T f , i.e. the sequence of pairs { ( l , m ) , ( l , m ) , . . . , ( l r , m r ) } . For instance,by applying the Newton-Puiseux algorithm to f ( x, y ) we may write y = a x n m + a x n m m + a x n m m m + . . . , at each branch; the pairs ( n i , m i ) are called the Puiseux pairs. Then the cable pairs ( l i , m i )are given by l i = n i − n i − m i + m i − n i − n i . The algorithm is explained in [30, Appendix toChapter I]. (cid:3) In the finer context of contact topology, the transverse link T f ⊆ ( S , ξ st ) is an iteratedcable with maximal self-linking number sl ( T f ) = sl , as it bounds the symplectic Milnor fiber M f ⊆ C of f ∈ C [ x, y ], equiv. the symplectic page of the contact open book [38, 55]. By thetransverse Bennequin bound [14], this self-linking must be equal to the Euler characteristc − χ ( M f ). A fact about the smooth isotopy class of links of singularities is their Legendriansimplicity: Proposition 2.2.
Let f ∈ C [ x, y ] be an isolated singularity and T f ⊆ ( S , ξ st ) . Thereexists a unique maximal Thurston-Bennequin Legendrian approximation Λ f ⊆ ( S , ξ st ) of thetransverse link T f .Proof. The classification of Legendrian representatives of iterated cables of positive torusknots is established in [68, Corollary 1.6], building on [39, 40]. The sufficent numericalcondition for Legendrian simplicity is m i +1 /l i +1 > tb ( K i ), where K i is the i th iterated cablein K ( l ,m ) , ( l ,m ) ,..., ( l r ,m r ) ⊆ S . The maximal Thurston-Bennequin equals tb ( K i ) = A r − B r ,where A r , B r ∈ N are defined in [68, Equation (2)], and satisfy m i l i > A i − B i . In particular,an algebraic link satisfies m i +1 /l i +1 > m i l i ≥ tb ( K i ), for all 1 ≤ i ≤ r −
1, and its max-tbrepresentative is unique. (cid:3)
Proposition 2.2 implies that there exists a unique
Legendrian link Λ f ⊆ ( S , ξ st ), up to contactisotopy, whose positive transverse push-off τ (Λ f ), as defined in [51, Section 3.5.3], is trans-verse isotopic to the transverse link T f . Note that two distinct Legendrian approximationsof a transverse link [34, Theorem 2.1] differ by Legendrian stabilizations, which necessarilydecrease the Thurston-Bennequin invariant. Remark 2.3.
Proposition 2.2 does not hold for K ⊆ ( S , ξ st ) an arbitrary smooth link. Forinstance, the smooth isotopy classes of the mirrors 5 , of the three-twist knot and theStevedore knot admit two distinct maximal-tb Legendrian representatives each [26, Section4]. That said, the knots 5 , are not links of singularities, as their Alexander polynomialsare not monic, and thus they are not fibered knots [80]. (cid:3) Proposition 2.2 allows us to canonically define a
Legendrian link associated to an isolatedsingularity:
Definition 2.4.
A Legendrian link Λ f ⊆ ( S , ξ st ) is associated to an isolated singularity f ∈ C [ x, y ] if it is a maximal-tb Legendrian link Λ f ⊆ ( S , ξ st ) whose positive transversepush-off τ (Λ f ) is transversely isotopic to the link of the singularity T f ⊆ ( S , ξ st ). (cid:3) Proposition 2.2 shows that the Legendrian isotopy class of a Legendrian link Λ f ⊆ ( S , ξ st )associated to an isolated singularity f ∈ C [ x, y ] is unique. Thus, we refer to Λ f ⊆ ( S , ξ st ) inDefinition 2.4 as the Legendrian link associated to the isolated singularity f ∈ C [ x, y ]. xample 2.5 (ADE Singularities) . Let us consider the three ADE families of simple isolatedsingularities [11, Chapter 2.5] . Their germs are given by ( A n ) f ( x, y ) = x n +1 + y , ( D n ) f ( x, y ) = xy + x n − , n ∈ N , ( E ) f ( x, y ) = x + y , ( E ) f ( x, y ) = x + xy , ( E ) f ( x, y ) = x + y . Figure 4.
The Legendrian link for the A n -singularity is the max-tb (2 , n +1)-torus link (Left). The Legendrian link for the D n -singularity is the link givenby the union of a max-tb (2 , n − The Legendrian link associated to the A n -singularity is the positive (2 , n + 1) -torus link, with tb = n − . These links are associated to the braid σ n +11 , as depicted in Figure 4 (Left). TheLegendrian link associated to the D n -singularity is the link consisting of the link associated tothe A n − -singularity and the standard Legendrian unknot, linked as in Figure 4 (Right). Thisis the topological consequence of the factorization f ( x, y ) = x ( y + x n − ) . These D n -links areassociated to the (rainbow closure of the) positive braid σ n − σ σ σ , n ≥ . The D -link isthe three-copy Reeb push-off of the Legendrian unknot, and the D -link is Legendrian isotopicto the A -link, i.e. a max-tb positive T (2 , -torus link. Figure 5.
The Legendrian links for the E , E and E simple singularities. The Legendrian links associated to the E and E singularities are the maximal-tb positive (3 , -torus Legendrian link and the Legendrian (3 , -torus link, as depicted in Figure 5. The E is a maximal-tb Legendrian link consisting of a trefoil knot and a standard Legendrianunknot, linked as in the center Legendrian front in Figure 5. This is implied by the f ( x, y ) = x ( x + y ) factorization of the E singularity. The Legendrian links for E , E and E canalso be obtained as the closure of the three braids σ n − σ σ σ , n = 6 , , . Figure 5 alsodepicts generators of the first homology group of the minimal genus Seifert surface; thesegenerate the first homology of each Milnor fiber, and the E , E and E Dynkin diagrams arereadily exhibited from their intersection pattern. (cid:3) he singularities f ( x, y ) = x a + y b , a ≥ , b ≥
6, or ( a, b ) = (4 , , (4 , a, b )-torus link, confer Remark 2.1. Two more instances areillustrated in the following: Example 2.6. ( Two Iterated Cables ) Consider the isolated curve singularity g ( x, y ) = x − x + 4 x y + 2 x y − y . The Puiseux expansion yields the Newton solution y = x / (1 + x / ) and thus Λ f ⊆ ( S , ξ st ) is the maximal-tb Legendrian representative of the (2 , -cable of the trefoil knot. This Leg-endrian knot is depicted in Figure 6 ( Left ) . The reader is invited to show that the Legendrianknot Λ f ⊆ ( S , ξ st ) of the singularity h ( x, y ) = x − x + 6 x y − x y + 2 x y + 3 x y − y , is the maximal-tb Legendrian representative of the (3 , -cable of the trefoil knot [52] , asdepicted in Figure 6 ( Right ) . ( For that, start by writing the relation as y ( x ) = x / + x / . ) (cid:3) Figure 6.
The Legendrian links Λ g and Λ h associated to the singularity g ( x, y ) = x − x + 4 x y + 2 x y − y , on the left, and the singularity h ( x, y ) = x − x + 6 x y − x y + 2 x y + 3 x y − y , on the right.2.2. A’Campo’s Divides and Their Conormal Lifts.
Let f ∈ C [ x, y ] be an isolatedsingularity, D ε ⊆ C a Milnor ball for this singularity [73, Corollary 4.5], ε ∈ R + , R = { ( x, y ) ∈ C : (cid:61) ( x ) = 0 , (cid:61) ( y ) = 0 } ⊆ C the real 2-plane and D ε = D ε ∩ R a real Milnor2-disk. Consider a real Morsification ˜ f t ( x, y ), t ∈ [0 , t ∈ (0 , f t ( x, y ) hasonly A -singularities, its critical values are real and the level set f − t (0) ∩ D ε , contains allthe saddle points of the restriction ( f t ) | D ε . The intersection D f = f − t (0) ∩ D ε ⊆ R , where˜ f = f , is known as the divide of the real Morsification ˜ f [3, 9, 61]. It is the image of aunion I of closed segments under an immersion i : I −→ R [53, 59, 60], and we assume itis a generic such immersion. By considering I ⊆ R as a wavefront, its biconormal lift [8] isa Legendrian link Λ ( D f ) in the contact boundary ( ∂ ( T ∗ R ) , λ st | ∂ ( T ∗ R ) ). See [2, 58] for theexistence and details of real Morsifications.The biconormal lift Λ ( D f ) ⊆ ∂ ( T ∗ R ) of the immersed curve D f to the (unit) boundary ofthe cotangent bundle T ∗ R can be constructed using the three local models:(i) The biconormal lift near a smooth interior point P ∈ D f is defined as { u ∈ T ∗ O p ( P ) : (cid:107) u q (cid:107) = 1 , T q D f ⊆ ker( u q ) for q ∈ D f ∩ O p ( P ) } , for an arbitrary fixed choice of metric in R , and neighborhood O p ( P ) ⊆ R . ii) The biconormal lift near an immersed point P ∈ D f is defined as the (disjoint) unionof the conormal lifts of each of its embedded branches through P .(iii) Finally, at the endpoint P ∈ D f , the biconormal lift is defined as the closure in T ∗ P R of one of the components of T ∗ P R \ { u ∈ T ∗ P R : (cid:107) u q (cid:107) = 1 , T P D f ⊆ ker( u P ) for q ∈ D f ∩ O p ( P ) } , where the tangent line T P D f is defined as the (ambient) smooth limit of the tangentlines T q i D f for a sequence { q i } i ∈ N of interior points q i ∈ D f convering to P ∈ D f .There are two such components, but our arguments are independent of such a choice. Remark 2.7.
