Braid Loops with infinite monodromy on the Legendrian contact DGA
BBRAID LOOPS WITH INFINITE MONODROMY ON THELEGENDRIAN CONTACT DGA
ROGER CASALS AND LENHARD NG
Abstract.
We present the first examples of elements in the fundamental group of the spaceof Legendrian links in ( S , ξ st ) whose action on the Legendrian contact DGA is of infiniteorder. This allows us to construct the first families of Legendrian links that can be shownto admit infinitely many Lagrangian fillings by Floer-theoretic techniques. These familiesinclude the first known Legendrian links with infinitely many fillings that are not rainbowclosures of positive braids, and the smallest Legendrian link with infinitely many fillingsknown to date. We discuss how to use our examples to construct other links with infinitelymany fillings, in particular giving the first Floer-theoretic proof that Legendrian ( n, m ) toruslinks have infinitely many Lagrangian fillings if n ≥ , m ≥ n, m ) = (4 , , (4 , Introduction
In this article, we construct Legendrian loops for several families of Legendrian links in thestandard contact 3-sphere ( S , ξ st ) and show that their monodromy action on their Legen-drian contact DGA is of infinite order. These are the first examples of such a Floer-theoreticphenomenon. We provide consequences of these results, including the first examples of Leg-endrian links in ( S , ξ st ) with a stabilized component that admit infinitely many Lagrangianfillings. To our knowledge, these are the first known examples of Legendrian links with in-finitely many Lagrangian fillings which are not the rainbow closure of a positive braid. Oneof these Legendrian links has 2 components and, with its Lagrangian fillings being of genus1, is arguably the smallest known Legendrian link, in terms of genus and components, withinfinitely many Lagrangian fillings. Finally, we show how to Floer-theoretically detect theexistence of infinitely many Lagrangian fillings for the max-tb Legendrian torus links ( n, m ), n ≥ , m ≥ n, m ) = (4 , , (4 , g ≥
2, we construct Weinstein 4-manifolds homotopic to the 2-sphere whose wrappedFukaya categories can distinguish infinitely many exact closed Lagrangian surfaces of thatgenus.The manuscript also develops technical results on the Legendrian contact DGA, of indepen-dent interest, needed for our argument. In particular, we present a combinatorial model forcomputing DGA morphisms associated to decomposable Lagrangian cobordisms L , where themorphisms are enhanced over integer group ring coefficients. We show that this is isomor-phic to the abstract enhancement previously developed by Karlsson, thus proving invarianceand allowing us to perform explicit computations over Z [ H ( L )]. This integrally enhancedpackage is then used to prove the above Floer-theoretical results concerning infinitely manyLagrangian fillings. Mathematics Subject Classification.
Primary: 53D10. Secondary: 53D15, 57R17. a r X i v : . [ m a t h . S G ] J a n .1. Context.
Legendrian links in contact 3-manifolds [Ben83, Ad90] are instrumental inthe study of 3-dimensional contact geometry [OS04, Gei08]. The study of their Lagrangianfillings yields non-trivial DGA representations of the Legendrian contact DGA associated toany Legendrian link, which themselves are effective invariants for distinguishing Legendrianrepresentatives in the same smooth type [Che02, Ng03, Siv11]. In particular, Floer theoryhas provided far-reaching methods to address questions on Legendrian links; for instance,along the lines of this paper, see [EP96, Etn03, K´al05, Cha10].Recently, the first examples of Legendrian links in ( S , ξ st ) which admit infinitely manyLagrangian fillings in ( D , λ st ) were discovered [CG20]. Indeed, [CG20, Corollary 1.5] showsthat the max-tb Legendrian ( n, m )-torus link Λ( n, m ) admits infinitely many Lagrangianfillings if n ≥ , m ≥ n, m ) = (4 , , (4 , D n -type depicted in Figure 1 (right) is one such class.This gives an alternative proof that the torus links in [CG20, Corollary 1.5] admit infinitelymany Lagrangian fillings. Figure 1.
The family of Legendrian links Λ n ⊂ ( S , ξ st ), n ≥
1, on the left.The Legendrian links of affine D n -type are depicted on the right, n ≥
4. Allof these have infinitely many fillings. The boxes indicate a series of positivecrossings.(ii) Second, since the appearance of [CG20], the articles [CZ20, GSW20a, GSW20b] havecontinued to develop various cluster and sheaf-theoretic methods that detect infinitely manyLagrangian fillings for a Legendrian link Λ ⊂ ( S , ξ st ). That said, the techniques presented in[CG20, CZ20, GSW20a, GSW20b] are currently only effective at studying Legendrian linkswhich are positive braids, i.e. when Λ ⊂ ( S , ξ st ) admits a Legendrian front given by therainbow closure of a positive braid. In the present manuscript we develop a Floer-theoreticargument which also applies to certain Legendrian links Λ ⊂ ( S , ξ st ) which are not therainbow closure of positive braids. For instance, we show that each of the Legendrian linksΛ n , n ∈ N , depicted in Figure 1 (left) admits infinitely many Lagrangian fillings. For n = 1,this yields a Legendrian link Λ which is not the rainbow closure of a positive braid because itcontains a tb = − ⊂ ( S , ξ st ) is deduced from a stronger result on Legendrian loops, Theorem 1.1, as e explain shortly. In particular, we provide the first examples of Legendrian loops whoseinduced monodromy action on the Legendrian contact DGA has infinite order.(iv) Finally, at a technical level, we study the lifts of the DGA maps induced by exact La-grangian cobordisms to Z -coefficients, which is required to argue the infinite order in ourargument. This is interesting on its own, as it provides correct signs for Floer theoreticalinvariants, such as augmentations, and it is a necessary ingredient for the study of clusterstructures on augmentation varieties and their holomorphic symplectic structures. In partic-ular, these results from this manuscript are used in the recent article [CGGS20] to constructa holomorphic symplectic structure on the augmentation varieties associated to Legendrianpositive braids. Figure 2.
On the left, Lagrangian projection of the Legendrian links Λ n , n ≥
1, and the purple box, which contains n positive crossings. The purple-boxLegendrian loop ϑ : S → L (Λ n ) is depicted by the dashed purple trajectory.On the right, Lagrangian projection of the Legendrian link Λ( (cid:101) D n ), n ≥ n −
2) positive crossings. The purple-boxLegendrian loop ϑ : S → L (Λ( (cid:101) D n )) is also illustrated by the dashed purpletrajectory.1.2. Main Results.
Let β be a positive braid, representing an element in the N -strandedpositive braid monoid Br + N , N ∈ N . We can associate a Legendrian link Λ( β ) ⊂ ( S , ξ st ) to β such that Λ( β ) is topologically the ( − β : this is achieved by placing β in a standard contact neighborhood of the standard Legendrian unknot in S of Thurston–Bennequin number tb = −
1. We now define the Legendrian links that we will study in thispaper.By definition, the (cid:101) D n –Legendrian link is the Legendrian Λ( β ( (cid:101) D n )) ⊂ ( S , ξ st ) associated to β ( (cid:101) D n ) = ( σ σ σ σ σ σ σ σ ) σ n − ∆ , n ≥ , where ∆ = σ ( σ σ )( σ σ σ ) is the 4-stranded half-twist. Figure 1 (right) shows a Legendrianfront projection for Λ( (cid:101) D n ), and the terminology will be explained in Section 2. Similarly, the Legendrian link Λ n ⊂ ( S , ξ st ) is the Legendrian link Λ n = Λ( β n ) associated tothe braid word β n = ( σ σ σ σ ) σ n , n ≥ . Figure 1 (left) shows a Legendrian front projection for Λ n . These are two distinct familiesof Legendrian links, with the exception of the accidental Legendrian isotopy Λ ∼ = Λ( (cid:101) D ).Finally, we will also consider the Legendrian links associated to the following braids: β ab = ( σ σ σ σ ) σ a σ b , a, b ∈ { , } . In characteristic different from 2. In particular, the correct signs are needed for arguing in characteristic0, the most studied case in cluster theory. The (cid:101) D n -Legendrian should be read as the affine D n -Legendrian. ote that Λ( β ) is Legendrian isotopic to Λ( (cid:101) D ). Following the above Dynkin-diagramnotation, Λ( β ) can also be referred to as the Λ( (cid:101) A , )-Legendrian link. From now onwards,we denote by H = { Λ n } n ≥ ∪ { Λ( (cid:101) D m ) } m ≥ ∪ { Λ( β ) , Λ( β ) , Λ( β ) } the set-theoretic union of the Legendrian links in the Λ n and Λ( (cid:101) D n ) families described aboveand the three Legendrians links Λ( β ) , Λ( β ) , Λ( β ). The Legendrian links in H allow usto tackle a wide range of additional Legendrian links, thanks to Corollary 1.3 below. Thisincludes torus links, as in Corollary 1.4, and the knots discussed in Section 7, see Remark1.5 below.Let L (Λ) be the space of Legendrian links isotopic to the Legendrian link Λ ⊂ ( S , ξ st ), withbase point an arbitrary but fixed Legendrian representative. In Section 2, for each of thelinks Λ ∈ H , we will define a certain loop ϑ of Legendrians based at Λ: that is, a continuousmap ϑ : ( S , pt) → ( L (Λ) , Λ). For instance, for the Legendrians in Figure 2, the loop arisesfrom moving the purple box around the link in the manner depicted. We will refer to thisLegendrian loop ϑ as the purple-box Legendrian loop.The graph of the Legendrian loop ϑ produces an exact Lagrangian concordance L ϑ in thesymplectization of ( S , ξ st ) from Λ to itself. Given any filling L ⊂ ( D , λ st ) of Λ, which wecan view as an exact Lagrangian cobordism from the empty link to Λ, we can concatenate L with any number of copies of L ϑ to produce an infinite family of fillings L L nϑ , n ∈ N , of Λ. What we will show is that for Λ ∈ H , we can choose a filling L of Λ such that all ofthese fillings L L nϑ are distinct.As discussed earlier, our method of proof involves the Legendrian contact DGA A Λ of Λ,which is an invariant of the Legendrian isotopy class of Λ ⊂ ( S , ξ st ), up to stable tame DGAisomorphism. The concordance L ϑ induces a DGA isomorphism A ( L ϑ ) : A Λ → A Λ while the filling L induces a DGA morphism (“augmentation”) ε L : A Λ → ( Z [ H ( L )] , , where ( Z [ H ( L )] ,
0) is the DGA with trivial differential, concentrated in degree 0. Functorial-ity then implies that the filling L L nϑ induces the augmentation ε L ◦ A ( L ϑ ) n . To distinguishthe fillings L L nϑ from each other, we will distinguish the augmentations ε L ◦ A ( L ϑ ) n , evenallowing for different choices of local systems on the fillings.To be precise, we say that the ϑ -orbit of the augmentation ε L is entire if for any k, l ∈ N distinct, there is no automorphism ϕ ∈ Aut( Z [ H ( L )]) such that ϕ ( ε L ◦ A ( L ϑ ) k ) = ε L ◦ A ( L ϑ ) l : A Λ → Z [ H ( L )] . The first result in our article is the following:
Theorem 1.1.
Let Λ ∈ H be a Legendrian link. The purple-box Legendrian loop ϑ : S →L (Λ) induces a DGA map A ( L ϑ ) : A (Λ) → A (Λ) of infinite order. In fact, there exists anexact Lagrangian filling L ⊂ ( D , λ st ) such that the ϑ -orbit of the corresponding augmentation ε L : A Λ → Z [ H ( L )] is entire. To our knowledge, Theorem 1.1 presents the first Legendrian loops which induce an infinite order action on the augmentations of a Legendrian contact DGA A Λ . Our work is a spiritualsuccessor to the work of T. K´alm´an [K´al05], who studied Legendrian loops for positive toruslinks Λ( n, m ) whose induced action on A (Λ( n, m )) has finite order ( n + m ).Theorem 1.1 implies the following: orollary 1.2. Let Λ ∈ H . Then the purple-box Legendrian loop ϑ generates an infinitesubgroup Z (cid:104) ϑ (cid:105) ⊂ π ( L (Λ)) . In addition, the graph of the Legendrian loop ϑ produces aLagrangian self-concordance of Λ which has infinite order as an element of the Lagrangianconcordance monoid based at Λ . Let us now focus on Lagrangian fillings. Theorem 1.1 implies that each of the Legendrian linksΛ ∈ H admits infinitely many Lagrangian fillings, up to Hamiltonian isotopy. More precisely,there exists a countably infinite collection { L i } i ∈ N of oriented embedded exact Lagrangianfillings L i ⊂ ( D , λ st ) of the Legendrian link Λ in the boundary S = ∂ D such that all L i are smoothly isotopic for i ∈ N , relative to a neighborhood of the boundary Λ, but none ofthe L i are Hamiltonian isotopic to each other; that is, if i (cid:54) = j , there exists no compactlysupported Hamiltonian isotopy { ϕ t } ∈ Ham c ( D , λ st ), ϕ = Id, such that ϕ ( L i ) = L j .We note that among the Legendrian links in H , four links —Λ , Λ( β ), Λ( β ), and Λ( β )—have a component which is a stabilized unknot with Thurston–Bennequin number −
3. (Infact Λ( β ) has two such components.) It follows that none of these four links is the rainbowclosure of a positive braid. We emphasize that the methods developed in [CG20, CZ20,GSW20a, GSW20b] for the detecting of infinitely many Lagrangian fillings only apply torainbow closures of positive braids, and thus our Floer-theoretic techniques provide newresults that we currently do not know how to address through cluster algebras [GSW20a,GSW20b] or the study of microlocal sheaves [CG20, CZ20]. We can use Legendrian links with infinitely many fillings to produce other Legendrian linkswith infinitely many fillings. Roughly speaking, if there is an exact Lagrangian cobordismfrom Λ − to Λ + and Λ − has infinitely many fillings, then Λ + does as well. (We only prove thisstatement subject to some important hypotheses; see Proposition 7.5 for the precise result.)In particular, we have the following consequence of Theorem 1.1. Corollary 1.3 (see Proposition 7.5) . Let Λ , Λ ⊂ ( S , ξ st ) be Legendrian links with Λ inthe list H , and suppose that there is a Lagrangian cobordism from Λ to Λ consisting of asequence of saddle moves at contractible Reeb chords of degree . Then the Legendrian link Λ admits infinitely many exact Lagrangian fillings, distinct up to Hamiltonian isotopy. As a special case, since there are such cobordisms to the max-tb Legendrian ( n, m ) torus linksΛ( n, m ) from Λ for n = 3 , m ≥
6, and from Λ( (cid:101) D ) for n, m ≥
4, we recover the followingresult of [CG20].
Corollary 1.4 ([CG20]) . The Legendrian torus links Λ( n, m ) each admit infinitely manyexact Lagrangian fillings if n ≥ , m ≥ or ( n, m ) = (4 , , (4 , . Remark 1.5.
As we will discuss in Section 7.2, among the universe of Legendrian links withinfinitely many fillings, a sensible notion of “simplicity” is given by the Thurston–Bennequinnumber, or equivalently the sum 2 g + m , where g is the genus of an exact Lagrangian fillingand m is the number of connected components of the link: the smaller 2 g + m is, the simplerthe link is. Among the Legendrian links that we can prove have infinitely many fillings, thesimplest by this measure is Λ( β ), which has ( m, g ) = (2 ,
1) and thus 2 g + m = 4.If we focus on Legendrian knots , rather than Legendrian links , Corollary 1.3 implies that, forinstance, the knot types 10 , m (10 ), m (10 ), 10 , and m (10 ) all have Legendrianrepresentatives with infinitely many fillings; see Proposition 7.7. Among these, the simplestis m (10 ), with g = 2 and 2 g + m = 5. Two of these knots, 10 and m (10 ), are positivebraid closures and indeed their Legendrian representatives are rainbow closures of positivebraids. We remark that the only other knots with crossing number ≤
10 that are positivebraid closures are the torus knots T (2 , T (2 , T (2 , T (3 , T (2 , T (3 , By Corollary 1.3, we can in fact construct an infinite family of links with infinitely many fillings that arenot the rainbow closure of a positive braid: Λ(( σ σ σ σ ) σ a σ ) for n ≥ N . onjectured that the (max-tb) Legendrian representatives of each of these knots has finitelymany fillings [Cas20, Conjecture 5.1]. (cid:3) The above results on Lagrangian fillings also have consequences in the study of Stein surfaces.For each g ∈ N and g ≥
6, the article [CG20] gave the first examples of Stein surfaceshomotopic to the 2-sphere S with infinitely many Hamiltonian isotopy classes of embeddedexact Lagrangian surfaces of genus g (and none of genus less than g ). The lower bound wasrecently improved to g ≥ Corollary 1.6.
Let g ∈ N and g ≥ . Then, there exists a Stein surface W homotopic tothe -sphere S which admits infinitely many Hamiltonian isotopy classes of embedded exactLagrangian surfaces of genus g . In addition, W contains no embedded exact Lagrangiansurfaces of genus h , h ≤ g − . In Corollary 1.6, the Stein surface W for g = 2 can be constructed by attaching a Weinstein2-handle to the standard symplectic 4-ball ( D , λ st ) along a max-tb Legendrian representativeof the smooth knot m (10 ). The results we prove also allow us achieve g = 1 if we allowourselves a bouquet of just two 2-spheres as the given homotopy type, instead of the 2-sphere S : Corollary 1.7.
The Stein surface W obtained by attaching two Weinstein 2-handles along Λ( β ) ⊂ ( ∂ D , ξ st ) , one per connected component, contains infinitely many Hamiltonianisotopy classes of embedded exact Lagrangian tori. Corollaries 1.6 and 1.7 are proven in Section 7. It remains an outstanding problem to con-struct a Legendrian knot with infinitely many distinct embedded Lagrangian 2-disk fillings(pairwise smoothly isotopic), or show no such knot exists. Organization.
Here is an outline of the rest of the paper. In Section 2, we review some nec-essary background and formally describe the Legendrian links discussed in this introduction.The Floer-theoretical core of the article is developed in Sections 3, 4, and 5. In particu-lar, Sections 3 and 4, jointly with Appendix A, develop a new combinatorial model for themaps between Legendrian contact DGAs with integral coefficients associated to a decompos-able exact Lagrangian cobordism. We believe these results are of independent interest for3-dimensional contact topology and Floer theory. We then apply these maps in Sections 6 toprove Theorem 1.1, and prove a number of corollaries and other ancillary results in Section 7.
Acknowledgements.
We thank Tobias Ekholm, Sheel Ganatra, Honghao Gao, EugeneGorsky, Yankı Lekili, Linhui Shen, and Daping Weng for illuminating conversations. R. Casalsis supported by the NSF grant DMS-1841913, the NSF CAREER grant DMS-1942363 andthe Alfred P. Sloan Foundation. L. Ng is partially supported by the NSF grants DMS-1707652and DMS-2003404. (cid:3) Legendrian Links and ϑ -loops In this section we describe the classes of Legendrian links Λ ⊂ ( R , ξ st ) and Legendrian loopsthat we study in this article. We begin in Section 2.1 with a review of Legendrian linksand exact Lagrangian cobordisms, and then proceed in Sections 2.2 and 2.3 to describe the The case of a link with infinitely many planar Lagrangian fillings (pairwise smoothly isotopic) mightalready be an interesting start. In terms of Stein surfaces, the analogue of the knot case would be to constructa Stein surface homotopic to S with infinitely many Hamiltonian isotopy classes of pairwise smoothly isotopicLagrangian 2-spheres. articular links of interest to us, which include the links in H presented in the introduction.We conclude in Section 2.4 by describing the purple-box Legendrian loops that are a keyingredient in our constructions.2.1. Legendrian links, exact Lagrangian cobordisms, and fillings.
Here we brieflyreview the basic geometric terminology that we will need for this paper. There is nowan extensive literature on exact Lagrangian cobordisms, including the papers cited in theintroduction, to which we refer the reader for further details; specifically, the paper [EHK16]has a full exposition of the setting we will use here.Rather than work with the contact manifold ( S , ξ st ) directly, it is convenient to remove apoint and work in the contact manifold ( R , ξ st ), where ξ st is the contact structure given bythe kernel of the standard contact 1-form α st := dz − y dx on R , endowed with Cartesiancoordinates ( x, y, z ) ∈ R . By definition, a link Λ ⊂ ( R , ξ st ) is Legendrian if it is everywheretangent to ξ st , or equivalently if α st | Λ = 0; all Legendrian links in this paper are oriented.As is customary, we will describe Legendrian links in ( R , ξ st ) by their front and Lagrangianprojections. These are the images of the link under the projections Π xz , Π xy : R → R to the xz and xy planes, respectively. Given a Legendrian link Λ, the Reeb chords of Λ are integralcurves of the Reeb vector field ∂ z with endpoints on Λ; these correspond to the crossingsof the Lagrangian projection Π xy (Λ). One numerical invariant associated to a Legendrianlink Λ is the Thurston–Bennequin number tb (Λ), which is the number of crossings of Π xy (Λ)counted with sign. Example 2.1.
The simplest Legendrian knot in R is the standard Legendrian unknot with tb = −
1, which we will denote by U . The front projection Π xz ( U ) is a “flying saucer” withtwo cusps, while the Lagrangian projection Π xy ( U ) is a “figure eight” diagram with a singlecrossing; see the top left of Figure 4. (cid:3) The symplectization of R is the 4-manifold R = R t × R equipped with the exact symplecticform ω st = dλ st with λ st = e t α st . Note that this symplectic manifold is symplectomorphic to( R , dλ ), where λ := ( x dy − y dx + x dy − y dx ), ( x , y , x , y ) ∈ R , is the radialLiouville form in R . Given that ( R , dλ st ) is symplectomorphic to the Liouville completionof the standard symplectic Darboux ball ( D , dλ ), we will also write λ st for λ and denoteby ( D , λ st ) the unique exact symplectic filling of ( S , ξ st ), with a radial primitive Liouvilleform.We will be interested in Lagrangian submanifolds of ( R , dλ st ), which are surfaces L ⊂ R suchthat ω st | L = 0. One class of Lagrangian submanifolds is given by cylinders over Legendrians:if Λ ⊂ R is Legendrian, then R × Λ ⊂ R is Lagrangian.More generally, suppose that Λ + , Λ − are Legendrian links in R . A Lagrangian cobordismfrom Λ − to Λ + is a Lagrangian L ⊂ R such that, for some T > L ∩ (( −∞ , − T ) × R ) = ( −∞ , − T ) × Λ − and L ∩ (( T, ∞ ) × R ) = ( T, ∞ ) × Λ + . The Lagrangian cobordism L is exact if there is a function f : L → R such that λ st | L = df and f is constant on each of the ends ( −∞ , − T ) × Λ − and ( T, ∞ ) × Λ + separately. AllLagrangian cobordisms considered in this paper will be oriented, embedded, and exact. Inthe special case where the negative end is empty, an exact Lagrangian cobordism from ∅ toΛ is called a filling of Λ. See Figure 3.We will be interested in Lagrangian cobordisms and fillings up to exact Lagrangian isotopy ,which is an isotopy through exact Lagrangian cobordisms that fixes the two cylindrical ends(or the positive cylindrical end, in the case of fillings). In the setting of R , this is the same asa Hamiltonian isotopy [FOOO09], which is an isotopy through Hamiltonian diffeomorphismsfixing the two ends ( −∞ , − T ) × R and ( T, ∞ ) × R . igure 3. A Lagrangian cobordism from Λ − to Λ + (left) and a filling of Λ(right). Remark 2.2.
Associated to a Legendrian link in R or a Lagrangian surface in R is its Maslov number , which takes values in Z . For a Lagrangian surface L , this is the greatestcommon divisor of the Maslov numbers of all closed loops in L , where the Maslov number of aloop in L is understood to be the Maslov number of the corresponding loop in the LagrangianGrassmannian of R . For a Legendrian link Λ, the Maslov number is the Maslov number ofthe surface R × Λ. All Legendrians and Lagrangians that we consider in this paper will haveMaslov number 0. (cid:3)
We will construct exact Lagrangian cobordisms out of key building blocks called elemen-tary cobordisms , due to [EHK16]. There are three types of elementary cobordisms betweenLegendrian links, which we describe in turn. (i) Isotopy cobordisms.
If Λ − and Λ + are Legendrian links that are related by a Leg-endrian isotopy Λ t , then the trace of this isotopy (the union of { t } × Λ t over all t ) can beperturbed to an exact Lagrangian cobordism from Λ − to Λ + , which we will call the isotopycobordism associated to this isotopy. The isotopy represents a path in the space of Legen-drian links from Λ − to Λ + , and homotopic paths lead to isotopy cobordisms that are exactLagrangian isotopic. (ii) Minimum cobordisms. Let U denote a standard Legendrian unknot as in Example 2.1.By [EP96], U has a filling by a Lagrangian 2-disk, which is necessarily exact, and this fillingis unique up to exact Lagrangian isotopy. Thus if Λ − is any Legendrian link and Λ + is thesplit union of Λ − and a standard unknot U , then there is an exact Lagrangian cobordismfrom Λ − to Λ + given by the union of the filling of U and the cylinder R × Λ − . This cobordismis called a minimum cobordism and corresponds topologically to the addition of a 0-handle. (iii) Saddle cobordisms. Let Λ + be a Legendrian link. Reeb chords of Λ + correspondto crossings in the Lagrangian projection Π xy (Λ + ). A Reeb chord is called contractible ifthere is a Legendrian isotopy of Λ + inducing a planar isotopy of Π xy (Λ + ) and ending in aLegendrian where the height of the Reeb chord is arbitrarily small. Suppose that we havea contractible Reeb chord a of Λ + that corresponds to a positive crossing of Π xy (Λ + ) (insymplectic terms, the Conley–Zehnder index of a is even). One can modify the diagramΠ xy (Λ + ) by replacing the corresponding crossing by its oriented resolution to produce theLagrangian projection of another Legendrian link Λ − ; see Figure 4. There is then an exactLagrangian cobordism from Λ − to Λ + called a saddle cobordism . This is sometimes called a pinch move because of what it looks like in the front projection, and we will also sometimesrefer to this as “resolving” the Reeb chord; it corresponds topologically to the addition of a1-handle. igure 4. Two elementary Lagrangian cobordisms, depicted in terms of their xy projections: a minimum cobordism (left) and a saddle cobordism (right).The dotted arrows go from the bottom to the top of the cobordisms. Thediagram on the top left is the standard Legendrian unknot U .We can build more cobordisms out of elementary pieces through the operation of concatena-tion . Suppose that L and L are exact Lagrangian cobordisms that go from Λ to Λ andfrom Λ to Λ , respectively. We can remove the top cylinder of L and the bottom cylinderof L and glue the resulting Lagrangians along their common boundary Λ to produce a newexact Lagrangian cobordism L L from Λ to Λ , the concatenation of L and L . Anexact Lagrangian cobordism is decomposable if it is the concatenation of some number ofelementary cobordisms. All of the cobordisms and fillings that we consider in this paper willbe decomposable. Now that we have reviewed the basic geometric concepts and terminology,let us delve into the specific objects of interest with a view towards the new contributions ofthis manuscript.2.2. Legendrian links associated to positive braids.