The restriction of the canonical projection π : ∂ ( T ∗ R ) −→ R is finite two-to-one onto the image of the interior points of I . The pre-image of π at (the image of) endpointscontains an open interval of the Legendrian circle fiber. For instance, the full conormal lift ofa point p ∈ R is Legendrian isotopic to the zero section S ⊆ ( J S , ξ st ), as is the conormallift of an embedded closed segment. (cid:3) These local models define the Legendrian biconormal lift Λ ( D f ) ⊆ ( ∂ ( T ∗ R ) , ξ st ) of thedivide of the Morsification ˜ f . Let ι : S −→ ( S , ξ st ) be a Legendrian embedding inthe isotopy class of the standard Legendrian unknot. A small neighborhood O p ( ι ( S ))is contactomorphic to the 1-jet space ( J S , ξ st ) ∼ = ( T ∗ S × R t , ker { λ st − dt } ), yielding acontact inclusion ι : ( J S , ξ st ) −→ ( S , ξ st ). Note that there exists a contactomorphismΨ : ( ∂ ( T ∗ R ) , ξ st ) −→ ( J S , ξ st ), where the zero section in the 1-jet space bijects to theLegendrian boundary of a Lagrangian cotangent fiber in T ∗ R . This leads to the following: Definition 2.8.
Let D f ⊆ R be the divide associated to a real Morsification of an isolatedsingularity f ∈ C [ x, y ]. The biconormal lift Λ( D f ) ⊆ ( S , ξ st ) is the image ι (Ψ(Λ ( D f ))).That is, the biconormal lift Λ( D f ) ⊆ ( S , ξ st ) is the satellite of the biconormal lift Λ ( D f ) ⊆ ( ∂ ( T ∗ R ) , ξ st ) with companion knot the standard Legendrian unknot in ( S , ξ st ). (cid:3) The central result in N. A’Campo’s articles [3, 4] is that the Legendrian link Λ( D f ) ⊆ S is smoothly isotopic to the transverse link T f , see also [60]. The formulation above, in terms ofthe satellite to the Legendrian unknot, is not necessarily explicit in the literature on dividesand their Legendrian lifts, but probably known to the experts, as it is effectively being usedin M. Hirasawa’s visualization [59, Figure 2]. See also the work of T. Kawamura [67, Figure2], M. Ishikawa and W. Gibson [53, 61] and others [25, 60]. The phrasing in Definition 2.8might help crystallize the contact topological characteristics of each object. Example 2.9. ( i ) The A -singularity admits two real Morsifications ˜ f ( x, y ) = x + y − and ˜ f ( x, y ) = x − y , with corresponding divides D = { ( x, y ) ∈ R : x + y − } , D = { ( x, y ) ∈ R : x − y = 0 } . The biconormal lift Λ ( D ) ⊆ ( ∂T ∗ R , ξ st ) consists of two copies of the Legendrian fibersof the fibration π : ∂T ∗ R −→ R . Each of these two copies is satellited to the standardLegendrian unknot, forming a maximal-tb Hopf link Λ( D ) ⊆ ( S , ξ st ) . Indeed, the secondLegendrian fiber can be assumed to be the image of the first Legendrian fiber under the Reebflow. Hence, the Legendrian link Λ( D ) ⊆ ( S , ξ st ) must consist of the standard Legendrianunknot union a small Reeb push-off. Similarly, the biconormal lift Λ ( D ) ⊆ ( ∂T ∗ R , ξ st ) equally consists of two copies of the Legendrian fibers of the fibration π : ∂T ∗ R −→ R , andthus both Legendrian links Λ( D ) , Λ( D ) are Legendrian isotopic in ( S , ξ st ) . ( ii ) The A -singularity f ( x, y ) = x + y admits the real Morsification ˜ f ( x, y ) = x ( x − y ,whose divide is D = { ( x, y ) ∈ R : x ( x −
1) + y = 0 } . The divide D ⊆ R with its co-orientations is depicted in Figure 7. The first row depicts a wavefront homotopy, which yields igure 7. A co-oriented divide D for the A -singularity f ( x, y ) = x + y , asa front for its Legendrian link Λ( D ) ⊆ ( ∂ ( T ∗ D ) , ξ st ). That is, the biconormallift of D is Λ( D ). Its satellite along the standard unknot is the (unique)max-tb Legendrian trefoil Λ(2 , ⊆ ( R , ξ st ). a Legendrian isotopy in ( ∂T ∗ R , ξ st ) . The second row starts by depicting the change of frontprojections induced by the contactomorphism Ψ , and performs the satellite to the standardLegendrian unknot. The resulting Legendrian Λ f ⊆ ( S , ξ st ) is the max-tb Legendrian trefoilknot Λ(2 , .In general, divides for A n -singularities are depicted in [44, Figure 4] . We invite the readerto study the A -singularity f ( x, y ) = x + y with its divide D = { ( x, y ) ∈ R : x ( x + x − x −
1) + y = 0 } and discover the corresponding Legendrian isotopy, as in Figure 7. The isotopy should endwith the max-tb Legendrian link Λ(2 , ⊆ ( S , ξ st ) , e.g. expressed as the (rainbow) closure ofthe positive braid σ , equiv. the ( − -framed closure of σ . The general case n ∈ N is similar. (cid:3) Proof of Theorem 1.1.
There is an interesting dissonance at this stage. The Legen-drian link Λ( D f ) ⊆ S in Definition 2.8 and the transverse link T f ⊆ S of the singularityare smoothly isotopic, yet certainly not contact isotopic. Their relationship is described bythe following: Proposition 2.10.
Let f ∈ C [ x, y ] be an isolated singularity and D f ⊆ R the divide asso-ciated to a real Morsification. The positive transverse push-off τ (Λ( D f )) ⊆ ( S , ξ st ) of theLegendrian link Λ( D f ) is contact isotopic to the transverse link T f ⊆ ( S , ξ st ) . In particular, Λ( D f ) ⊆ ( S , ξ st ) is Legendrian isotopic to the Legendrian link Λ f ⊆ ( S , ξ st ) associated tothe singularity f ∈ C [ x, y ] . (cid:3) Proof.