We now describe the specific Leg-endrian links in ( R , ξ st ) that we will consider in this paper. These are a natural family ofLegendrian links associated to positive braids, topologically given by the closures of thesebraids with one full negative twist.Let Br + N be the monoid of positive braids in N -strands, N ∈ N . Consider an element β ∈ Br + N and a braid word β := l ( β ) (cid:89) j =0 σ i j , i j ∈ [1 , N − , where l ( β ) is the length of β ∈ Br + N , equivalently its number of crossings. There is a well-defined (up to isotopy) Legendrian link (cid:101) Λ( β ) ⊂ ( J S , ξ st ) associated to β (cf. [EV18]). Bydefinition, the Legendrian (cid:101) Λ( β ) ⊂ ( J S , ξ st ) is the Legendrian link whose front in S × R (image of the projection map J S = T ∗ S × R → S × R ) consists of the N horizontalstrands S × { j } , j = 1 , . . . , N , where (positive) crossings are added left to right accordingto the braid word β . Figure 5 depicts (cid:101) Λ( β ) with an explicit example: the S -coordinate ishorizontal, and the two vertical yellow walls are identified with each other.Given a Legendrian link (cid:101) Λ( β ) ⊂ ( J S , ξ st ), we denote by Λ( β ) ⊂ ( R , ξ st ) the Legendrian linkobtained by satelliting (cid:101) Λ( β ) along the standard Legendrian unknot U ⊂ R . To be precise,( J S , ξ st ) is contactomorphic to a standard contact neighborhood O p (Λ ) ⊂ ( R , ξ st ) ofΛ ⊂ ( R , ξ st ), and Λ( β ) is the image of (cid:101) Λ( β ) under the resulting inclusion J S (cid:44) → R ; thisis a special case of the Legendrian satellite construction [NT04].Figure 6 (left) shows the front projection for the Legendrian link Λ( β ). The transition fromΠ xz (Λ) to Π xy (Λ) (“resolution”) can be performed as in [Ng03, Proposition 2.2] and it is igure 5. The front for the Legendrian link (cid:101) Λ( β ) ⊂ ( J S , ξ st ) associatedto the braid word β (left). Explicit example of (cid:101) Λ( β ) associated to the braidword β = σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ ∈ Br +6 (right).depicted in Figure 6. In this Lagrangian projection, a combinatorial advantage is that Reebchords for Λ( β ) ⊂ ( R , ξ st ) are in bijection with the (positive) crossings of Π xy (Λ( β )). Figure 6.
The Lagrangian projection of a Legendrian link Λ (right) obtainedby resolving a front projection (left). In the Lagrangian projection, the cross-ings of the braid word β are all positive crossings in the diagram. The arrowsindicate the orientation of the link.It will significantly simplify our computations of the Legendrian contact DGA and associatedmonodromy to use not Λ( β ) but a link that is Legendrian isotopic to Λ( β ). Definition 2.3.
Let β be a positive braid. Consider the link diagram in R given by theblackboard-framed satellite closure of β around the figure-eight unknot diagram Π xy ( U ), asdepicted in the rightmost diagram of Figure 7. If this diagram is the Lagrangian projectionof a Legendrian link, then we call this Legendrian link the ( − -closure of β . (cid:3) It is apparent that Λ( β ) and the ( − β represent smoothly isotopic links, as theyare both the ( − β . Furthermore, their Lagrangian projectionsare regularly homotopic: an isotopy between them is indicated in Figure 7. The first step inthis isotopy is just a planar isotopy moving the (cid:0) N (cid:1) negative crossings to the left of β to thetop of the diagram. We then use a sequence of Reidemeister II and III moves to obtain thesquare-grid configuration of crossings shown in the blue box in Figure 7 (right). However,the smooth isotopy from the center diagram to the right diagram does not always represent aLegendrian isotopy—and in particular the right diagram does not even necessarily representa Legendrian link—as we illustrate by an example. Example 2.4.
Consider the Legendrian link Λ( e ) where e is the trivial 2-stranded braid[ ∅ ] ∈ Br +2 . Following the resolution procedure as in Figure 6, we find that the Lagrangianprojection of Λ( e ) is the exact Lagrangian L ⊂ R depicted in Figure 8.(i). In what follows,we use D. Sauvaget’s calculus [Sau04, Section II.2] for exact Lagrangian projections – see also[Lin16, Section 2] for an introduction. Let A, B, C, P , P ∈ R + be the areas of the bounded igure 7. An isotopy from the Lagrangian projection obtained by the reso-lution of a front projection (left) to the ( − β .regions R \ L , as shown in Figure 8.(i); we may and do assume that we have B < C . Thetwo area constraints for this projection read P = A + B, P = A + C. In order to perform a Reidemeister III in the region with area B , we first empty the area inthat region, leading to Figure 8.(ii), and the corresponding exactness constraints are satisfied: P = 0 + ( A + B ) , P = ( A + B ) − ( A + C ) . The Reidemeister III move leads to Figure 8.(iii) and an additional Reidemeister II move,creating a canceling pair of crossings, to Figure 8.(iv). A second Reidemeister III move,which is admissible due to the zero area in its triangular region, yields Figure 8.(v). Thearea constraints are still satisfied, as they coincide with those in Figure 8.(ii). These movesconcatenate to a Hamiltonian isotopy from Figure 8.(i) to Figure 8.(v), through exact La-grangians. Now, we claim that the transition from Figure 7 (Center) to Figure 7 (right)cannot exist through exact Lagrangians: the resulting Lagrangian – shown in Figure 8 – isnot an exact Lagrangian. This can be directly seen by the area constraints: α = γ + δ + ε, α + β + ε = γ, α, β, γ, δ, ε ∈ R + , which imply δ + β + 2 ε = 0, contradicting positivity of the areas δ, β, ε ∈ R + . Alternatively,it is rather immediate that the two curves in Figure 8.(vi) bound an immersed annulus, withpositive area. Hence, the conclusion is that a constraint on β needs to be imposed, shouldwe want to work with a Legendrian link through a Lagrangian projection of the form shownin Figure 7 (right). (cid:3) We will want to consider braids where the isotopy in Figure 7 is legal. In the followingdefinition, let π : J S → T ∗ S denote the projection to the first factor, where J S = T ∗ S × R z with the standard contact form dz − λ st . Definition 2.5.
Let β ∈ Br + N , N ∈ N , and consider its smooth braid closure c ( β ) in S × R ∼ = T ∗ S , depicted as a (horizontal) link diagram. Then β ∈ Br + N is said to be admissible if c ( β )is the Lagrangian projection of a Legendrian link Λ ⊂ ( J S , ξ st ): that is, if there exists aLegendrian link Λ ⊂ ( J S , ξ st ) such that c ( β ) = π (Λ) as link diagrams, where crossings aretaken into account. (cid:3) As we now explain, if β is admissible, then the isotopy in Figure 7 is legal and in particularit makes sense to refer to the ( − β . Proposition 2.6.
Suppose that β is admissible. Then the diagram on the right of Figure 7 isthe Lagrangian projection of a Legendrian link in R , and the sequence of moves in Figure 7represents a Legendrian isotopy. igure 8. The sequence of Lagrangian projections discussed in Example2.4. The transitions from (i) to (v) are all realized by Hamiltonian isotopies,preserving the exactness of the immersed Lagrangian. Item (vi) displays anexample of a Lagrangian which is not exact. The consequence of (v) not beingnecessarily Legendrian isotopic to (vi) leads to Definition 2.5.
Proof.
Suppose β is admissible, and let Λ be the Legendrian link in J S whose Lagrangianprojection is c ( β ). If we cut Λ at a point in S then we obtain a Legendrian braid in J ([0 , front projection is β . It follows that this remains true when we satellite these braids around thestandard Legendrian unknot U ⊂ R . The satellite of the latter braid is Λ( β ) as defined inSection 2.2, which in the Lagrangian projection is the leftmost diagram in Figure 7. On theother hand, one directly sees (without passing to the front projection) that the Lagrangianprojection of the satellite of Λ is the rightmost diagram in Figure 7. The result follows. (cid:3) Example 2.4 shows that not every braid β ∈ Br + N is admissible. Let us introduce a sufficiencycriterion for a braid β ∈ Br + N to be admissible. For that, let∆ N = N − (cid:89) i =1 N − i (cid:89) j =1 σ j ∈ Br + N denote the half-twist on N strands, i.e., the Garside element of the N -stranded braid groupBr N . Proposition 2.7.
Any positive braid containing a half-twist is admissible, i.e. if β , β arebraids in Br + N , then β ∆ N β is admissible. roof. Since admissibility depends only on the closure of the braid in the solid torus, wemay move β to the beginning of the braid; it thus suffices to show that if β ∈ Br + N then β ∆ N is admissible. For this, consider the standard front for (cid:101) Λ( β ∆ N ) in S × R and deformit, scanning left-to-right, using the resolution procedure in [Ng03, Section 2.1]: see [Ng03,Figure 3] and Figure 9 (left). The procedure described in [Ng03] uses a front projection in R q × R z , instead of S q × R z , but can still be used with this latter base S × R by using thehalf-twist ∆ N ∈ Br + N , which is part of the braid β ∆ N by hypothesis. Indeed, this is depictedin Figure 9 (left), where the half-twist is shown in the yellow box. The Lagrangian projectionassociated to this deformed front is depicted in Figure 9 (right), where the half-twist ∆ N now appears thanks to the crossings associated to the (green) Reeb chords that appear atthe right-most part of the front in S q × R z . This concludes the statement. (cid:3) Figure 9.
Deforming the front projection for (cid:101) Λ( β ∆ N ) in S × R (left) so thatthe corresponding Lagrangian projection in T ∗ S is as shown on the right.For future reference we note the following variant on Proposition 2.7. Each crossing in theLagrangian projection in T ∗ S of a Legendrian link in J S corresponds to a Reeb chordof the link. A Reeb chord is called contractible if its height can be made arbitrarily smallwithout changing the Lagrangian projection of the link (up to planar isotopy). Proposition 2.8. If β , β are braids in Br + N , then any crossing coming from β or β inthe admissible braid β ∆ N β is contractible.Proof. First consider the special case where β consists of a single crossing. We claim thatthis crossing is contractible. Indeed, a slight variant on the construction from Figure 9involving swapping two of the strands in the yellow box in the front projection gives thedesired contractible crossing: see Figure 10. Figure 10.
A variant on the argument from Proposition 2.7. Here two ofthe strands passing through the yellow ∆ N box in the front projection swapplaces, producing a contractible crossing in the Lagrangian projection: thered dot on the right, corresponding to the single crossing of β . n the general case, cut the closure of the braid β ∆ N β at the specified crossing. Push ∆ N to the end of the resulting braid by a sequence of Reidemeister III moves. From the abovespecial case, we can realize the resulting braid as the Lagrangian projection of a Legendrianlink in such a way that the distinguished crossing is contractible. Then push ∆ N back intoits original position without disturbing a neighborhood of the contractible crossing; this is abraid isotopy and thus corresponds to a Legendrian isotopy by the classification of positiveLegendrian braids [EV18]. (cid:3) A class of Legendrian ( − -closures. The Legendrian links that we use in this man-uscript are particular examples of ( − w ( β ) ∈ S N be the permutation given by the Coxeter projection w : Br N → S N of β ontothe symmetric group, where the relations σ i = 1 are imposed for the Artin generators i = 1 , . . . , N −
1. Suppose that the bijection w ( β ) : [1 , N ] → [1 , N ] has a fixed point i , for some i ∈ [1 , N ]. Then the Legendrian Λ( β ) contains a connected component Λ( β ) i which is a standard Legendrian unknot. Since Λ( β ) i ⊂ ( R , ξ st ) is a Legendrian link, thereexists a neighborhood O p (Λ( β ) i ), disjoint from Λ( β ) \ Λ( β ) i , which is contactomorphic to O p (Λ( β ) i ) ∼ = ( J Λ( β ) i , ξ st ), where the contactomorphism sends Λ( β ) i ⊂ O p (Λ( β ) i ) to thezero section Λ( β ) i ⊂ ( J Λ( β ) i , ξ st ).Now, let γ ∈ Br + M be a positive M -stranded braid, M ∈ N . Let us denote by Λ( γ ) i ⊂O p (Λ( β ) i ) the Legendrian link obtained by satelliting (cid:101) Λ( γ ) ⊂ ( J S , ξ st ) along the standardLegendrian unknot Λ( β ) i ⊂ O p (Λ( β ) i ). Definition 2.9.
Let β ∈ Br + N be such that i ∈ [1 , N ] is a fixed point of w ( β ), and γ ∈ Br + M be an M -stranded braid, M ∈ N . The Legendrian link Λ( β, i ; γ ) ⊂ ( R , ξ st ) is the Legendrianlink (Λ( β ) \ Λ( β ) i ) ∪ Λ( γ ) i ⊂ ( R , ξ st ), where Λ( γ ) i ⊂ O p (Λ( β ) i ) is embedded in an arbitrarilybut fixed neighborhood of the component Λ( β ) i . Colloquially, Λ( β, i ; γ ) is the result ofsatelliting the braid γ around the component of the Legendrian link Λ( β ) labeled by i . (cid:3) The Legendrian links in Theorem 1.1 are of the form Λ( β, i ; γ ) for γ ∈ Br +2 and β ∈ Br + N ,where N = 2 ,
3. For instance, the Legendrian links Λ n come from setting β = σ and γ = σ n with N = M = 2: Λ n ∼ = Λ( σ , σ n ) . Similarly, the Legendrian links Λ( (cid:101) D n ), n ≥
4, come from setting β = ( σ σ σ σ ) σ and γ = σ n − with N = 3 , M = 2:Λ( (cid:101) D n ) ∼ = Λ(( σ σ σ σ ) σ , σ n − ) . Remark 2.10.
As noted in the introduction, the Legendrian links Λ( (cid:101) D n ), n ≥
4, are alsothe rainbow closures of the positive braids η n = ( σ σ σ σ σ σ σ σ ) σ n − , n ≥ , η n ∈ Br +4 . The brick diagram [Rud92, BLL18] associated to this positive braid word η n coincides withthe Coxeter–Dynkin diagrams (cid:101) D n associated to the affine Coxeter group of D -type. Thisaffine Coxeter diagram also arises from two natural constructions starting with η n . First, thequiver associated to the positive braid η n , according to the algorithm in [BFZ05], and second,as the diagram for the intersection form associated to a set of (distinguished) generators inthe first homology group of a minimal-genus Seifert surface associated to the link given by η n [Mis17, BLL18]. In addition, the augmentation variety associated to Λ( (cid:101) D n ) admits a clusterstructure of (cid:101) D n -type. These reasons lead us to the notation Λ( (cid:101) D n ) and referring to thesebraids as the (maximal-tb) affine D n -Legendrian links. (cid:3) Definition 2.9 is rather direct diagrammatically. Indeed, given the front diagram for Λ( β ) ⊂ ( R , ξ st ) shown in Figure 6 (left), a front diagram for Λ( β, i ; γ ) ⊂ ( R , ξ st ) is obtained by aking the M -copy Reeb push-off of the i -th component of Λ( β ), corresponding to the i -thstrand in Br + N , and inserting the front diagram for (cid:101) Λ( γ ). This is shown in Figure 11. Figure 11.
Front projections for the Legendrian links Λ( β ) ⊂ ( R , ξ st ) (left)and Λ( β, i ; γ ) ⊂ ( R , ξ st ) (right).Similarly, this construction is depicted in the Lagrangian projection in Figure 12. Figure 12.
Lagrangian projections for the Legendrian links Λ( β ) ⊂ ( R , ξ st )(left) and Λ( β, i ; γ ) ⊂ ( R , ξ st ) (right). These are the Lagrangian projectionsthat we use in order to compute the Legendrian contact DGA.The crucial property of the Legendrian links Λ( β, i ; γ ) ⊂ ( R , ξ st ) is the existence of a specificcontact isotopy ϕ t : ( R , ξ st ) → ( R , ξ st ), t ∈ [0 , ϕ (Λ( β, i ; γ )) = Λ( β, i ; γ ) and ϕ t | R \O p (Λ( β ) i ) = Id for all t ∈ [0 , The purple-box Legendrian loop.
Let β ∈ Br + N , γ ∈ Br + M and consider the Legen-drian link Λ( β, i ; γ ) ⊂ ( R , ξ st ). We construct a Legendrian loop ϑ : S → L (Λ( β, i ; γ )) basedat Λ( β, i ; γ ), whose action on the Legendrian contact DGA of Λ( β, i ; γ ) will be studied inSection 5, and subsequently lead to Theorem 1.1. Intuitively, the Legendrian loop ϑ will fixthe components of the Legendrian link Λ( β ) which do not belong to the satellite Λ( γ ) ⊂ Λ( β ),and induce a rotation of Λ( γ ) corresponding to one full revolution of the S direction in J S .Let us provide the details for its rigorous description.Consider the component Λ( β ) i ⊂ Λ( β ) with a standard neighborhood O p (Λ( β ) i ) and theLegendrian link (cid:101) Λ( γ ) ⊂ O p (Λ( β ) i ). Fix a contactomorphism O p (Λ( β ) i ) ∼ = ( J S θ , ker( dz − p θ dθ )) , where ( J S θ , ker( dz − p θ dθ )) is the 1-jet space with coordinates ( θ, p θ ) ∈ T ∗ S , z ∈ R .Fix the standard round metric in S , and choose R ∈ R + such that (cid:101) Λ( γ ) ⊂ B R , where B R = D R ( T ∗ S ) × [ − R, R ] ⊂ T ∗ S × R , with D R ( T ∗ S ) being the radius R (open) diskbundle.Now, consider the Hamiltonian p θ : J S θ → R and its associated contact vector field X p θ = − ∂ θ . Let ε ∈ R + , and choose a smooth cut-off function χ : J S → R such that χ | B R + ε ≡ , χ | B R +2 ε \ B R + ε ≡ . he contact vector field X ϑ associated to the Hamiltonian χ · p θ : J S → R restricts to − ∂ θ in the tube B R containing (cid:101) Λ( γ ), and it vanishes away from B R . The contact flow of X ϑ yieldsa compactly supported contact isotopy (cid:101) Θ t : ( J S , ξ st ) → ( J S , ξ st ), which we parametrizesuch that t = 1 is the smallest t ∈ R + with (cid:101) Θ t ( (cid:101) Λ( γ )) = (cid:101) Λ( γ ) pointwise. Definition 2.11.
Let β ∈ Br + N be such that i ∈ [1 , N ] is a fixed point of w ( β ), and let γ ∈ Br + M be an M -stranded braid, M ∈ N . The Θ t -contact isotopy associated to Λ( β, i ; γ ) ⊂ ( R , ξ st ), t ∈ [0 , (cid:101) Θ t : O p (Λ( β ) i ) → O p (Λ( β ) i ), t ∈ [0 , O p (Λ( β ) i ). A Legendrian loop ϑ : S → L (Λ( β, i ; γ )) is said to be a ϑ -loop if it is obtained as Θ t (Λ( β, i ; γ )), t ∈ [0 , t -contact isotopy associated toΛ( β, i ; γ ) ⊂ ( R , ξ st ). (cid:3) We will also call the Legendrian ϑ -loops in Definition 2.11 purple-box Legendrian loops , asthey are obtained by moving the purple box which contains the braid γ clockwise arounduntil it comes back to itself. Figure 13 (left) provides a schematic picture of such a ϑ -loop. Figure 13.
Left: a Legendrian ϑ -loop for the Legendrian link Λ( β, i ; γ ) ⊂ ( R , ξ st ), where the purple γ -box moves around clockwise around the β -boxand comes back to itself using the upper strands). Right: the local move,consisting of a sequence of l ( γ ) Reidemeister III moves, which we use in orderto push the purple γ -box, right to left, through the β -box.From a computational viewpoint, it is important to stress that a Legendrian ϑ -loop can bedescribed in the Lagrangian projection strictly in terms of Reidemeister III moves and planarisotopies. In precise terms, a Legendrian ϑ -loop consists of two pieces:(i) Transferring the purple γ -box through the β -box, through a sequence of ReidemeisterIII moves. Indeed, it suffices to notice that moving the purple γ -box through onestrand is achieved by l ( γ ) consecutive Reidemeister III moves, one per each crossingof γ . This local move, past one strand, is shown in Figure 13 (right). Thus, thepurple γ -box can be pushed through the β -box, right to left, by performing l ( β ) · l ( γ )Reidemeister III moves.(ii) Moving the purple γ -box from the left of the β -box to its right using the upper strands .This is achieved by a planar isotopy, which moves the purple γ -box up and to theright (leaving the β -box beneath and passing above it), and then applying N · l ( γ )Reidemeister III moves to make the purple γ -box go around the pig-tailed loop untilit returns to its initial position.Hence, using a total of l ( γ ) · ( N + l ( β )) Reidemeister III moves in the Lagrangian projection,we can realize the Legendrian ϑ -loops in Definition 2.11. One could instead use the resolution of the front projection as in Figure 6, and similarly push the purple γ -box around the front projection; this is e.g. what K´alm´an does in [K´al05]. However, this version of theisotopy requires the use of both Reidemeister III and II moves. Our setup does not require Reidemeister IImoves and this consequently simplifies our computations with the Legendrian contact DGA. emark 2.12. Legendrian ϑ -loops can be considered as elements in π ( L (Λ( β, i ; γ ))), or wecan graph them in the symplectization as Lagrangian self-concordances L ϑ ⊂ ( R × R , λ st )from the Legendrian link Λ( β, i ; γ ) ⊂ ( R , ξ st ) to itself. Most interestingly, given an exactLagrangian filling L ⊂ ( D , λ st ) of Λ( β, i ; γ ) ⊂ ( S , ξ st ), we can concatenate L with L ϑ , at theconvex end of L and the concave end of L ϑ . One may ask whether concatenating Lagrangianfillings with L ϑ yields new Lagrangian fillings not Hamiltonian isotopic to L . Theorem 1.1shows that there are Legendrian links where concatenating certain Lagrangian fillings with k consecutive copies of L ϑ yields (infinitely many) pairwise distinct Lagrangian fillings, fordifferent values of k ∈ N . (cid:3) Example 2.13.
Legendrian ϑ -loops behave differently depending on the choice of braids β ∈ Br + N and γ ∈ Br + M . For example, if γ ∈ Br +1 is the trivial 1-stranded braid, thenthe ϑ -loop is constant on the entire link Λ( β, i ; γ ) ⊂ ( R , ξ st ), regardless of the choice of β ∈ Br + N . On the other hand, if we choose the braid β to be 1-stranded and the purple box γ = σ n +21 ∈ Br +2 to be 2-stranded, then we recover K´alm´an’s Legendrian loop of (2 , n )-toruslinks [K´al05]. In this case, [K´al05, Theorem 1.3] shows that the action of the ϑ -loop onthe degree-0 Legendrian contact homology of Λ( β, i ; γ ) is nontrivial but of finite order. SeeSection 5.3 for further discussion of the K´alm´an loop. (cid:3) Legendrian Contact DGAs and Cobordism Maps
In this section, we review the definition of the Legendrian contact DGA, with particularattention paid to integer and group-ring coefficients and the role of spin structures. We thenproceed to discuss maps between DGAs induced by exact Lagrangian cobordisms, includingexact Lagrangian fillings. There is now a reasonably large literature about these cobordismmaps, beginning with work of Ekholm, Honda, and K´alm´an [EHK16] defining the maps over Z ; we will need to compute a lift of these maps to Z , which abstractly exists by workof Karlsson [Kar17, Kar20]. In this section we will present a framework that will allowus to perform explicit combinatorial computations of the cobordism maps over Z , buildingthem out of maps corresponding to particular elementary cobordisms. The maps for theseelementary cobordisms are then presented in the following section, Section 4.3.1. The Legendrian contact DGA.
The Legendrian contact DGA, also known as theChekanov–Eliashberg DGA, has been well-studied in the literature, especially in the settingof ( R , ξ st ). For the definition of the DGA in this setting, we refer the reader e.g. to [Che02]for the original definition over Z , [ENS02] for the definition over Z [ t ± ] (see also the survey[EN18]), and [NR13, NRS +
20] for an upgraded definition with multiple base points. Herewe will briefly review the definition that we will use, with Z coefficients and multiple basepoints.Let Λ be an oriented Legendrian link in ( R , ξ st ) equipped with a number of base points, suchthat there is at least one base point on each component. We will assume that Λ is sufficientlygeneric that the xy projection Π xy (Λ) in R is immersed with only transverse double pointsingularities, and no base point lies at one of these double points. We label the crossings ofΠ xy (Λ), which correspond to Reeb chords of Λ, as a , . . . , a r , and decorate each base pointwith a monomial of the form ± s ± i . Let { s , . . . , s q } be the collection of indeterminates thatappear in the labeling of the base points. To this decorated oriented Legendrian link Λ, wecan associate the Legendrian contact DGA ( A Λ , ∂ ), as follows. Generators.
The algebra A Λ is the unital tensor algebra over the coefficient ring Z [ s ± , . . . , s ± q ]generated by a , . . . , a r . (One can lift this to the “fully noncommutative” algebra where thecoefficients s ± i do not commute with Reeb chords a i , and in our computations we will some-times order our monomials accordingly. However, for the purposes of this paper, we willalways assume that coefficients and Reeb chords commute.) rading. We assume for simplicity that each component of Λ has rotation number 0, whichwill be the case for the Legendrian links we study. The algebra A Λ is then graded overthe integers Z ; if Λ has a single component, then this grading is well-defined, while if Λ hasmultiple components, the grading depends on some additional choices. We will fix the gradingby choosing a collection of distinguished base points, one on each component, such that theoriented tangent vectors to Π xy (Λ) at these points are all parallel in R . Label these basepoints by t , . . . , t m , where m is the number of components of Λ and the base point t j is onthe j -th component. Consider a Reeb chord a ∈ A Λ that ends on component r ( a ) and beginson component c ( a ); we define a capping path γ a along Λ to be the concatenation of a pathfrom the beginning point (undercrossing) of a to t c ( a ) , and a path from t r ( a ) to the endingpoint of a , following the orientation of Λ for both paths. As we traverse γ a , the unit tangentvector to Π xy ( γ a ) changes continuously from the tangent vector to the undercrossing at a tothe tangent vector to the overcrossing; let r ( γ a ) ∈ R denote the number of counter-clockwiserevolutions around S that the tangent vector makes during this process, and note that r ( γ a ) (cid:54)∈ Z because of transversality. Then the grading of a is defined to be −(cid:100) r ( γ a ) (cid:101) ∈ Z .We also place all the marked point monomials s i in grading 0, which completes the gradingof A Λ . Differential.
In order to set up the differential ∂ on A Λ , we first decorate the four quadrantsat each crossing of Π xy (Λ) by two signs, a Reeb sign and an orientation sign. At eachcrossing, two opposite quadrants have Reeb sign + and the others have Reeb sign − , whilethe orientation signs depend on whether the crossing is positive (even degree) or negative(odd degree): for positive crossings, two quadrants have orientation sign + and two have − ,while for negative crossings, all four quadrants have orientation sign +. See Figure 14. Figure 14.