In A’Campo’s isotopy [3, Section 3] from the link associated to the divide to the link ofthe singularity, the key step is the almost complexification of the Morsification ˜ f : R −→ R .This replaces the R -valued function ˜ f by an expression of the form˜ f C : T ∗ R −→ C , ˜ f C ( x, u ) := ˜ f ( x ) + id ˜ f ( x )( u ) − χ ( x ) H ( f ( x ))( u, u ) , which is a C -valued function, where u = ( u , u ) ∈ R are Cartesian coordinates in the fiber.Here H ( f ( x )) is the Hessian of f , which is a quadratic form, and χ ( x ) is a bump functionwith χ ( x ) ≡ D f ⊆ R and χ ( x ) ≡ esults in [3], see also [60, 61], imply that the transverse link of the singularity is isotopicto the intersection T ε R ∩ f − C (0) ⊆ ( T ε R , ξ st ) of the ε -unit cotangent bundle with the 0-fiber of (cid:101) f C , ε ∈ R + small enough. It thus suffices to compare this transverse link to theLegendrian lift Λ( D f ) ⊆ ( T ε R , ξ st ), which we can check in each of the two local models:near a smooth interior point of the divide D f and near each of its double points. Note thatthe case of boundary points can be perturbed to that of smooth interior points, as in thefirst perturbation in Figure 7. We detail the computation in the first local model, the caseof double points follows similarly.The contact structure ( T ε R , ξ st ) admits the contact form ξ st = ker { cos( θ ) dx − sin( θ ) dx } ,( x , x ) ∈ R and θ ∈ S is a coordinate in the fiber – this is the angular coordinate in the( u , u )-coordinates above. The divide can be assumed to be cut locally by D = { ( x , x ) ∈ R : x = 0 } ⊆ R , as we can write ˜ f ( x , x ) = x , and thus its bi-conormal Legendrian liftis Λ( D ) = { ( x , x , θ ) ∈ R × S : x = 0 , θ = ± π/ } . Note that the tangent space T ( x ,x ) Λ( D ) of Λ( D ) is spanned by ∂ x , which satisfies (cid:104) ∂ x (cid:105) = ker { cos( θ ) dx − sin( θ ) dx } , as cos( θ ) = 0 at θ = ± π/ . Since the model is away from a double point, ˜ f C ( x, u ) := x + i (0 , · ( u , u ) t = x + iu becomes the standard symplectic projection R × R −→ R onto the second (symplectic)factor. The zero set is thus x = 0 and u = 0 and so the intersection with T ε R is κ = { ( x , x , θ ) ∈ R × S : x = 0 , θ = 0 , π } , as the points with | u | = ε are at θ -coordinates θ = 0 , π . The tangent space T κ = (cid:104) ∂ x (cid:105) isspanned by ∂ x , which is transverse to the contact structure along κ :(cos( θ ) dx − sin( θ ) dx )( ∂ x ) = ± , at θ = 0 , π. It evaluates positive for θ = 0 and negative for θ = π , which corresponds to each of the twobranches in the biconormal lift. It is readily verified [51, Section 3.1] that κ is the transversepush-off, positive and negative , of Λ( D ), e.g. observe that the annulus { ( x , x , θ ) ∈ R × S : x = 0 , ≤ θ ≤ π } is a (Weinstein) ribbon for the Legendrian segment { ( x , x , θ ) ∈ R × S : x = 0 , θ = π/ } . (cid:3) Proposition 2.10 implies that real Morsifications ˜ f yield models for the Legendrian linkΛ f ⊆ ( S , ξ st ) of a singularity f ∈ C [ x, y ], as introduced in Definition 2.4. That is, given anisolated plane curve singularity f ∈ C [ x, y ], the Legendrian link Λ f ⊆ ( S , ξ st ) is Legendrianisotopic to the Legendrian lift Λ( D ˜ f ) ⊆ ( S , ξ st ) of a divide D ˜ f ⊆ R of a real Morsification,and thus we now directly focus on studying the Legendrian links Λ( D ˜ f ) ⊆ ( S , ξ st ).Let us now prove Theorem 1.1. For that, we use N. A’Campo’s description [4] of the set ofvanishing cycles associated to a divide of a real Morsification. For each double point p i ∈ D in the divide D = D ˜ f , there is a vanishing cycle ϑ p i . For each bounded region of R \ D ,which we label by q j , there is a vanishing cycle ϑ q j . First, we visualize those vanishing cyclesby perturbing the divide D ⊆ R to a divide D (cid:48) ⊆ R , as depicted in Figure 8.(i) and (ii).The lift of D (cid:48) only uses one conormal direction at a given point. This perturbation is a fronthomotopy and thus produces a Legendrian isotopy of the associated Legendrian link. This mimicks S. Donaldson’s construction of Lefschetz pencils, where the boundary of a fiber is a transverselink at the boundary, see also E. Giroux’s construction of the contact binding of an open book [54, 55]. The orientation for the negative branch is reversed when considering the global link κ . igure 8. (Left) Two front homotopies from the pieces of a divide to a(generic) Legendrian front. The vanishing cycle ϑ p is drawn in the Lagrangianbase R . (Right) A perturbation of a divide for the E -singularity. Thevanishing cycles ϑ p coming from the double points of the divide are drawn inyellow, and the vanishing cycles ϑ q coming from each of the three boundedinterior regions are drawn in red.Once the perturbation has been performed, we can draw the curves ϑ p i , ϑ q j as in Figure 8.For instance, Figure 8.(iii) depicts the case of the E -singularity with a particular choiceof divide D and its perturbation D (cid:48) , with ϑ p i in yellow and ϑ q j in red. That is, for eachdouble point, the curve ϑ p i is a closed simple curve exactly through the four new doublepoints in D (cid:48) . For each closed region, ϑ q j is a simple closed curve which (exactly) passesthrough the double points at the perturbed boundary in D (cid:48) of the region q j . The algorithmin [4] implies that a singular model of the topological Milnor fiber of f is obtained as R union the conical Lagrangian conormal L ( D (cid:48) ) of the perturbed divide D (cid:48) . This Lagrangianconormal intersects the unit cotangent bundle of T ∗ R at Λ( D (cid:48) ) and thus, being conical,the information of L ( D (cid:48) ) is equivalent to that of Λ( D (cid:48) ). In addition, [4] guarantees that thecurves ϑ p i , ϑ q j are vanishing cycles for the real Morsification ˜ f .At this stage, the key fact that we use from A’Campo’s algorithm is that our choice ofimmersion of the divide D (cid:48) ⊆ R , given by the perturbation, exhibits Lagrangian 2-disks D p i , D q j ⊆ R such that ∂ D p i = ϑ p i and ∂ D q j = ϑ q j . For ϑ p i , this follows from Figure 8.(i),where the 2-disk D p i is (a small extension of) the square given by the four double pointsin D (cid:48) appearing in the perturbation of p i ∈ D . For ϑ q j , the 2-disk D q j is chosen to be asmall extension of the bounded region itself. These disks are (exact) Lagrangian because R ⊆ ( T ∗ R , λ st ) is exact Lagrangian. The Liouville vector field in ( T ∗ R , λ st ) vanishesat R and is tangent to L ( D (cid:48) ). Hence, the inverse flow of the Liouville field retracts theWeinstein pair ( R , Λ( D (cid:48) )) to L ( D (cid:48) ) union the zero section R . This shows that L ( D (cid:48) ) ∪ R is a Lagrangian skeleton of the Weinstein pair ( R , Λ( D )). Now, the Lagrangian skeletonhas an open boundary at the unbounded part of R , which can be trimmed [97] to the disks D p i , D q j ⊆ R . Thus, the union of the conical Lagrangian L ( D (cid:48) ) and the Lagrangian 2-disks D p i , D q j ⊆ R is a Lagrangian skeleton of the Weinstein pair ( R , Λ( D (cid:48) )), as required. (cid:3) Lagrangian Skeleta.
Arboreal Lagrangian skeleta L ⊆ ( W, λ ) for Weinstein 4-manifoldsare defined in [76, 97]. Given a Weinstein manifold W = W (Λ), the arborealization proce-dure in [97] yields an arboreal Lagrangian skeleton L ⊆ ( W, λ ) with ∂ L (cid:54) = ∅ . Intuitively,those Lagrangian skeleta are obtained by attaching 2-handles to D along a (modification ofa) front for Λ, and thus roughly contain the same information as a front π (Λ) ⊆ R for Λ.Let Λ ⊆ ( S , ξ st ) be a Legendrian link and ( W, λ ) a Weinstein manifold. efinition 2.11. A compact arboreal Lagrangian skeleton L ⊆ C for a Weinstein pair( C , Λ) is said to be closed if ∂ L = Λ. A compact arboreal Lagrangian skeleton L ⊆ W fora Weinstein manifold ( W, λ ) is said to be closed if ∂ L = ∅ .The Lagrangian skeleta in Theorem 1.1 and Corollary 1.2 are arboreal and closed. Forreference, we denote the two Cal-skeleta associated to a real Morsification ˜ f of an isolatedplane curve singularity f ∈ C [ x, y ] by L ( ˜ f ) := M f ∪ ϑ ( ˜ f ) | ϑ ( ˜ f ) | (cid:91) i =1 D , L ( ˜ f ) := M f ∪ ϑ ( ˜ f ) | ϑ ( ˜ f ) | (cid:91) i =1 D . The former L ( ˜ f ) is a Lagrangian skeleton for the Weinstein pair ( C , Λ f ), and the latterfor the Weinstein 4-manifold W (Λ f ). The notation M f stands for the surface obtainedby capping each of the boundary components of the Milnor fiber M f with a 2-disk. Thenotation L ( f ) and L ( f ) will stand for any Cal-skeleton obtained from a real Morsification ˜ f as in Theorem 1.1 and Corollary 1.2. Remark 2.12.