In the top row, the Reeb signs (left diagram) and orientationsigns (two right diagrams) at a crossing. Quadrants that have − orientationsign are shaded, while all other quadrants have + orientation sign. In thebottom row, two examples of disks in ∆( a ). Both disks have sgn = +1 (onthe right, the corner with negative orientation sign cancels the − in − s − )and they contribute +1 and + s − a a s − a , respectively, to ∂ ( a ).The differential now counts immersions of a disk D with boundary punctures to R , mappingthe boundary of D to Π xy (Λ), such that a neighborhood of each boundary puncture ismapped to one of the four quadrants at a crossing of Π xy (Λ). We call such a disk an immerseddisk for short; each corner of an immersed disk is a positive (+) corner or a negative ( − ) corner depending on the Reeb sign of the quadrant. For a Reeb chord a , define ∆( a ) tobe the set of immersed disks (up to reparametrization) with a single + corner at a and no ther + corners. To any such disk ∆ ∈ ∆( a ), we can define two quantities. One is thesign sgn(∆) ∈ {± } , given by the product of the orientation signs over all corners of ∆,multiplied by the signs of any base points traversed by the boundary of the disk (+1 for anybase point labeled by s ± i and − − s ± i ). The other is the word w (∆) ∈ A Λ , which is the product, in order, of the Reeb chords at the − corners and thebase points that are encountered as we traverse the boundary of the disk counterclockwise,beginning and ending at the corner at a . A base point labeled by ± s ± i contributes s ± i if itis traversed along the orientation of Λ and s ∓ i if it is traversed oppositely. The differential ∂ ( a ) is now defined to be: ∂ ( a ) := (cid:88) ∆ ∈ ∆( a ) sgn(∆) w (∆) . See Figure 14 for an example.
Remark 3.1 (multiple base points) . In order to count augmentations over Z , it is importantthat each component of Λ have at least one base point. Adding extra base points beyondone per component changes the DGA in a simple way. First note that moving a base pointlabeled ± s ± i along Λ and through a crossing a has the effect of replacing a by ( ± s ± i ) ± a :that is, the algebra is the same before and after the move, and the differential changes byconjugation by the automorphism that sends a to ( ± s ± i ) ± a and fixes all other generators.Thus if we have multiple base points on a single component, then up to a Z [ s ± , . . . , s ± q ]-algebra isomorphism of the DGA, we can assume that all of the base points lie on the samesegment of Π xy (Λ). In this case we can replace the multiple base points by a single basepoint labeled by the product of their labels, and the differential is unchanged. (cid:3) Remark 3.2 (dependence on spin structure) . In the differential over Z of the Legendriancontact DGA ( A Λ , ∂ ) of a link Λ, the signs depend on a choice of spin structure on Λ, as laidout by the construction of Ekholm, Etnyre, and Sullivan [EES05]. For each S connectedcomponent of the Legendrian link Λ, there are two spin structures: the Lie group spinstructure, induced by the fact that the 1-sphere S is a Lie group, and the null-cobordant spin structure, induced by the fact that S bounds a 2-disk D and we can restrict the uniquespin structure on D to the boundary S . Here we review the discussion in [EES05] abouthow the choice of spin structure affects the differential ∂ in ( A Λ , ∂ ).Choose one base point on each of the m components of Λ, so that A Λ is an algebra over R = Z [ t ± , . . . , t ± m ], and write ∂ comb for the combinatorial differential on A Λ as definedabove. The set of spin structures on Λ is an affine space based on H (Λ , Z ) ∼ = Z m ; of interestto us will be two spin structures differing by (1 , . . . , all components of Λ. We will write ∂ Lie and ∂ NC for the geometric differentials on A Λ corresponding to these two spin structures. Thetwo differentials ∂ Lie and ∂ NC depend on a number of auxiliary choices, including cappingoperators for Reeb chords—see Section 3.4 below for further discussion—but up to R -algebraisomorphism, ( A Λ , ∂ Lie ) and ( A Λ , ∂ NC ) are well-defined.The combinatorial differential ∂ comb comes from the Lie group spin structure on Λ. To beprecise, in [EES05, Theorem 4.32] it is shown that one can make choices so that ∂ Lie agreeswith our definition of ∂ comb with signs as in Figure 14, except that for positive crossings(the left diagram on the top right of Figure 14), the opposite two quadrants are shaded. This change of shading corresponds to the R -algebra isomorphism of A Λ sending each Reebchord a to − a for even-graded Reeb chords and + a for odd-graded Reeb chords, and so thisisomorphism sends ( A Λ , ∂ comb ) ∼ = −→ ( A Λ , ∂ Lie ). The superscript ∂ NC stands for Null-Cobordant. In fact [EES05, Theorem 4.32] presents two choices of signs for ∂ Lie , of which we are describing one;however, it was subsequently proven in [Ng10] that the two choices lead to isomorphic DGAs. or cobordisms, the null-cobordant spin structure is more natural than the Lie group spinstructure. To compute ∂ NC , we can appeal to [EES05, Theorem 4.29] (see also Remark 4.35from the same paper), which implies that changing the spin structure by ( c , . . . , c m ) ∈ Z m has the effect of replacing t i by ( − c i t i for i = 1 , . . . , m . In particular, define the Z -algebraisomorphism φ : A Λ → A Λ by φ ( a ) = a for all Reeb chords a and φ ( t i ) = − t i for all i ; then φ : ( A Λ , ∂ Lie ) ∼ = −→ ( A Λ , ∂ NC ) . More generally, suppose that we have multiple base points on each component of Λ as inRemark 3.1, each decorated by a monomial of the form s ± i . Then, since no base pointintroduces a sign, the resulting combinatorial DGA ( A Λ , ∂ comb ) over Z [ s ± , . . . , s ± q ] hassigns corresponding to the Lie group spin structure. Now suppose that S is any subset ofthese base points. If we replace the decoration s ± i of each base point in S by − s ± i , weobtain a new differential ∂ S on A Λ . Then ∂ S gives the differential corresponding to the spinstructure that differs from the Lie group spin structure by ( c , . . . , c m ) ∈ Z m , where c i is thenumber of base points in S that lie on component i . In particular, if S has an odd numberof points on each component , then we have an isomorphism of DGAs over Z [ s ± , . . . , s ± q ]:( A Λ , ∂ S ) ∼ = ( A Λ , ∂ NC ) . (cid:3) Link automorphisms.
In the case where Λ is a multi-component Legendrian link,rather than a knot, there is a structure on the Legendrian contact DGA of Λ that is hiddenin the knot case. This is the “link grading” first introduced by K. Mishachev [Mis03], whichessentially gives the DGA the structure of a path algebra (the “composable algebra”) ona graph whose vertices are components of Λ and whose edges are Reeb chords of Λ. Thisstructure leads to a family of automorphisms of the DGA of the Legendrian link Λ, whichwe call link automorphisms . These will feature in our discussion at various points, and wediscuss them now in detail.Let Λ = Λ ∪ · · · ∪ Λ m be an m -component Legendrian link. For any Reeb chord a of Λ, define r ( a ) , c ( a ) ∈ { , . . . , m } to be the number of the component containing the endpoint (for r ( a ))or beginning point (for c ( a )) of a . The key observation of Mishachev is the following: in theDGA for Λ, any term in the differential ∂a of a Reeb chord a must be of the form a i · · · a i k ,where r ( a ) = r ( a i ) , c ( a i ) = r ( a i ) , . . . , c ( a i k − ) = r ( a i k ) , c ( a i k ) = c ( a ). This motivates thefollowing definition. Definition 3.3.
Let Λ be an m -component Legendrian link and ( A Λ , ∂ ) its DGA. A linkautomorphism of Λ is an algebra automorphism Ω : A Λ → A Λ of the following form: thereexist units u , . . . , u m in the coefficient ring of A Λ such that for all Reeb chords a ,Ω( a ) = u r ( a ) u − c ( a ) a. (cid:3) The following is an immediate consequence of Mishachev’s observation.
Proposition 3.4.
Let Λ be a Legendrian link. Any link automorphism Ω : A Λ → A Λ is achain map of the Legendrian DGA ( A Λ , ∂ ) . (cid:3) In addition, Mishachev’s link grading structure is preserved by Legendrian isotopy, as canbe checked by keeping track of components in the DGA chain maps induced by Legendrian From a geometric viewpoint, S indicates points where we add a π -rotation to the Lie group trivializationof the stabilized tangent bundle to S . Doing this an odd number of times on each component yields thenull-cobordant trivialization. See [EES05, Remark 4.35]. sotopy. See [Mis03], and see Section 4.1 below for explicit formulas for these chain maps.As a consequence, link automorphisms persist under Legendrian isotopy: Proposition 3.5.
Suppose Λ and Λ (cid:48) are Legendrian isotopic links with respective DGAs ( A Λ , ∂ ) and ( A Λ (cid:48) , ∂ ) . Suppose that Ψ : ( A Λ , ∂ ) → ( A Λ (cid:48) , ∂ ) is the DGA map induced by aLegendrian isotopy. If Ω : A Λ → A Λ is a link automorphism of Λ , then there is a corre-sponding link automorphism Ω (cid:48) : A (cid:48) Λ → A (cid:48) Λ of Λ (cid:48) such that Ω (cid:48) ◦ Ψ = Ψ ◦ Ω .Proof. The numbering of the components of the Legendrian link Λ induces a correspondingnumbering of the components of the link Λ (cid:48) . If Ω is defined by Ω( a ) = u r ( a ) u − c ( a ) a for some( u , . . . , u m ), then we define Ω (cid:48) in the same way: Ω (cid:48) ( a ) := u r ( a ) u − c ( a ) a . Since Ψ preserves thelink grading, it follows that it intertwines Ω and Ω (cid:48) , as desired. (cid:3) Link automorphisms will appear in our discussion in two related ways. First, they naturallyarise when considering the family of augmentations induced by an exact Lagrangian filling,as we will next describe in Section 3.3. Second, in Section 4.2 below, we describe a formulafor the cobordism map over Z associated to a saddle cobordism; our proof that the formula iscorrect is indirect and essentially reduces to arguing that there is only one possible candidatefor the cobordism map that is actually a chain map over Z . However, the existence of linkautomorphisms forces us to qualify this statement, since composing a chain map with a linkautomorphism produces another chain map. See Proposition 4.8 and Appendix A.3.3. The geometric map induced by an exact Lagrangian cobordism.
Suppose thatΛ is a Legendrian link in ( R , ξ st ) and that L ⊂ ( R , λ st ) is a Lagrangian filling of Λ. Then L induces an augmentation of the Legendrian contact DGA ( A Λ , ∂ ). More precisely, thefilling L equipped with a rank 1 local system induces an augmentation; put another way, thefilling gives a family of augmentations and the additional choice of a local system picks outone of these. In the setting of ( R , ξ st ), the study of augmentations coming from fillings wasinitiated by Ekholm, Honda, and K´alm´an [EHK16], who proved that an exact filling inducesan augmentation over the group ring Z [ H ( L )] through a count of rigid holomorphic disks inthe symplectization of R with boundary on L . Karlsson [Kar20] subsequently lifted Z to Z by showing that the relevant moduli spaces of holomorphic disks can be coherently oriented.We summarize all of this work as follows. Theorem 3.6 ([EHK16, Kar20]) . Suppose that L is an (oriented, embedded, exact) La-grangian filling of the Legendrian link Λ ⊂ ( R , ξ st ) with Maslov number . Then L inducesa DGA map ε L : ( A Λ , ∂ ) → ( Z [ H ( L )] , where Z [ H ( L )] lies entirely in grading . ( The map ε L is referred to as an augmentation. ) Furthermore, if L and L (cid:48) are Lagrangian fillings of Λ which are isotopic through exact La-grangian fillings of Λ , then the corresponding augmentations ε L and ε L (cid:48) are DGA homotopicmaps. (cid:3) Remark 3.7.
For the definition of DGA homotopic maps, see e.g. [K´al05, EHK16, NRS + A Λ is supported entirelyin nonnegative degree. In this setting, two DGA maps ( A Λ , ∂ ) → ( Z [ H ( L )] ,
0) are DGAhomotopic if and only if they are equal. Thus if two fillings
L, L (cid:48) produce augmentations to Z [ H ( L )] that are distinct (under an isomorphism identifying H ( L ) and H ( L (cid:48) )), then we an use Theorem 3.6 to conclude that L, L (cid:48) are not exact Lagrangian isotopic (or, equivalentlyin this setting, not Hamiltonian isotopic). (cid:3)
Remark 3.8.
The augmentation ε L depends on a choice of spin structure on the filling L , asexplained in [Kar20]. If we change the spin structure by an element ϑ ∈ H ( L ; Z ), then wecan define an isomorphism Z [ H ( L )] → Z [ H ( L )] by x (cid:55)→ ( − ϑ ( x ) x , and the augmentationchanges by composition with this isomorphism. This does not change the augmentation upto equivalence, in the sense of Definition 3.9 below. (cid:3) It will be convenient for us to enlarge the coefficient ring Z [ H ( L )] to incorporate link au-tomorphisms, as introduced in Section 3.2 above. Suppose that Λ is a Legendrian linkwith m components. Recall that given units u , . . . , u m , we can define a link automorphismΩ : ( A Λ , ∂ ) → ( A Λ , ∂ ). Any augmentation of ( A Λ , ∂ ) can be composed with this link au-tomorphism to produce another augmentation, and so a single augmentation produces an( m − s , . . . , s m − ,where we define s i = u i /u m for i ≤ m −
1. We restate this observation as follows.Consider the ring Z [ H ( L )][ s ± , . . . , s ± m − ] ∼ = Z [ H ( L ) ⊕ Z m − ]. Then the augmentation ε L : ( A Λ , ∂ ) → ( Z [ H ( L )] ,
0) lifts to an augmentation˜ ε L : ( A Λ , ∂ ) → ( Z [ H ( L ) ⊕ Z m − ] , a of Λ ending on component r ( a ) and beginning oncomponent c ( a ), we define ˜ ε L ( a ) := u r ( a ) u − c ( a ) ε L ( a ), where u i = s i for i ≤ m − u m = 1.The augmentation ˜ ε L to Z [ H ( L ) ⊕ Z m − ] incorporates both the geometry of the filling L and link automorphisms; henceforth we will view it as “the” augmentation coming from thefilling L and will drop the tilde. We will also not need the distinction between generators of H ( L ) and generators of Z m − . It is then convenient to recast the augmentation ε L in thefollowing definition. Definition 3.9. A k -system of augmentations of Λ is an algebra map ε : A Λ → Z [ s ± , . . . , s ± k ]such that ε ◦ ∂ = 0. By definition, two k -systems of augmentations ε : A Λ → Z [ s ± , . . . , s ± k ] , ε (cid:48) : A Λ → Z [( s (cid:48) ) ± , . . . , ( s (cid:48) k ) ± ]are considered to be equivalent if there exists a Z -algebra isomorphism ψ : Z [ s ± , . . . , s ± k ] → Z [( s (cid:48) ) ± , . . . , ( s (cid:48) k ) ± ] such that ε (cid:48) = ψ ◦ ε . Note that the space of such isomorphisms isparametrized by Z k × GL k ( Z ). (cid:3) Finally, we now recast Theorem 3.6 for our purposes in the following proposition; note thatif L has genus g then H ( L ) ⊕ Z m − has rank 2 g + 2 m − Proposition 3.10.
Let Λ be an m -component Legendrian link. Let L be a connected, ori-entable exact Lagrangian filling of Λ of genus g and Maslov number . Then L gives rise toa (2 g + 2 m − -system of augmentations of Λ , and this system is well-defined, independentof choices, up to equivalence. Furthermore, if all Reeb chords of Λ have nonnegative degree,then isotopic fillings of Λ give rise to equivalent systems of augmentations. (cid:3) Signs and functoriality of the cobordism map.
In order to establish our mainresults, such as Theorem 1.1, we will apply Proposition 3.10 to systems of augmentationsthat we will explicitly compute for particular fillings. For that, we will divide our fillings intoelementary cobordism pieces, calculate the cobordism map for each elementary piece, andcompose the resulting cobordism maps, using the fact that the cobordism map is functorial.This functoriality over Z is established in the work of Karlsson [Kar20], and we summarizein this subsection the results from [Kar20] that we need. iven an orientable exact Lagrangian cobordism L between Λ + and Λ − , we choose a spinstructure on L that restricts on each component of Λ + and Λ − to the null-cobordant spinstructure. Note that there are | H ( (cid:98) L ; Z ) | such spin structures, where (cid:98) L is the closed surfaceobtained from L by gluing a disk to each boundary component, and any such spin structurewill do. Besides a spin structure on L , the other pieces of auxiliary data that Karlsson usesto define the cobordism maps are systems of capping operators for Λ + and Λ − satisfyingcertain technical conditions. These capping operators are used by Karlsson to define thesigns in the DGAs ( A Λ + , ∂ NC+ ) and ( A Λ − , ∂ NC − ), where ∂ NC ± are the differentials associated tothe null-cobordant spin structures on Λ ± , as well as the signs in the cobordism map betweenthe DGAs.In our setting, for any Legendrian Λ with the Lie group spin structure, a suitable system ofcapping operators has been constructed in [EES05, Section 4.5], compare [Kar20, Remark2.9]. These capping operators give precisely the signs for the DGA differential on Λ that wehave presented combinatorially in Section 3.1 above and written as ∂ comb , see Remark 3.2.However, for the cobordism maps we need the signs from the null-cobordant rather than theLie group spin structure. As explained in Remark 3.2, we can express this combinatoriallyby choosing a set S of marked points on Λ with an odd number of marked points on eachcomponent, resulting in a differential ∂ S on A Λ such that we have an isomorphism φ S : ( A Λ , ∂ S ) ∼ = −→ ( A Λ , ∂ NC ) . To return to the setting of a cobordism L between Λ + and Λ − , Theorem 2.5 in [Kar20] givesa DGA map over Z , Φ L : ( A Λ + , ∂ NC ) → ( A Λ − , ∂ NC ). If we choose sets of marked points S ± on Λ ± with an odd number of marked points on each component of Λ ± , then Φ L induces aDGA map from ( A Λ + , ∂ S + ) to ( A Λ − , ∂ S − ). We also denote this map by Φ L , and it satisfiesthat the following diagram commutes:( A Λ + , ∂ S + ) φ S + ∼ = (cid:47) (cid:47) Φ L (cid:15) (cid:15) ( A Λ + , ∂ NC ) Φ L (cid:15) (cid:15) ( A Λ − , ∂ S − ) φ S− ∼ = (cid:47) (cid:47) ( A Λ − , ∂ NC ) . Furthermore, the cobordism maps Φ L constructed by Karlsson are functorial. To state thisproperty, suppose that L and L are exact Lagrangian cobordisms that go from Λ to Λ and from Λ to Λ (from bottom to top), respectively. We can concatenate these to producean exact cobordism L L from Λ to Λ . As before, equip L , L with spin structuresthat restrict to the null-cobordant spin structures on their boundaries. Choices of cappingoperators on Λ , Λ , Λ now produce DGA maps Φ L : ( A Λ , ∂ NC ) → ( A Λ , ∂ NC ), Φ L :( A Λ , ∂ NC ) → ( A Λ , ∂ NC ), and Φ L L : ( A Λ , ∂ NC ) → ( A Λ , ∂ NC ), and [Kar20, Theorem2.6] states that: Φ L ◦ Φ L = Φ L L . Let us choose collections of marked points S , S , S on Λ , Λ , Λ such that each componenthas an odd number of marked points (as usual). Then, we can use the isomorphisms be-tween ( A Λ i , ∂ S i ) and ( A Λ i , ∂ NC ) to produce DGA maps Φ L , Φ L , Φ L L between the DGAs A Λ i , ∂ S i ) such that the following diagram commutes:( A Λ , ∂ S ) φ S ∼ = (cid:47) (cid:47) Φ L (cid:15) (cid:15) Φ L L (cid:38) (cid:38) ( A Λ , ∂ NC ) Φ L (cid:15) (cid:15) Φ L L (cid:120) (cid:120) ( A Λ , ∂ S ) φ S ∼ = (cid:47) (cid:47) Φ L (cid:15) (cid:15) ( A Λ , ∂ NC ) Φ L (cid:15) (cid:15) ( A Λ , ∂ S ) φ S ∼ = (cid:47) (cid:47) ( A Λ , ∂ NC ) . (3.1)Note that all of the horizontal maps in this diagram are algebra maps over the relevantcoefficient ring. Colloquially, they send each homology coefficient s i to s i , and not to − s i .This discussion above is summarized in the following result. Proposition 3.11.
Given an exact Lagrangian cobordism L between Λ + and Λ + , and choicesof marked points S ± on Λ ± with an odd number on each component, we can write the cobor-dism map Φ L as a DGA map from ( A Λ + , ∂ S + ) to ( A Λ − , ∂ S − ) . If we have exact cobordisms L from Λ to Λ and L from Λ to Λ , and marked points S , S , S on Λ , Λ , Λ with anodd number on each component, then the cobordism maps for L , L , and their concatenation L L satisfy Φ L ◦ Φ L = Φ L L . (cid:3) System of augmentations for a decomposable filling.
All the Lagrangian fillingsthat we consider in this paper are decomposable in the sense of [EHK16] (see Section 2.1).For a decomposable filling, one can explicitly construct the corresponding system of augmen-tations by composing the cobordism maps induced by each of the elementary cobordisms.These elementary cobordism maps are described in Sections 4.1 and 4.2 below. To combinethem into the desired system of augmentations, we additionally need to keep track of basepoints and discuss how they produce the parameters in the system of augmentations. Thisis the content of the discussion that now follows. First, consider a general exact Lagrangian cobordism L between Legendrians Λ + and Λ − ,inducing a chain map Φ L between the DGAs of Λ + and Λ − . Recall from Section 3.1 that inthe setting of the DGA of a Legendrian Λ, it is convenient to choose base points on Λ anduse these points to keep track of the homology classes of the boundaries of the holomorphicdisks that contribute to the differential. In a similar manner, we will keep track of homologyclasses contributing to Φ L by placing arcs on L and counting intersections of holomorphicdisks with these arcs.To this end, suppose that we have a collection of oriented arcs and circles on L , such thatall circles lie in the interior of L , the endpoints of all arcs lie on Λ + ∪ Λ − , and the arcsare transverse to Λ + ∪ Λ − at their endpoints. Label these arcs γ , . . . , γ k . Some subset { γ i , . . . , γ i p } has at least one endpoint on Λ + , and we view these endpoints as base points onΛ + ; similarly some subset { γ j , . . . , γ j q } has at least one endpoint on Λ − , and we view theseendpoints as base points on Λ − . The chain map Φ L between the DGAs of Λ + and Λ − isdefined by counting a finite collection of holomorphic disks with boundary on L and boundarypunctures mapping to Reeb chords for Λ + and Λ − ; we make the (generic) assumption thatour curves γ i intersect the boundaries of these disks transversely, and that no endpoint of anarc γ i lies at the endpoint of a Reeb chord of Λ + or Λ − .In this setting, Φ L is a map of algebras over the coefficient ring Z [ s ± , . . . , s ± k ]. Moreprecisely, the DGA for Λ + equipped with the base points from γ i , . . . , γ i p has coefficient ring Z [ s ± i , . . . , s ± i p ], and we can tensor this DGA over Z [ s ± i , . . . , s ± i p ] with Z [ s ± , . . . , s ± k ] to We note that a similar treatment of base points on Lagrangian cobordisms and the induced DGA maps(over Z ) appears in [GSW20a, section 2]. btain a DGA over Z [ s ± , . . . , s ± k ], which we write as ( A Λ + , ∂ + ). Similarly we can define theDGA ( A Λ − , ∂ − ) over Z [ s ± , . . . , s ± k ]. Then we can define the chain map Φ L : A Λ + → A Λ − as a map of Z [ s ± , . . . , s ± k ]-algebras: each holomorphic disk ∆ contributing to Φ L is giventhe coefficient s n (∆)1 · · · s n k (∆) k ∈ Z [ s ± , . . . , s ± k ], where n i (∆) counts the number of signedintersections of ∂ ∆ with the curve γ i .We now apply this discussion to describe how to concretely construct a system of augmen-tations for a Legendrian link Λ associated to a connected, decomposable exact Lagrangianfilling L of Λ. Let m denote the number of components of Λ and g the genus of L . By assump-tion, L is a union of 0-handles (minimum cobordisms) and 1-handles (saddle cobordisms); let k denote the number of 0-handles, and note that it follows that there are 2 g + m + k − (cid:96) L to produce a newcobordism L (cid:48) whose top end is Λ and whose bottom end is a k -component unlink Λ , suchthat L (cid:48) is assembled out of just the 1-handles of L .We can view L (cid:48) through slices from top to bottom, so that it becomes a movie of embeddedLegendrian links (except at finitely many times) starting with Λ, at the top, and ending withthe k -component unlink Λ , at the bottom. In the Lagrangian projection, each saddle moveis then represented by replacing a (contractible) crossing by its 0-resolution. We can nowadd base points to this movie as follows. Place base points t , . . . , t m on the m componentsof Λ. Each time we pass through a saddle, add two more base points labeled s i and − s − i .All base points persist to the bottom of the cobordism, Λ . See Figure 15. Figure 15.
On the left, placing a pair of base points at the bottom of a saddlecobordism, representing opposite sides of an arc passing through the saddlepoint in the cobordism. On the right, dividing a decomposable filling of Λinto elementary pieces: from top to bottom, a sequence of saddle cobordismsending at an unlink Λ , and then filling in each unknot component. n the Lagrangian cobordism L (cid:48) , the base points t , . . . , t m trace out arcs joining Λ to Λ ,while for each i = 1 , . . . , (cid:96) , the base points s i , − s − i together trace out an arc joining Λ toitself. We call these arcs τ , . . . , τ m and σ , . . . , σ (cid:96) , respectively. Orient the τ i arcs upwards,and orient the σ i arcs so that in each slice the arc is oriented upwards at the point labeled s i and downwards at the point labeled − s − i . This places the decomposable cobordism betweenΛ and Λ in the general picture described above of a cobordism equipped with oriented arcs.Label the slices of L (cid:48) from bottom to top by Λ , Λ , . . . , Λ (cid:96) = Λ, and divide L (cid:48) into saddlecobordisms L , . . . , L (cid:96) , where L j is the piece of L (cid:48) between Λ j − and Λ j ; note that the saddleof L j is associated to the arc σ (cid:96) +1 − j . Each Λ j is equipped with a collection of base pointseach labeled by either t i or ± s ± i . For j = 1 , . . . , (cid:96) , let ( A Λ j , ∂ comb ) denote the DGA ofΛ j over Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] with the differential ∂ comb defined combinatorially as inSection 3.1 (note that some of the s i parameters may not correspond to base points of Λ j and thus may not appear in the definition of ∂ comb ).We next relate the DGAs ( A Λ j , ∂ comb ) to the discussion from Section 3.4. To this end, foreach j = 0 , . . . , (cid:96) , we identify a subset S j of the base points on Λ j such that each componentof Λ j contains an odd number of points in S j ; we abbreviate this condition by calling such asubset odd-cardinality . We define S j by backwards induction on j . Let S (cid:96) be the collection ofall of the base points t , . . . , t m on Λ (cid:96) = Λ, and note that this is odd-cardinality. Given S j ,each base point on Λ j descends to a corresponding base point on Λ j − , and so we may view S j as a collection of base points on Λ j − . On Λ j − , we can add to S j one more base point,from the two new base points labeled by ± s ± (cid:96) +1 − j , such that the resulting collection S j − isodd-cardinality: if Λ j − has one more component than Λ j , then the choice of this extra basepoint is forced by the odd-cardinality condition, while if Λ j − has one fewer component thanΛ j , then we can choose either.The choice of base points S j on Λ j produces a differential ∂ S j on A Λ j as follows: first removethe − signs at the front of any base points on Λ j labeled by − s − i , so that all base pointsare labeled by t i or s ± i ; then negate any base point in S j , and let ∂ S j be the resultingcombinatorial differential as in Remark 3.2. Note that each t i is negated in this process,while exactly one of s i or s − i is negated, depending on which of these base points is in S j .Thus we can define a Z -algebra isomorphism ψ : Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] → Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ]by ψ ( t i ) = − t i for all i = 1 , . . . , m and ψ ( s i ) = ± s i for each i = 1 , . . . , (cid:96) (with the signdetermined by whether s i or s − i is in S ), which extends to a map ψ : A Λ j → A Λ j byspecifying ψ ( a ) = a for all Reeb chords a . This map ψ now intertwines the differentials ∂ comb and ∂ S j : ψ : ( A Λ j , ∂ comb ) ∼ = −→ ( A Λ j , ∂ S j ) . We can combine this with the isomorphism φ S j : ( A Λ j , ∂ S j ) ∼ = −→ ( A Λ j , ∂ NC ) from Section 3.4to obtain an isomorphism φ S j ◦ ψ from ( A Λ j , ∂ comb ) to the DGA ( A Λ j , ∂ NC ) with the null-cobordant spin structure.Recall from Section 3.4 that since each S j is odd-cardinality, each cobordism L j induces acobordism map Φ L j : ( A Λ j , ∂ S j ) → ( A Λ j − , ∂ S j − ). By combining this with the isomorphism ψ , we can view the cobordism map as a map ( A Λ j , ∂ comb ) → ( A Λ j − , ∂ comb ), which we alsowrite as Φ L j , so that the following diagram commutes:( A Λ j , ∂ comb ) ψ ∼ = (cid:47) (cid:47) Φ Lj (cid:15) (cid:15) ( A Λ j , ∂ S j ) Φ Lj (cid:15) (cid:15) ( A Λ j − , ∂ comb ) ψ ∼ = (cid:47) (cid:47) ( A Λ j − , ∂ S j − ) . imilarly, we can view the cobordism map Φ L (cid:48) as a DGA map ( A Λ , ∂ comb ) → ( A Λ , ∂ comb ).By the functoriality property from Proposition 3.11, we haveΦ L (cid:48) = Φ L ◦ · · · ◦ Φ L (cid:96) : ( A Λ , ∂ comb ) → ( A Λ , ∂ comb ) . We obtain the filling L of Λ from the cobordism L (cid:48) by filling in the k components of theunlink Λ with disjoint Lagrangian disks. Each disk filling produces a unique augmentation,as we record in the following statement. Proposition 3.12.