In the context of low-dimensional topology, the 2-complexes underlying theseLagrangian skeleta are often referred to as
Turaev’s shadows , following [100, Chapter 8]. Inparticular, it is known how to compute the signature of a (Weinstein) 4-manifold from anyCal-skeleton by using [100, Chapter 9]. Similarly, the SU (2)-Reshetikhin-Turaev-Witten in-variant of the 3-dimensional (contact) boundary can be computed with the state-sum formulain [100, Chapter 10]. It would be interesting to explore if such combinatorial invariants canbe enhanced to detect information on the contact and symplectic structures. (cid:3) Augmentation Stack and The Cluster Algebra ofFomin-Pylyavskyy-Shustin-Thurston
In the article [44], the authors develop a connection between the topology of an isolatedsingularity f ∈ C [ x, y ] and the theory of cluster algebras. In concrete terms, they associatea cluster algebra A ( f ) to an isolated singularity. An initial cluster seed for A ( f ) is givenby a quiver Q ( D ˜ f ) coming from the AΓ-diagrams of a divide D ˜ f of a real Morsification of f . Equivalently, by [4, 58], the quiver Q ( D ˜ f ) is the intersection quiver for a set of vanishingcycles associated to a real Morsification of f . The conjectural tenet in [44] is that differentchoices of Morsifications lead to mutation equivalent quivers and, conversely, two quiversassociated to two real Morsifications of the same complex topological singularity must bemutation equivalent.There are two varieties associated to a cluster algebra, the X -cluster variety and the A -cluster variety [43, 56, 92]. In the case of the cluster algebra A ( f ) from [44], one can askwhether either of these varieties has a particularly geometric meaning. Our suggestion isthat either of these cluster varieties is the moduli space of exact Lagrangian fillings for theLegendrian knot Λ f ⊆ ( R , ξ st ). Equivalently, they are the moduli space of (certain) objectsof a Fukaya category associated to the Weinstein pair ( C , Λ f ); for instance, the partiallywrapped Fukaya category of C stopped at Λ f . In this sense, these cluster varieties are mirrorto the Weinstein pair ( R , Λ f ). Focusing on the Legendrian link Λ f ⊆ ( R , ξ st ), let us thensuggest an alternative route from a plane curve singularity f ∈ C [ x, y ] to a cluster algebra A ( f ), following Definition 2.4 and Proposition 2.2 and 2.10.Starting with f ∈ C [ x, y ], consider the Legendrian Λ f ⊆ ( R , ξ st ), where ( R , ξ st ) is identi-fied as the complement of a point in ( S , ξ st ) and the Legendrian DGA A (Λ f ), as defined by The difference between X - and A -varieties should be the decorations we require for the Lagrangian fillings. In the context of plabic graphs [44, Section 6], the zig-zag curves [56, 88] also provide a front for theLegendrian link Λ f . . Chekanov in [24] and see [35]. Then we define A ( f ) to be the coordinate ring of functionson the augmentation variety A (Λ f ) of the DGA A (Λ f ). Technically, the DGA A (Λ f ) allowsfor a choice of base points, and the augmentation variety depends on that. Thus, it is moreaccurate to define: Definition 3.1.
Let f ∈ C [ x, y ] be an isolated singularity, the augmentation algebra A ( f )associated to f is the ring of k -regular functions on the moduli stack of objects ob(Aug + (Λ f ))of the augmentation category Aug + (Λ f ). (cid:3) The Aug + (Λ) augmentation category of a Legendrian link Λ ⊆ ( R , ξ st ) is introduced in [81].An exact Lagrangian filling defines an object in the category Aug + (Λ), and the morphismsbetween two such objects are given by (a linearized version of) Lagrangian Floer homology.In fact, there is a sense in which any object in Aug + (Λ) comes from a Lagrangian filling[85, 86], possibly immersed, and thus ob(Aug + (Λ)) is a natural candidate for a moduli spaceof Lagrangian fillings. The algebra A ( f ) is known to be a cluster algebra [49] in characteristictwo. The lift to characteristic zero can be obtained by combining [22] and [49].By Proposition 2.2, A ( f ) is a well-defined invariant of the complex topological singularity.For these Legendrian links Λ = Λ f , the Couture-Perron algorithm [29] implies that thereexist a Legendrian front π (Λ f ) ⊆ R given by the ( − β ∆ ,where ∆ is the full twist; equivalently the front is the rainbow closure of the positive braid β [20]. Hence, there is a set of non-negatively graded Reeb chords generating the DGA A (Λ f )and ob(Aug + (Λ f )) coincides with the set of k -valued augmentations of A (Λ f ) where exactly one base point per component has been chosen, k a field. The articles [22, 63] provide anexplicit and computational model for ob(Aug + (Λ f )), and thus A ( f ), as follows.First, suppose that Λ = Λ f is a knot. Then, A ( f ) is the algebra of regular functions of theaffine variety X ( β ) := {B ( β ∆ ) + diag i ( β ) ( t, , , . . . ,
1) = 0 } ⊆ C | β ∆ | +1 , where B are the ( i ( β ) × i ( β ))-matrices defined in [22, Section 3] and Computation 3.2 below, i ( β ) is the number of strands of β, ∆, and | β ∆ | is the number of crossings of β ∆ . In the caseΛ f is a link with l components, the space ob(Aug + (Λ f )) is a stack , with isotropy groups ofthe form ( C ∗ ) k . If the tenet [44, Conjecture 5.5] holds, the affine algebraic type of the aug-mentation stack ob(Aug + (Λ f )) of a Legendrian link should recover the Legendrian link Λ f and the complex topological type of the singularity f . Here is how to compute ob(Aug + (Λ f )). Computation 3.2.
Let Λ = Λ f be an algebraic knot, we can find a set of equations for theaffine variety ob( Aug + (Λ f )), essentially using [64], see also [22]. Consider a positive braid β ◦ ∈ Br + n such that the ( − β ◦ is a front for Λ = Λ( β ◦ ). For k ∈ [1 , n − n × n matrix P k ( z ), with variable z ∈ C :( P k ( z )) ij = i = j and i (cid:54) = k, k + 11 ( i, j ) = ( k, k + 1) or ( k + 1 , k ) z i = j = k + 10 otherwise;Namely, P k ( z ) is the identity matrix except for the (2 × k and k + 1, where it is ( z ). Suppose that the crossings of β ◦ , left to right, are Throughout the text, exact Lagrangian fillings are, if needed, implicitely endowed with a C ∗ -local system. Namely, it is isomorphic to a quotient of X ( β ) × ( C ∗ ) l by a non-free ( C ∗ ) l − -action. Note that β ◦ can be written in the form β ◦ = β ∆ . k , . . . , σ k s , s = | β ◦ | ∈ N , σ i ∈ Br + n the Artin generators. Then the augmentation stackob(Aug + (Λ f )) is cut out in C s × C ∗ = Spec[ z , z , . . . , z s , t, t − ] by the n equations(3.1) diag n ( t, , , . . . ,
1) + P k ( z ) P k ( z ) · · · P k s ( z s ) = 0 . The matrix P k ( z ) P k ( z ) · · · P k s ( z s ) is denoted by B ( β ◦ ). Equations 3.1 provide a compu-tational mean to an explicit description of the affine varieties ob(Aug + (Λ f )) that yield thecluster algebra A ( f ). (cid:3) Example 3.3.
Consider the plane curve singularity described by f ( x, y ) = − x y − x y − x y + 6 x y − x y + x − x + x + y == (cid:0) x y − x y + x − x − y (cid:1) (cid:0) x y + 4 x y + x − x − y (cid:1) The Puiseux expansion yields y ( x ) = x / + x / and using the Couture-Perron algorithm [29] , or [44, Definition 11.3] , a positive braid word associated to this singularity is β = ( σ σ σ σ σ σ σ σ ) σ ( σ σ σ σ σ σ σ σ ) σ σ The Legendrian Λ f ⊆ ( R , ξ ) is the rainbow closure of β , and the ( − -framed closure of β ◦ = β ∆ . Note that Λ f is a knot, and thus we will use one base point t ∈ C ∗ in the computation of X ( β ) = ob(Aug + (Λ f )) . Following Computation 3.2 above, we can write equations for affinevariety X ( β ) as a subset X ( β ) ⊆ C × C ∗ . We use coordinates ( z , z , . . . , z ; t ) ∈ C × C ∗ , ( z , z , . . . , z ) corresponding to the crossings of β and ( z , . . . , z ) account for the crossings of ∆ ∈ Br +3 . There are a total of 16 equations, the first three of which read asfollows: z + z z + ( z + ( z + z z ) z ) z + ( z + z z + ( z + z z ) z ) z +( z z + ( z + z z ) z + ( z + z z + ( z + z z ) z ) z + 1) z = − t − z + z z + ( z z + z z + ( z + z z ) z + 1) z + ( z + z z + ( z + z z ) z +( z z + z z +( z + z z ) z +1) z ) z +( z +( z + z z ) z +( z z + z z +( z + z z ) z +1) z +( z + z z +( z + z z ) z +( z z + z z +( z + z z ) z +1) z ) z ) z +( z z + z z +( z + z z ) z +( z + z z + ( z z + z z + ( z + z z ) z + 1) z ) z + ( z + z z + ( z + z z ) z +( z z + z z +( z + z z ) z +1) z ) z +( z +( z + z z ) z +( z z + z z +( z + z z ) z +1) z +( z + z z + ( z + z z ) z + ( z z + z z + ( z + z z ) z + 1) z ) z ) z + 1) z = 0 z z + ( z + z z ) z + ( z z + z ( z + z z ) + 1) z + ( z z z + z + ( z + z z ) z +( z z + z ( z + z z )+1) z ) z +( z + z z +( z z + z ( z + z z )+1) z +( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z ) z +( z z +( z + z z ) z +( z z + z ( z + z z )+1) z +( z z z + z +( z + z z ) z + ( z z + z ( z + z z ) + 1) z ) z ) z ) z + ( z z + ( z + z z ) z + ( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z ) z +( z z +( z + z z ) z +( z z + z ( z + z z )+1) z +( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z ) z ) z +( z + z z +( z z + z ( z + z z )+1) z +( z z z + z + ( z + z z ) z + ( z z + z ( z + z z ) + 1) z ) z + ( z z + ( z + z z ) z + ( z z + z ( z + z z )+1) z +( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z ) z ) z ) z +1) z +( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z +( z z +( z + z z ) z +( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z ) z +1) z +( z z +( z + z z ) z +( z z + z ( z + z z )+1) z +( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z ) z ) z +( z + z z +( z z + z ( z + z z )+1) z +( z z z + z + ( z + z z ) z + ( z z + z ( z + z z ) + 1) z ) z + ( z z + ( z + z z ) z + ( z z + z ( z + z z )+1) z +( z z z + z +( z + z z ) z +( z z + z ( z + z z )+1) z ) z ) z ) z ) z = 0 We have chosen this example as a continuation of [29, Example 5.3] and [44, Figure 6]. he remaining 13 equations are longer, but can be readily obtained. This hopefully illustratesthat the method is computationally immediate. (cid:3) Remark 3.4. (i) One may consider the moduli stack ob(Sh s Λ f ( R )) of sheaves with mi-crolocal rank-1 along Λ f , instead of ob(Aug + (Λ f )). By [81], there is an equivalence ofcategories Aug + (Λ f ) ∼ = Sh f ( R ). The stack ob(Sh f ( R )) is a X -cluster variety; theassociated A -cluster variety in the cluster ensemble is the moduli of framed sheaves[92]. In short, the cluster algebra A ( f ) could have been defined in terms of the mod-uli space of constructible sheaves microlocally supported in Λ, instead of Floer theory.(ii) The Aug + -category is Floer-theoretical in nature, e.g. its morphisms are certain Floerhomology groups. It would have also been natural to consider the partially wrappedFukaya category W ( C , Λ f ), as defined [48, 98], or the infinitesimal Fukaya category F uk ( C , Λ) [78, 74]. These are Floer-theoretical Legendrian invariants associated toΛ f , and thus the singularity f ∈ C [ x, y ], which might be of interest on their own.4. A few Computations and Remarks
Consider the derived dg-category Sh Λ ( M ) of constructible sheaves in a closed smooth ma-nifold M microlocally supported at a Legendrian link Λ ⊆ ( ∂T ∞ M, ξ st ), e.g. as introducedin [95, Section 1]. Equivalently, one may consider a conical Lagrangian L ⊆ T ∗ M instead ofΛ ⊆ ( T ∞ M, ξ st ); in practice, the input data is a wavefront π (Λ) ⊆ M [8]. Let µ sh denotethe sheaf of microlocal sheaves defined in [77, Section 5]. There are two situations we con-sider, depending on whether the focus is on the Weinstein pair ( C , Λ f ) or on the Weinstein4-manifold W (Λ f ):(i) Sheaf Invariants of the Weinstein pair ( C , Λ f ) . The category of microlocalsheaves µ sh( L ( f )) is an invariant of ( C , Λ f ), as established in [57, 77, 95]. Inthis case, the global sections µ sh( L ( f )) is a category equivalent to the more famil-iar Sh Λ( f ) ( R ). For simplicity, we focus on the moduli stack S ( f ) ⊆ ob(Sh Λ( f ) ( R ))of simple sheaves, whose microlocal support is rank one, microlocally supported inthe Legendrian link of an isolated plane curve singularity f : C −→ C . See [66,Section 7.5] or [57, Section 1.10] for a detailed discussion on simple sheaves. In ourcase Λ = Λ( f ), S ( f ) is an Artin stack of finite type [95, Prop. 5.20], and typicallyis an algebraic variety or a G -quotient thereof, with G = ( C ∗ ) k or GL( k, C ). Notethat µ sh( L ( f )) is equivalent to the wrapped Fukaya category of C stopped at Λ f [47].(ii) Sheaf Invariants of the Weinstein 4-manifold W (Λ f ) . The category µ sh( L ( f ))of microlocal sheaves [77] on a Lagrangian skeleton L ( f ) ⊆ W (Λ f ) is an invariantof W (Λ f ), up to Weinstein homotopy [77] and up to symplectomorphism [47]. Thiscategory is Sh ϑ ( f ) ( M f ), or µloc ( L ( f )), in the notation of [93], i.e. the global sectionsof the Kashiwara-Schapira sheaf of dg-categories [93, Prop. 3.5] on the Lagrangianskeleton L ( f ). For simplicity, we focus on the moduli stack Θ( f ) ⊆ µ sh( L ( f )) ofsimple sheaves as well. Note that µ sh( L ( f )) is equivalent to the wrapped Fukayacategory of W (Λ f ) by [47]. Even if the equations themselves, being rather long, may not be particularly enlightening. The cluster algebra structure for A ( f ) defined by [49] is obtained by pulling-back the cluster algebrastructure of the open Bott-Samelson cell associated to β . There should exist a cluster algebra structure on A ( f ) defined strictly in Floer-theoretical terms. Thanks go to V. Shende for helpful discussions on sheaf invariants. Invariance up to Weinstein homotopy [27], and also symplectomorphism of Liouville pairs. The category µ sh( L ( f )) is likely not an invariant of the Weinstein 4-manifold W (Λ f ) itself. he moduli stack S ( f ) in (i) is isomorphic to the stack of simple sheaves in ob(Sh ϑ ( f ) ( M f )).This is because the union of R ⊆ T ∗ R and the Lagrangian cone of Λ ⊆ ( T + R , ξ st ) isa Lagrangian skeleton for the relative Weinstein pair ( C , Λ), so is L ( f ) by Theorem 1.1,and ob(Sh ϑ ( f ) ( M f )) is an invariant of the Weinstein pair ( C , Λ), independent of the choiceof Lagrangian skeleton. Thus, the difference between S ( f ) and Θ( f ) is at the boundary,which for S ( f ) might give monodromy contributions (and these become trivial on Θ( f )). Inother words, since L ( f ) is obtained from L ( f ) by attaching 2-disks (to close the boundaryof the Milnor fiber M f ), the category µ sh( L ( f )) is a homotopy pull-back of µ sh( L ( f )). Inparticular, the moduli stacks of simple microsheaves are related as above. Remark 4.1.
There are currently two methods for computing S ( f ): either by direct means,as exemplified in [95], or by using the equivalence of categories Aug + (Λ( f )) ∼ = Sh s Λ f ( R ) from[81, Theorem 1.3], the latter being denoted by C (Λ f ) in [81]. Thanks to the computationaltechniques available for augmentation varieties, the moduli of objects ob( Aug + (Λ( f ))) isreadily computable for ( − f ) could be computed directly, or by means of the isomorphismto the wrapped Fukaya category of W (Λ f ). (cid:3) In this section, we take to opportunity to build on [77, 93] and perform an actual computationfor a class of Cal-Skeleta coming from Theorem 1.1.
Figure 9.
A Cal-skeleta L ( f n +1 ) for the Weinstein 4-manifolds W (Λ( A n +1 )).4.1. Cal-Skeleta for A n -Singularities. Consider the A n -singularity f n ( x, y ) = x n +1 + y .The Legendrian Λ( A n ) ⊆ ( R , ξ st ) associated to the singularity is the max-tb Legendrian(2 , n + 1)-torus link. By Theorem 1.1, a Lagrangian skeleton L ( f n ) for the Weinstein pair( C , Λ f ) is obtained by attaching n / − ( − n / (cid:98) n − (cid:99) –genussurface along an A n -Dynkin chain of embedded curves. Similarly, Corollary 1.2 impliesthat a Lagrangian skeleton L ( f n ) for the Weinstein 4-manifold W n = W (Λ( A n )) is givenby attaching n (cid:98) n − (cid:99) –genus surface along an A n -Dynkin chain, as depicted inorange in Figure 10, see also Figure 9.Let us compute Θ( f n ) for n ∈ N even, so that Λ( A n ) is a knot; the n ∈ N odd case issimilar. The key technical tool is the Disk Lemma [65, Lemma 4.2.3]. The complement M f \ ϑ ( f ) of the vanishing cycles is a 2-disk, and the category of local systems is just C -mod. Thus, the moduli of simple constructible sheaves on M f microlocally supported on(the Legendrian lift of) the vanishing cycles ϑ ( f ) consists of a vector space V = C and maps x , x , . . . , x n ∈ End ( V ), one associated to each vanishing cycle. This is depicted in Figure10 for n = 2 ,
6, and note that n = | ϑ ( f ) | . Denote by L ( f n ) ⊆ T ∗ M f the Lagrangian skeletongiven by M f union the conormal lifts of ϑ ( f ). These maps are not necessarily invertible in µ sh( L ( f n ) ). Should the reader be willing to use the surgery formula, this wrapped Fukaya category may be presentedas modules over the Legendrian DGA of Λ f . (This is only informative and not needed for the present purposes.) igure 10. The Cal-skeleta L ( f ) for the Weinstein 4-manifolds W (Λ( A ))and W (Λ( A )). The relative Cal-skeleta L ( f ) for the corresponding Weinsteinpairs ( C , Λ( A )) and ( C , Λ( A )) are obtained by introducing one punctureto the surfaces.The skeleton L ( f n ) is obtained by attaching n Lagrangian 2-disks to L ( f n ) , i.e. L ( f n ) isthe homotopy push-out of L ( f n ) and the disjoint union of n L ( f n ) is given by the homotopy pull-back of the categoryof microlocal sheaves on L ( f n ) and the category of microlocal sheaves on n disjoint 2-disks(which are just copies of C -mod). Attaching a 2-disk along a vanishing V i cycle in ϑ ( f ), i ∈ [1 , n ], has the effect of trivializing the “monodromy” corresponding map x i , as explainedin [93] and [65, Section 4.2]. Here, the monodromy is given by restricting a microlocal sheafto (an arbitrarily small neighborhood of) V i . Note that in this restriction, we land into a1-dimensional Lagrangian skeleton given by a circle V i ∼ = S union conical segments comingfrom the adjacent vanishing cycles. Let us call γ i the composition of maps from cone ( x i )to itself obtained by going around V i , each of the maps coming from traversing a segment.Then, the trivialization is a homotopy to the identity, and it translates into adding a map α i such that x i α i − γ i . Example 4.2.