Let U denote the standard Legendrian unknot with a collection of basepoints with labels l , . . . , l p (where typically each label is of the form ± s ± i or ± t ± i ). If l · · · l p = − then the DGA ( A U , ∂ U ) has a unique augmentation.Proof. Let a denote the Reeb chord of U . If l , . . . , l q are the base points on one lobe ofthe figure eight in Π xy ( U ) and l q +1 , . . . , l p are the base points on the other, then ∂ U ( a ) = l · · · l q + l − p · · · l − q +1 . The condition for ε to be an augmentation is that ε ( ∂ U ( a )) = 0, inwhich case ε is uniquely determined since ε ( a ) = 0 for grading reasons. (cid:3) Now let w , . . . , w k denote the product of the labels of the base points on each of the k com-ponents of the unlink Λ , and write R for the ring R = ( Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ]) / ( w = · · · = w k = − by disks yields an augmentation ε : A Λ → R. Composing with Φ L (cid:48) now gives the augmentation of Λ induced by L :Φ L = ε ◦ Φ L (cid:48) : A Λ → R. We will already call this Φ L the combinatorial system of augmentations of Λ induced by L ,even though it will not become fully combinatorial until we present the combinatorial cobor-dism maps for isotopy cylinders and saddle cobordisms in Section 4. This is to temporarilydistinguish Φ L from the geometric system of augmentations of Λ from Proposition 3.10. Infact, the two systems agree up to equivalence, as we will show next.3.6. The systems of augmentations agree.
In this subsection, we prove that the com-binatorial and geometric systems of augmentations of a decomposable filling L are equiv-alent. This result generalizes a result of Y. Pan from [Pan17b, section 3], which uses Z coefficients and treats the case where Λ has a single component. We use the same no-tation as in the previous subsection: Φ L is a map from A Λ to R , where R is the ring R = ( Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ]) / ( w = · · · = w k = −
1) with w , . . . , w k being words asso-ciated to the k minima of L . The desired equivalence is shown in the following result, whichwill be proven momentarily: Proposition 3.13.
Suppose that the filling L of Λ is connected. Then we have R ∼ = Z [ Z g +2 m − ] and consequently Φ L is a (2 g + 2 m − -system of augmentations of Λ . Fur-thermore, up to equivalence, Φ L agrees with the geometric system of augmentations fromProposition 3.10. The crucial consequence of Proposition 3.13 is that since geometric systems of augmentationsare invariant under Hamiltonian isotopy of the filling, the same is true of the combinatorialsystem of augmentations Φ L . This is the fact that will allow us to distinguish fillings througha combinatorial calculation of their augmentations. Indeed, the following result is a directconsequence of Propositions 3.10 and 3.13: Proposition 3.14.
Let L be a connected filling of Λ , and suppose that all Reeb chords of Λ have nonnegative degree. Then the combinatorial system of augmentations Φ L of A Λ isinvariant, up to equivalence, under exact Lagrangian isotopy of L . (cid:3) he argument for Proposition 3.13 above occupies the remainder of this section. Proof of Proposition 3.13.
By functoriality, Φ L and the system of augmentations from Propo-sition 3.10 agree over Z . What we need to do is keep track of the homology coefficients thatappear in the definitions of the two families of augmentations, and show that the two agreeup to equivalence. Thus, we reduce mod 2 and work with group rings over Z . In the courseof tracking the homology coefficients, we will see that the abelian group generated multi-plicatively by t , . . . , t m , s , . . . , s (cid:96) with relations w = · · · = w k = 1 is isomorphic to a freeabelian group with 2 g + 2 m − R ∼ = Z [ Z g +2 m − ].As in Section 3.5, let τ i and σ i denote the oriented arcs on L (cid:48) corresponding to t i and s i . Themap Φ L counts intersections with τ i and σ i ; what we will show is that these counts keep trackof homology classes in H ( L ) along with link automorphisms. If a is a degree-0 Reeb chordof Λ, let M ( a ) denote the moduli space of (rigid) holomorphic disks ∆ with boundary on L and a single positive boundary puncture mapping to a . We may assume that L is generic, sothat none of the minima of L lies on the boundary of a holomorphic disk in any of the M ( a ).Recall that L (cid:48) is obtained from L by removing a neighborhood of each minimum of L . Bymaking these neighborhoods sufficiently small, we may assume that the boundary of eachof the holomorphic disks ∆ ∈ M ( a ) lies entirely in L (cid:48) and does not intersect the negativeboundary Λ of L (cid:48) : that is, ∂ ∆ is an oriented arc on L (cid:48) with endpoints at the endpoints of a .The cobordism map Φ L (cid:48) is then given as follows, for all degree 0 Reeb chords a of Λ:Φ L (cid:48) ( a ) = (cid:88) ∆ ∈M ( a ) w (∆) ∈ Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] , where(3.2) w (∆) = m (cid:89) i =1 t ∂ ∆ ∩ τ i ) i (cid:96) (cid:89) i =1 s ∂ ∆ ∩ σ i ) i . By the discussion preceding the proposition, the augmentation Φ L is the composition of Φ L (cid:48) with the quotient map Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] → Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] / ( w = · · · = w k = 1).We want to compare Φ L with the geometric setup from Section 3.3. Recall from Theorem 3.6that L induces an augmentation ε L : A Λ → Z [ H ( L )]. This map agrees over Z with Φ L but the group-ring coefficients are given by: ε L ( a ) = (cid:88) ∆ ∈M ( a ) exp([ ∂ ∆]) ∈ Z [ H ( L )] . The notation here is as follows. Choose a capping path γ a for each Reeb chord a of Λ: apath in the connected surface L whose endpoints are the same as the endpoints of γ a . Foreach disk ∆ ∈ M ( a ), close up the arc ∂ ∆ by adding the reverse of γ a to give a closed loop ∂ ∆ = ( ∂ ∆) ∪ ( − γ a ). Then ∂ ∆ represents a homology class in H ( L ), and we denote thisclass in Z [ H ( L )] by exp([ ∂ ∆]) (the exponential changes addition to multiplication).We specify particular capping paths γ a as follows. For i = 0 , . . . , (cid:96) −
1, let L >i := L i +1 ∪· · ·∪ L (cid:96) denote the portion of L above Λ i , and L >(cid:96) := Λ. Note that L >(cid:96) has m components while L > has 1 component, and there are exactly m − i for which L > ( i − has 1 fewercomponent than L >i . For notational simplicity we will assume that these are the largestpossible values: i = (cid:96) − m + 2 , . . . , (cid:96) . (A similar argument holds in general.) In this casethe first m − σ , . . . , σ m − are the cores of these 1-handle attachments, and we write σ ∨ , . . . , σ ∨ m − for the corresponding cocores. (More explicitly, begin at the i -th saddle, place one point oneach strand of the crossing above this saddle, and trace this pair of points upwards throughthe cobordism to Λ to produce σ ∨ i .) The paths σ ∨ , . . . , σ ∨ m − join the m components of Λ to ach other. For each Reeb chord a of Λ, we can now choose the capping path γ a to lie onΛ ∪ σ ∨ ∪ · · · ∪ σ ∨ m − and to avoid the base points t , . . . , t (cid:96) on Λ. By construction, among thearcs τ , . . . , τ m , σ , . . . , σ (cid:96) , the only ones that γ a intersects are some subset of σ , . . . , σ m − determined by which components of Λ contain the endpoints of a .Since the arcs σ ∨ , . . . , σ ∨ m − form a tree connecting the components of Λ, it is easy to verifythat we can find units u , . . . , u m ∈ Z [ s ± , . . . , s ± m − ] such that for each i = 1 , . . . , m − σ ∨ i ends on component r ( i ) and begins on component c ( i ) of Λ, then s i = u r ( i ) u − c ( i ) .Furthermore, ( u , . . . , u m ) are well-defined once we specify u m = 1, and the induced map Z [ s ± , . . . , s ± m − ] → Z [ u ± , . . . , u ± m − ] is an isomorphism. It now follows by the constructionof the capping paths γ a that if a is any Reeb chord of Λ and r ( a ) , c ( a ) ∈ { , . . . , m } are thecomponents of the ending and beginning points of a , then m − (cid:89) i =1 s γ a ∩ σ i ) i = u r ( a ) u − c ( a ) . Suppose for now that L has exactly one minimum; the general case will be considered after-ward. Then Λ is a single-component unknot U , and the product of the labels of the basepoints on Λ is t · · · t m since the s i base points cancel in pairs. Note that the abelian groupgenerated by t , . . . , t m , s , . . . , s (cid:96) with a single relation t · · · t m = 1 is free on m + (cid:96) − g + 2 m − L maps to the ring Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] / ( t · · · t m = 1) ∼ = Z [ Z g +2 m − ].When L has one minimum, the relative homology H ( L, Λ) is generated by σ m , . . . , σ (cid:96) and τ − τ , . . . , τ m − τ . (Strictly speaking all of these arcs end on Λ ; we extend these arcs byadding arcs in the disk filling Λ , so that any endpoint on Λ is replaced by an endpoint at theminimum of L .) Since H ( L ) is dual to H ( L, Λ), we can compute the homology class [ ∂ ∆]for ∆ ∈ M ( a ) by counting intersections with the generating set of H ( L, Λ). To be precise,we can identify Z [ H ( L )] ∼ = Z [ t ± , . . . , t ± m , s ± m , . . . , s ± (cid:96) ], and under this isomorphism wehaveexp[ ∂ ∆] = m (cid:89) i =2 t ∂ ∆ ∩ τ i ) − ∂ ∆ ∩ τ ) i (cid:96) (cid:89) i = m s ∂ ∆ ∩ σ i ) i = ( t · · · t m ) − ∂ ∆ ∩ τ ) m (cid:89) i =2 t ∂ ∆ ∩ τ i ) i (cid:96) (cid:89) i = m s ∂ ∆ ∩ σ i ) i . We now compare this to the formula for w (∆) in equation (3.2): w (∆) | t =( t ··· t m ) − = (cid:32) m (cid:89) i =1 t ∂ ∆ ∩ τ i ) i (cid:33)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =( t ··· t m ) − m − (cid:89) i =1 s ∂ ∆ ∩ σ i ) i (cid:96) (cid:89) i = m s ∂ ∆ ∩ σ i ) i = ( t · · · t m ) − ∂ ∆ ∩ τ ) m (cid:89) i =2 t ∂ ∆ ∩ τ i ) i m − (cid:89) i =1 s γ a ∩ σ i ) i (cid:96) (cid:89) i = m s ∂ ∆ ∩ σ i ) i = u r ( a ) u − c ( a ) exp[ ∂ ∆] , where in the second equality we use the fact that ∂ ∆ and γ a have the same endpoints and σ i is a separating curve in L . Now, we extend ε L : A Λ → Z [ H ( L )] ∼ = Z [ t ± , . . . , t ± m , s ± m , . . . , s ± (cid:96) ]by a link automorphism to˜ ε L : A Λ → Z [ H ( L ) ⊕ Z m − ] ∼ = Z [ t ± , . . . , t ± m , s ± m , . . . , s ± (cid:96) , u ± , . . . , u ± m − ]defined by ˜ ε L ( a ) := u r ( a ) u − c ( a ) ε L ( a ), as in Section 3.3. Then, we have˜ ε L ( a ) = (cid:88) ∆ ∈M ( a ) u r ( a ) u − c ( a ) exp([ ∂ ∆]) = (cid:88) ∆ ∈M ( a ) w (∆) | t =( t ··· t m ) − . hat is, the following diagram commutes: A Λ ˜ ε L (cid:47) (cid:47) Φ L (cid:43) (cid:43) Z [ t ± , . . . , t ± m , s ± m , . . . , s ± (cid:96) , u ± , . . . , u ± m − ] ∼ = (cid:15) (cid:15) Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] / ( t · · · t m = 1) . This shows that the combinatorial cobordism map Φ L and the geometric cobordism map ˜ ε L agree up to isomorphism when L has one minimum.Now suppose that L has k > σ i have endpoints at the minima; since L is connected,there is a spanning tree of k − σ (cid:96) − k +2 , . . . , σ (cid:96) . Now imagine deforming L by homotopy equivalence by successively contracting each arc σ (cid:96) − k +2 , . . . , σ (cid:96) to a point.The result is a new surface (cid:101) L with a single minimum, which inherits the arcs τ , . . . , τ m and σ i , i ≤ (cid:96) − k + 1. The geometric cobordism map ε L is defined homologically and does notchange when we replace L by (cid:101) L .We now examine what happens to the cobordism map Φ L as we pass from L to (cid:101) L . Recallthat Φ L is an augmentation taking values in the ring Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] / ( w = · · · = w k = 1), where for j = 1 , . . . , k , w j is the word given by the product of the arcshaving an endpoint at the j -th minimum (each endpoint contributes t ± i or s ± i dependingon the orientation of the corresponding arc at the minimum). At the step where we contract σ i , note that s i appears in exactly two words w i and w i corresponding to the endpointsof σ i . We use the relation for one of these words, w i = 1, to solve for s i , and substituteinto w i = 1; the result is exactly the relation corresponding to the new minimum givenby contracting σ i . Once we have contracted all of σ (cid:96) − k +2 , . . . , σ (cid:96) , we are left with a singleword w for the unique remaining minimum, and this process gives an isomorphism betweenthe coefficient ring Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] / ( w = · · · = w k = 1) for L and the ring Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) − k +1 ] / ( w = 1) for (cid:101) L . In particular, note that the abelian groupgenerated by t , . . . , t m , s , . . . , s (cid:96) with relations w = · · · = w k = 1 is again free on 2 g +2 m − L has one minimum. Figure 16.