Consider the map x in Figure 10 (Left), which is depicted transversely tothe vanishing cycle V . The restriction of a microlocal sheaf to a neighborhood of V gives amicrolocal sheaf for the skeleton S ∪ T ∗ , + p S ⊆ T ∗ S , where T ∗ , + p S is the positive half of thecotangent fiber at a point p ∈ S . Such a microlocal sheaf is described by a (complex of ) vectorspace(s) and an endomorphism. In this case the vector space is V = C and this endomorphismis identified with γ = x . Hence, trivializing along V adds a map α ∈ End ( C ) , which wecan think of as a variable α ∈ C , such that x α + 1 = − x . Similarly, trivializing along V ,with γ = − α , adds a variable α ∈ C such that x α = − α . Hence Θ( f ) is the affinevariety Θ( f ) = { ( x, y, z ) ∈ C : xyz + x − z − } . This affine variety appears in the study of isomonodromic deformations of the Painlev´e Iequation [102, Section 3.10] , see also [18, Section 5] . (cid:3) The vanishing cycles V , V n have simpler monodromies γ , γ n , as they only intersect one othervanishing cycle. Adding the 2-disks to the skeleton L ( f n ) along V , V n yields a category ofmicrolocal sheaves whose moduli space of simple objects is described by that of L ( f n ) andthe two equations x α + 1 = − x and x n α n + 1 = − α n − . For each of the middle vanishingcycles V i , 2 ≤ i ≤ n −
1, we have the monodromy γ i = α i − x i +1 . In consequence, attachingthe n L ( f n ) along all the curves V i , i ∈ [1 , n ], leads to the moduli spaceΘ( f ) ∼ = { ( x i , α i ) ∈ ( C ) n : x α +1 = − x , x n α n +1 = − α n − , x j α j = α j − x j +1 , j ∈ [2 , n − } . Remark 4.3.
Consider ( n + 3)-tuples of vectors ( v , . . . , v n +3 ) ∈ C , modulo GL ( C ), theequations for Θ( f ) above can be read directly by writing the ( n + 3)-tuple as We had written “monodromy” in quotations because it is not a priori necessarily invertible. (cid:19) , (cid:18) (cid:19) , (cid:18) − x (cid:19) , (cid:18) α x (cid:19) , (cid:18) α x (cid:19) , (cid:18) α x (cid:19) , (cid:18) α x (cid:19) , . . . , (cid:18) α n − x n (cid:19) , (cid:18) α n − (cid:19) , and imposing v i ∧ v i +1 = 1, where we have use the GL ( C ) gauge group to trivialize thefirst two vectors, and one component of the third and last vectors. P. Boalch [18] names thismoduli stack after Y. Sibuya [96]. Note that [18, Section 5] points out that some of theseequations were initially discovered by L. Euler in 1764 [41]. In the context of open Bott-Samelson cells [92, 94], these spaces appear as the open positroid varieties { p ∈ Gr(2 , n + 3) : P i,i +1 ( p ) (cid:54) = 0 } , where P i,j is the Pl¨ucker coordinate given by the minor at the i and j columns,and the index i is understood Z / ( n + 3)-cyclically. (cid:3) Finally, we notice that the cohomology H ∗ (Θ( f ) , C ), or that of H ∗ ( S ( f ) , C ), can be aninteresting invariant [95, Section 6]. For the case of A n -singularities, we can use the fact thatthese are actually cluster varieties of A n -type in order to compute their cohomology using [69].For n = 2 m ∈ N even, and removing any C ∗ -factors coming from frozen variables, one obtainsthat the Abelian graded cohomology group is isomorphic to Q [ t ] /t m +1 , | t | = 2. In general,the mixed Hodge structure for these moduli spaces can be non-trivial, but for singularities of A n -type, these cohomologies are of Hodge-Tate type, and entirely concentrated in H k, ( k,k ) . Remark 4.4.
It would be valuable to understand the relation between sheaf invariants ofa singularity f ∈ C [ x, y ], such as µ sh( L ( f )) and µ sh( L ( f )), and classical invariants fromsingularity theory [3, 9, 10]. In particular, it could be valuable to develop more systematicmethods to compute µ sh( L ( f )) and µ sh( L ( f )) both directly and from a divide. (cid:3) Structural Conjectures on Lagrangian Fillings
Let Λ ⊆ ( S , ξ st ) be a max-tb Legendrian link. The classification of embedded exact La-grangian fillings L ⊆ ( D , λ st ) with fixed boundary Λ, up to Hamiltonian isotopy, is a centralquestion. The only Legendrian Λ for which a complete classification exists is the standardunknot [32]. In this case, the standard Lagrangian flat disk is the unique filling: there isprecisely one exact Lagrangian filling, up to Hamiltonian isotopy. The recent developments[20, 22, 23, 49] show that such finiteness is actually rare: e.g. the max-tb torus links ( n, m )admit infinitely many exact Lagrangian filling, up to Hamiltonian isotopy, if n, m ≥
4. Thisfinal section states and discusses Conjectures 5.1 and 5.4, which might help in the classifica-tion of exact Lagrangian fillings of Legendrian links.
Geometric Strategy . Given Λ ⊆ ( S , ξ st ), we would like to know whether it admits finitelymany Lagrangian fillings or not, and in the finite case provide the exact count. Theorem1.1 provides insight for the class of Legendrian links Λ ⊆ ( S , ξ st ) that are algebraic linksand, more generally, arise from a divide. Indeed, Lagrangian fillings for Λ can be constructedby using the Lagrangian skeleta for the Weinstein pair ( C , Λ) built in the statement. Forinstance, the inclusion of the Milnor fiber M ˜ f ⊆ L ˜ f provides an exact Lagrangian filling,and performing Lagrangian disk surgeries along the Lagrangian 2-disks in L ˜ f \ M ˜ f , whichbound vanishing cycles, will potentially yield new Lagrangian fillings. This strategy can beimplemented in certain cases but, in general, one must be able to find an embedded Lagrangiandisk in the new Lagrangian skeleton (with an embedded boundary curve), in order to performthe next Lagrangian disk surgery. Curves being immersed rather than embedded , might apriori represent a challenge. This geometric scheme has the following algebraic incarnation. Equivalently, the existence of curves with zero algebraic intersection but non-empty geometricintersection. The vanishing cycles can be organized as a quiver Q , the additional data of a superpotential ( Q, W )should be helpful in solving the disparity between immersed and embedded curves in the Milnor fiber. lgebraic Strategy . Consider the intersection quiver Q ϑ ( ˜ f ) of vanishing cycles for a realMorsification ˜ f , Lagrangian disk surgeries induce mutations of the quiver [93] and the (mi-crolocal) monodromies of a local system serve as cluster X -variables [23, 94]. Thus, thecluster algebra A ( Q ( f )) associated to the quiver, as it appears in [44], governs possible exactLagrangian fillings for the Legendrian link Λ. That is, a Lagrangian filling L ⊆ ( D , λ st )yields a cluster chart for this algebra [49, 94], and the Lagrangian skeleta from Theorem 1.1provide a geometric realization for the quiver in the form of an exact Lagrangian filling withambient Lagrangian disks ending on it.The recent developments [20, 49, 93, 94] and the existence of the Lagrangian skeleta inTheorem 1.1 shyly hint towards the fact that, possibly, Lagrangian fillings are classified bythe cluster algebra A ( Q ( f )). That is, every cluster chart in A ( Q ( f )) is induced by precisely one exact Lagrangian filling. It should be emphasized that this is not known for anyΛ ⊆ ( R , ξ st ) except the standard Legendrian unknot. It is possible that the case of theHopf link Λ( A ) can be solved by building on the techniques in [89], which classifies exactLagrangian tori near the Whitney sphere . Having informed the reader on the currentlyavailable evidence, the following conjectural guide might be helpful. Conjecture 5.1 (ADE Classification of Lagrangian Fillings) . Let Λ ⊆ ( R , ξ st ) be the Leg-endrian rainbow closure of a positive braid such that the mutable part of its brick quiver isconnected. Then one of the following possibilities occur:
1. Λ is smoothly isotopic to the link of the A n -singularity.Then Λ has precisely n +2 (cid:0) n +2 n +1 (cid:1) exact Lagrangian fillings.
2. Λ is smoothly isotopic to the link of the D n -singularity.Then Λ has precisely n − n (cid:0) n − n − (cid:1) exact Lagrangian fillings.
3. Λ is smoothly isotopic to the link of the E , E or the E -singularities.Then Λ has precisely 833, 4160, and 25080 exact Lagrangian fillings, respectively.
4. Λ has infinitely many exact Lagrangian fillings.
The following comments are in order:(i) In [45], S. Fomin and A. Zelevinsky classify cluster algebras of finite type. Thisis an ADE-classification, parallel to the classification of simple singularities [9], theCartan-Killing classification of semisimple Lie algebras, finite crystallographic rootsystems (via Dynkin diagrams) and the like. Thus, Conjecture 5.1 first states thatΛ will have finitely many exact Lagrangian fillings, up to Hamiltonian isotopy, if andonly if the associated quiver is ADE.(ii) The case of Λ = Λ f an algebraic link associated to a non-simple singularity f ∈ C [ x, y ]of a plane curve follows from [20], and the case of a Legendrian Λ with a non-ADEunderlying quiver has recently been proven in [50]. These approaches are based onthe following fact: if there exists an embedded exact Lagrangian cobordism fromΛ − to Λ + and Λ − admits infinitely many Lagrangian fillings, then so does Λ + . See[22, 83] and [20, Section 6]. This itself initiates the quest for finding the smallest Legendrian link which admits infinitely many exact Lagrangian fillings. At present,if we measure the size of a link Λ as π (Λ) + g (Λ), g (Λ) the (minimal) genus of a(any) embedded Lagrangian filling, the smallest known Legendrian link has g (Λ) = 1 That is, two Lagrangian fillings inducing the same cluster chart in A ( Q ( f )) are Hamiltonian isotopic and every cluster chart is induced by at least one Lagrangian filling. See also [28], which appeared during the writing of this manuscript. nd two components π (Λ) = 2. Intuitively, it is the geometric link corresponding tothe ˜ A , cluster algebra.(iii) According to (ii) above, the missing ingredient for Conjecture 5.1 is showing that(1), (2) and (3) hold. For the A n -case (1), it is known that there are at least thestated Catalan number worth of exact Lagrangian fillings, distinct up to Hamilton-ian isotopy. This was originally proven by Y. Pan [84] and subsequently understoodin [94, 99] from the perspective of microlocal sheaf theory. It remains to show thatany exact Lagrangian filling of Λ( A n ) is Hamiltonian isotopic to one of those; the firstunsolved case is the Hopf link Λ( A ) having exactly two embedded exact Lagrangianfillings. For the Λ( D n ) , Λ( E ) , Λ( E ) and Λ( E ) cases in Conjecture 5.1, one needsto first find the corresponding number of distinct Lagrangian fillings, and then showthese are all. The construction part should be relatively accessible, in the spirit ofeither [23, 84, 94], and it is reasonable to suspect that these many fillings can bedistinguished using either augmentations or microlocal monodromies. (iv) The numbers appearing in Conjecture 5.1.(i)-(iii) are the number of cluster seeds forthe corresponding cluster algebra. Precisely, consider a root system of Cartan-Killingtype X n , e , . . . , e n its exponents and h the Coxeter number. Then the numbers inConjecture 5.1 are N ( X n ) = (cid:81) ni =1 ( e i + h + 1)( e i + 1) − for X n = A n , D n , E , E , E .The brick graph of a positive braid is defined in [13, 91], it can be enhanced to a quiver,which we call the brick quiver, following the algorithm in [92, Section 3.1] or [49, Section4.2], which itself generalizes the wiring diagram construction in [16, 42]. Remark 5.2.
The hypothesis of the mutable part of its brick quiver being connected isnecessary. We could otherwise add a meridian to any positive braid, which would createa disconnected quiver; the resulting cluster algebra would be a product with A , whichpreserves being of finite type. It stands to reason that adding a meridian to a Legendrianlink Λ would yield a Legendrian link Λ ∪ µ with exactly twice as many Lagrangian fillings. Itis clear that there are at least twice as many Lagrangian fillings for Λ ∪ µ , as there are twodistinct Lagrangian cobordisms from Λ to Λ ∪ µ . The simplest case is Λ = Λ the standardLegendrian unknot and Λ ∪ µ ∼ = Λ( A ) the Hopf link, which should have 2 = 2 · A ), so thatΛ( A ) ∪ µ ∼ = Λ( D ), in line with Λ( D ) conjecturally having 4 = 2 · (cid:3) Note that the article [22] has provided the first examples of Legendrian links Λ ⊆ ( S , ξ st )which are not rainbow closures of positive braids and yet they admit infinitely many La-grangian fillings, up to Hamiltonian isotopy. These Legendrian links have components whichare stabilized, not max-tb, and thus they cannot be rainbow closures of any positive braid.It would be interesting to extend Conjecture 5.1 to a larger class of links, possibly including( − Remark 5.3.
To the author’s knowledge, [32, 84], Theorem 1.1, and the recent [20, 23, 22, 49,50], constitute the current evidence towards Conjecture 5.1. That said, parts of Conjecture5.1 might have appeared in the symplectic folklore in one form or another. The adventof Symplectic Field Theory led to the mantra of “pseudoholomorphic curves or nothing” ,the subsequent arrival of microlocal sheaf theory to symplectic topology led to “sheaves or In particular, this would show that the two possible Polterovich surgeries [87] of a 2-dimensional La-grangian node are the only two exact Lagrangian cylinders near the node, up to Hamiltonian isotopy. Showing these exhaust all fillings, up to Hamiltonian isotopy, is another matter, possibly much morechallenging. That is, if pseudoholomorphic invariants cannot distinguish two objects, they must be equal. othing”. In the current zeitgeist, cluster algebras provide a new algebraic invariant thatone might hope to be complete. In this sense, I would like to mention Y. Eliashberg, D.Treumann, H. Gao, D. Weng and L. Shen as some of the colleagues which might have alsodiscussed or hinted towards parts of Conjecture 5.1. (cid:3)
Finally, an ADE-classification is often part of a larger classification , involving a few addi-tional families. For instance, simple Lie algebras are classified by connected Dynkin diagrams,which are A n , D n , E , E , E , known as the simply laced Lie algebras, and B n , C n , F and G .These latter cases, B n , C n , F and G , are interesting on their own right. For instance, simplesingularities are classified according to A n , D n , E , E , E , and B n , C n , F then arise in theclassification of simple boundary singularities [9, Chapter 17.4], as shown in [10, Chapter 5.2].(See also D. Bennequin’s [15, Section 8] and [7].) In general, the tenet is that B n , C n , F and G arise when classifying the same objects as in the ADE-classification with the additional data of a symmetry. This a perspective (and technique) called folding , ubiquitous in thestudy of B n , C n , F , G , which is developed in [46, Section 2.4] for the case of cluster algebras.Let us consider a Legendrian Λ ⊆ ( R , ξ st ), a Lagrangian filling L ⊆ ( R , λ st ), ∂L = Λ,and a finite group G acting faithfully on ( R , λ st ) by exact symplectomorphisms, inducingan action on the boundary piece ( R , ξ st ) by contactomorphisms. For instance, s : R −→ R , s ( x, y, z, w ) = ( − x, − y, z, w ) is an involutive symplectomorphism which restricts to thecontactomorphism ( x, y, z ) (cid:55)→ ( − x, − y, z ) on its boundary piece ( R , ker { dz − ydx } ). Letus define an exact Lagrangian G -filling of Λ to be an exact Lagrangian filling L of Λ suchthat G ( L ) = L and G (Λ) = Λ setwise. Also, by definition, we say Λ ⊆ ( R , ξ st ) admits a G -symmetry if there exists a faithful action of G by contactomorphisms on ( R , ξ st ) suchthat G (Λ) = Λ setwise. Examples of such symmetries can be readily drawn in the frontprojection, as shown in Figure 11 for Λ( A ) , Λ( D ) , Λ( E ) and Λ( D ). Following the tenetabove, the following classification might be plausible: Conjecture 5.4 (BCFG Classification of Lagrangian Fillings) . Let Λ( β ) ⊆ ( S , ξ st ) theLegendrian rainbow closure of a positive braid β :
1. ( B n ) If Λ( β ) = Λ( A n − ) , the Z -symmetry ( x, z ) −→ ( − x, z ) for the front depictedin Figure 11 lifts to a Z -symmetry of Λ( A n − ) . Then Λ( A n − ) has precisely (cid:0) nn (cid:1) exact Lagrangian Z -fillings.
2. ( C n ) If Λ( β ) = Λ( D n +1 ) , the Z -symmetry ( x, z ) −→ ( − x, z ) for the front depictedin Figure 11 lifts to a Z -symmetry of Λ( D n +1 ) . Then Λ( D n +1 ) has precisely (cid:0) nn (cid:1) exact Lagrangian Z -fillings.
3. ( F ) If Λ( β ) = Λ( E ) , the Z -symmetry ( x, z ) −→ ( − x, z ) in the front depicted inFigure 11 lifts to a Z -symmetry of Λ( E ) . Then Λ( E ) has precisely exact La-grangian Z -fillings.
4. ( G ) If Λ( β ) = Λ( D ) , the Z -symmetry in the front depicted in Figure 11 lifts to a Z -symmetry of Λ( D ) . Then Λ( D ) has precisely exact Lagrangian Z -fillings. For the G -case in Conjecture 5.4.(4), it might be helpful to notice that the D -singularity istopologically equivalent to f ( x, y ) = x + y . The Z -symmetry cyclically interchanges the As with the previous two cases, there is no particularly hard evidence for “cluster algebras or nothing”. The larger classification is an ABCDEFG-classification, which admittedly does not roll off the tongue. The study of boundary singularities can be understood as the study of singularities taking into accounta certain Z -symmetry. igure 11. Legendrian fronts for Λ( A n − ) , Λ( D n +1 ) , Λ( E ) , Λ( D ) with G -symmetries, G = Z , Z . The upper row exhibits these symmetric fronts asdivides of the associated singularities, and the lower row depicts them in thestandard front projection ( x, y, z ) (cid:55)→ ( x, z ) for a Darboux chart ( R , ξ st ).three linear branches of this singularity. In particular, we can draw a front for the LegendrianΛ( D ) as the (3 , β = ( σ σ ) . For the B n -case in Conjecture 5.4.(1), the construction of (cid:0) nn (cid:1) distinct Lagrangian Z -fillingslikely follows from adapting [84]. Indeed, in the Z -invariant front for Λ( A n − ), as depictedin Figure 11, there are n crossing to the left, equivalently right, of the Z -symmetry axis. Wecan construct a Z -filling of Λ( A n − ) by opening those n crossings in any order, with the rulethat we simultaneously open the corresponding Z -symmetric crossing. Should one distin-guish these Z -fillings via their augmentations, as in [84], an appropriate G -equivariant Floertheoretic invariant (e.g. G -equivariant DGA and its augmentations) needs to be defined. Theperspective of microlocal sheaves [99] yields combinatorics closer to those of triangulations[45, Section 12.1], modeling A n -cluster algebras, and thus might provide a simpler route todistinguish these fillings. In either case, Conjecture 5.4 calls for a G -equivariant theory ofinvariants for Legendrian submanifolds of contact manifolds.5.1. Some Questions.
We finalize this section with a series of problems on Weinstein 4-manifolds and their Lagrangian skeleta. To my knowledge, there are several unansweredquestions at this stage, including checkable characterizations of Weinstein 4-manifolds of theform W (Λ f ), where Λ f is the Legendrian link of an isolated plane curve singularity. Hereare some interesting, yet hopefully reasonable, problems: Problem 1 . Find a characterization of Legendrian links Λ ⊆ ( S , ξ st ) for which ( C , Λ), or W (Λ), admits a Cal-skeleton. (Ideally, a verifiable characterization.) Problem 2 . Find necessary and sufficient conditions for a Lagrangian skeleton L ⊆ ( W, λ ) toguarantee that the Stein manifold (
W, λ ) is an affine algebraic manifold. Similarly, charac-terize Legendrian links Λ ⊆ ( S , ξ st ) such that W (Λ) is an affine algebraic variety. The Z -action should coincide with the loop Ξ ◦ ( δ − ◦ Ξ ◦ δ ) from [20, Section 2]. The naive count of 312-pattern avoiding permutations from [31, 84] would indicate that there are n (cid:0) nn (cid:1) such Lagrangian Z -fillings, instead of (cid:0) nn (cid:1) . Thus, should Conjecture 5.4 hold, there must be an additionalrule for Z -fillings (not just those in [84, Lemma 3.10]), possibly related to the fact that the crossing closestto the Z -axis is different from the rest. ote that the standard Legendrian unknot Λ ∼ = Λ( A ) ⊆ ( S , ξ st ) and the max-tb Hopf linkΛ( A ) ⊆ ( S , ξ st ) yield affine Weinstein manifolds, as we have W (Λ ) ∼ = { ( x, y, z ) ∈ C : x + y + z = 1 } , W (Λ( A )) ∼ = { ( x, y, z ) ∈ C : x + y + z = 1 } . By [21, Section 4.1], the trefoil Λ( A ) is also an example of such a Legendrian link, as W (Λ( A )) ∼ = { ( x, y, z ) ∈ C : xyz + x + z + 1 = 0 } . Heuristic computations indicate that Λ( A ) and Λ( D ) also have this property. See [70, 71]for a source of necessary conditions, and [90] for (topological) skeleta of affine hypersurfaces. Problem 3 . Find necessary and sufficient conditions for a Lagrangian skeleton L ⊆ ( W, λ ) toguarantee that the Stein manifold (
W, λ ) is flexible. (Again, a verifiable characterization.)Similarly, characterize Λ ⊆ ( S , ξ st ) such that W (Λ) is flexible.Note that affine manifolds W ⊆ C N might be flexible [21, Theorem 1.1]. In particular, itcould be fruitful to compare Lagrangian skeleta of X m = { ( x, y, z ) ∈ C : x m y + z = 1 } for m = 1 and m ≥
2, e.g. the ones provided in [90].
Problem 4 . Suppose that a Weinstein 4-manifold W = W (Λ) is obtained as a Lagrangian2-handle attachment to ( D , ω st ). Given a Cal-skeleton L ⊆ ( W, λ ), devise an algorithm tofind one such possible Legendrian Λ ⊆ ( ∂ D , ξ st ). Problem 5 . Let L ⊆ ( W, λ ) be a closed exact Lagrangian surface. Study whether thereexists a Cal-skeleton L ⊆ ( W, λ ) such that L ⊆ L . In addition, study whether there existsa Legendrian handlebody Λ ⊆ ( k S × S , ξ st ), so that W = W (Λ), and L is obtained bycapping a Lagrangian filling of a Legendrian sublink of Λ.See [103] for an interesting construction in the case of Bohr-Sommerfeld Lagrangian subman-ifolds and see [33] for a general discussion on regular Lagrangians. The nearby Lagrangianconjecture holds for W = T ∗ S , T ∗ T , thus the answer is affirmative in these cases. Problem 6 . Characterize which cluster algebras A can arise as the ring of functions of theaugmentation stack of a Legendrian link Λ ⊆ ( S , ξ ).By using double-wiring diagrams [16], (generalized) double Bruhat cells satisfy this property[92]. It is proven in [22, 49] that the cluster algebras A ( ˜ D n ) of affine D n -type have thisproperty. Heuristic computations indicate that the affine types ˜ A p,q also verify this [22]. Itmight be reasonable to conjecture that cluster algebras of surface type all have this property. Problem 7 . Let a (Λ) be the number of A -arboreal singularities of a Cal-skeleton L ⊆ ( W, λ ).Find the number a ( W ) := min L ⊆ W a ( L ), where L ⊆ W runs amongst all possible Cal-skeleta. In particular, characterize Weinstein 4-manifolds ( W, λ ) with a ( W ) = 0. Problem 8 . Develop a combinatorial theory of symplectomorphisms in Symp(
W, dλ ) in termsof Cal-skeleta L ⊆ ( W, λ ).This is being developed in the case dim( W ) = 2 by using A’Campo’s tˆete-`a-tˆete twists[5, Section 3], see also [6, Section 5]. A (symplectic) mapping class in Symp( W, dλ ) isa composition of Dehn twists in this 2-dimensional case. This is no longer the case indim( W ) = 4, e.g. due to the existence of Biran-Giroux’s fibered Dehn twists, confer [101,Section 3] and [104, Section 2]. Note that π (Symp( W )) might be infinite even if W containsno exact Lagrangian 2-spheres [20]. Problem 9 . Compare Cal-skeleta L ⊆ ( W , λ ), L ⊆ ( W , λ ) for exotic Stein pairs W , W .That is, W is homeomorphic to W , but not diffeomorphic. In particular, investigate skeletalcorks : combinatorial modifications on a Cal-skeleton that can produce exotic Stein pairs.In [79], H. Naoe uses Bing’s house [17] to study some such corks. Not closed in this case. See [27] for flexible Weinstein manifolds. In the 4-dimensional case above, we might just define flexibleas being of the form W = W (Λ) where Λ is a stabilized knot. roblem 10 . Find a contact analogue of Turaev’s Shadow formula [100, Chapter 10] for thecontact 3-dimensional boundary in terms of the combinatorics of a Cal-skeleton L ⊆ ( W, λ ).That is, find a contact invariant of ( ∂W, λ | ∂W ) which can be computed in terms of thecombinatorics of L ⊆ ( W, λ ). References [1] Norbert A’Campo. Sur la monodromie des singularit´es isol´ees d’hypersurfaces complexes.
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