Sliding the arc ∂ ∆ to avoid intersections with σ i , and then con-tracting σ i .Now in (cid:101) L , the boundaries ∂ ∆ of some holomorphic disks may pass through the minimum.To restore transversality, we perturb each ∂ ∆ as follows: at the step where we contract σ i ,we homotop ∂ ∆ near any intersection with σ i so that it wraps around one of the endpointsof σ i instead; see Figure 16. This removes any intersections of ∂ ∆ with σ i , and it does notchange the word w ( ∂ ∆) as given in (3.2) because of the relations w j = 1. The end resultis the surface (cid:101) L where all boundaries ∂ ∆ are disjoint from the minimum of (cid:101) L , and we havereduced to the case of a single minimum. This completes the proof. (cid:3) Remark 3.15.
The above proof shows that the augmentation/cobordism map Φ L : A Λ → Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] / ( w = · · · = w k = 1) sends the product t · · · t m to 1, since the roduct w · · · w k is equal to t · · · t m : each σ arc contributes endpoints that cancel, and each τ arc ends at exactly one of the minima.We can lift this statement to Z coefficients: if A Λ is the DGA of Λ with the Lie group spinstructure, then Φ L : A Λ → Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ] / ( w = · · · = w k = 1) sends t · · · t m to ( − m . This follows from a result of Leverson [Lev17] that any augmentation of A Λ to afield (whether or not it comes from a filling) must send t · · · t m to ( − m , whence this mustbe true of Φ L . (cid:3) Cobordism Maps for Elementary Cobordisms
Given a decomposable Lagrangian filling L of a Legendrian link Λ, we have described inSection 3 the general theory of how to build a system of augmentations for L . In order toapply this theory, we will use combinatorial formulas for cobordism maps corresponding toelementary cobordisms, which we can then compose to produce a formula for the cobordismmap of an arbitrary decomposable filling. Of the three elementary cobordisms in Section 2.1,we have already discussed the DGA map for a minimum cobordism; see Proposition 3.12.In this section we present combinatorial formulas for the cobordism maps for the other twoelementary cobordisms: isotopy cobordisms and saddle cobordisms.The map for an isotopy cobordism (Section 4.1) is not new and dates back originally towork of K´alm´an [K´al05]. The map for a saddle cobordism (Section 4.2) occupies the bulkof Section 4, with some technical details postponed to Appendix A. It builds on work ofEkholm–Honda–K´alm´an [EHK16], but introduces two new features:1. A combinatorial lift to integer coefficients Z ,2. A formula that (even) over Z works for some saddle cobordisms (where the combi-natorial EHK map over Z does not).In order to lift the saddle cobordism map to Z , rather than directly constructing explicitorientations of the relevant moduli spaces, we use an ad hoc argument that allows us todeduce signs for a particularly simple saddle cobordism from the fact, due to work of Karlsson[Kar20], that the map must be a chain map over Z . In fact we conclude a slightly weakerresult: namely, we show that the cobordism map agrees with our combinatorial formulaup to a link automorphism. Nevertheless, this additional choice of link automorphism willnot affect our computations, and the statement we obtain is sufficient for the purposes ofcalculating augmentations for fillings. This is explained in Section 4.3.4.1. The cobordism map for a Legendrian isotopy.
In this subsection we review thecobordism map for an isotopy cobordism. Suppose that Λ + and Λ − are Legendrian linksrelated by a Legendrian isotopy. There is then a quasi-isomorphism between the DGAs( A Λ + , ∂ ) and ( A Λ − , ∂ ), as first constructed by Chekanov [Che02] over Z and then lifted to Z in [ENS02]. More precisely, these quasi-isomorphisms are DGA maps that are constructedfor certain elementary Legendrian isotopies, to be described below. Any general Legendrianisotopy can be broken down into a sequence of elementary isotopies, and we compose theDGA maps for the elementary pieces to produce a DGA map for the isotopy.This picture fits in a natural way with cobordism maps. Given a Legendrian isotopy betweenΛ + and Λ − , let L denote the corresponding Lagrangian cobordism between Λ + and Λ − .Then Ekholm–Honda–K´alm´an [EHK16, section 6.3] show that over Z , the cobordism mapΦ L : ( A Λ + , ∂ ) → ( A Λ − , ∂ ) agrees with the DGA map associated to the isotopy; note that byfunctoriality, it suffices to show this when L is the cobordism for an elementary isotopy. Thisresult was subsequently upgraded to Z coefficients by the combined work of K´alm´an [K´al05],who showed that the map of DGAs over Z associated to an isotopy (a path in the space ofLegendrian links) is invariant under homotopy of the path; Ekholm–K´alm´an [EK08], whoshowed that over Z , this DGA map gives the differential for the Legendrian contact DGA f the Legendrian surface given by the lift of L ; and Karlsson [Kar20, section 6], who showedthat one can assign signs to the differential of this Legendrian surface to induce signs for thecobordism map Φ L . For our purposes, we summarize this work as follows. Proposition 4.1 ([K´al05, EK08, EHK16, Kar20]) . Suppose that Λ + and Λ − are relatedby an elementary Legendrian isotopy, with corresponding Lagrangian cobordism L . Choosebase points, a spin structure, and capping operators on Λ + ; these induce, via the isotopy,a corresponding choice of base points, spin structure, and capping operators on Λ − . Then,the cobordism map Φ L : ( A Λ + , ∂ ) → ( A Λ − , ∂ ) is equal to the DGA map for the isotopy asconstructed in [Che02, ENS02] . By “elementary Legendrian isotopy”, we will mean one of the following three isotopies be-tween Legendrian links with base points, all described in terms of their xy projections: • Base point moves: fix the xy projection and move a base point across a crossing, • Reidemeister III moves (triple point moves), • Reidemeister II moves.Any Legendrian isotopy can be decomposed into these elementary isotopies, along with planarisotopies of the xy projection in R .In the remainder of this subsection, we review the combinatorial formulas from [Che02,ENS02] for the DGA maps for elementary isotopies. As usual, to compute the cobordismmap for a general Legendrian isotopy, we can divide the isotopy into elementary isotopiesand compose the resulting cobordism maps.4.1.1. Base point moves.
Suppose that Λ and Λ (cid:48) are Legendrian links that are related by abase point move: outside of a neighborhood of a Reeb chord a , their xy projections agree,and inside this neighborhood, a base point moves across the crossing. See Figure 17. Thenthe DGA map for this move is Ψ : ( A Λ , ∂ ) → ( A Λ (cid:48) , ∂ (cid:48) ) defined as follows: Ψ acts as theidentity on all Reeb chords besides a and on all base point variables including s , andΨ( a ) = sa (left diagram) Ψ( a ) = as − (right diagram) . Note that Ψ is an isomorphism, and the DGA map for the reverse of one of these base pointmoves is Ψ − . Figure 17.
A base point move.We observe that if we move a base point (or collection of base points) all the way arounda component of Λ until it returns to where it started, the corresponding automorphism of( A Λ , ∂ ) is the identity map. (This uses the fact that the variable s associated to the basepoint commutes with Reeb chord generators of A Λ ; in the fully noncommutative settingwhere s does not commute with Reeb chords, the automorphism is conjugation by s .) Asa consequence, when calculating the cobordism map for an isotopy cobordism L , we do notneed to specify an arc on L joining corresponding base points on the ends of L , since anytwo choices of such an arc will yield the same map. .1.2. Reidemeister III moves.
Suppose that Λ and Λ (cid:48) are related by a Reidemeister III move:see Figure 18. There are two types of Reidemeister III moves, III a (left diagram) and III b (right diagram); these are called “Move II” and “Move I” in [ENS02], respectively, and “L1a”and “L1b” in [EHK16]. There is a one-to-one correspondence between the Reeb chords of Λand Λ (cid:48) , with the correspondence between the three crossings involved in the move shown inFigure 18. Under this identification, A Λ and A Λ (cid:48) are identical. Figure 18.
The two types of Reidemeister III moves.The DGA map for Reidemeister III a , which we will actually not need in this paper, is simplythe identity map on A Λ . To describe the DGA map for Reidemeister III b , let σ ∈ {± } denote the product of the orientation signs of the three quadrants of Π xy (Λ) indicated inFigure 18: this is +1 or − A Λ , δ ) → ( A Λ (cid:48) , δ (cid:48) ) is defined to be the identity on allReeb chords except for a and on all base point variables, andΨ( a ) = a + σa a . Reidemeister II moves.
Figure 19.
A Reidemeister II move.The DGA maps for a Reidemeister II move are more involved than for the other elementaryisotopies. Suppose that Λ and Λ (cid:48) are related by a Reidemeister II move, with Π xy (Λ (cid:48) ) havingtwo more crossings than Π xy (Λ), as shown in Figure 19.Let a , . . . , a r be the Reeb chords of Λ, and let b , b denote the two new Reeb chords of Λ (cid:48) .Write ( A Λ , ∂ ) and ( A Λ (cid:48) , ∂ (cid:48) ) for the DGAs of Λ and Λ (cid:48) . Let | b | = i = | b | + 1 in A Λ (cid:48) , andconstruct the stabilization ( S ( A Λ ) , ∂ ) by adding two generators e , e with | e | = i = | e | + 1to A Λ and extending the differential ∂ by ∂ ( e ) = e , ∂ ( e ) = 0. There is a chain isomorphismΨ : A Λ (cid:48) → S ( A Λ ) whose definition we recall below. We can then compose Ψ − with theinclusion map i : A Λ → S ( A Λ ) to get a chain map Ψ − ◦ i : A Λ → A Λ (cid:48) . In the other direction,we can compose Ψ with the projection map p : S ( A Λ ) → A Λ sending each generator of A Λ to itself and sending e , e to 0, to get a chain map p ◦ Ψ : A Λ (cid:48) → A Λ : A Λ i (cid:47) (cid:47) Ψ − ◦ i (cid:36) (cid:36) S ( A Λ ) p (cid:111) (cid:111) A Λ (cid:48) . Ψ ∼ = (cid:111) (cid:111) p ◦ Ψ (cid:99) (cid:99) hen Ψ − ◦ i and p ◦ Ψ are the cobordism maps for the cobordisms from Λ (cid:48) to Λ and fromΛ to Λ (cid:48) , respectively, induced by the Reidemeister II isotopy.We will need the precise definition of Ψ from [ENS02], and we recall it now. Let a , . . . , a r be the Reeb chords of Λ, ordered in increasing height. Inductively construct a sequence ofalgebra isomorphisms Ψ , Ψ , . . . , Ψ r : A Λ (cid:48) → S ( A Λ ) as follows. By inspecting Figure 19, wesee that there is a bigon for Λ (cid:48) with + corner at b and − corner at b , and so we can write ∂ (cid:48) ( b ) = σb + v where σ ∈ {± } and v counts disks with + corner in the leftmost quadrantat b . The map Ψ (written as Φ in [ENS02]) is defined byΨ ( b ) = e Ψ ( b ) = σ ( e − v ) Ψ ( a (cid:96) ) = a (cid:96) . Given Ψ (cid:96) − , we define Ψ (cid:96) = g (cid:96) ◦ Ψ (cid:96) − , where g (cid:96) : S ( A Λ ) → S ( A Λ ) is the identity on allgenerators except a (cid:96) , and g (cid:96) ( a (cid:96) ) = a (cid:96) + H ( ∂a (cid:96) − Ψ (cid:96) − ∂ (cid:48) a (cid:96) ) . Here H is the map on S ( A Λ ) (a module map, not an algebra map) defined by H ( w ) = 0 if w is any word that either does not contain e or e , or for which the leftmost e i appearing in w is e , and H ( w e w ) = ( − | w | +1 w e w if w does not contain e or e . Finally, Ψ = Ψ r .Noting that for each (cid:96) , Ψ( a (cid:96) ) = Ψ (cid:96) ( a (cid:96) ) = g (cid:96) ( a (cid:96) ), we can restate the definition of Ψ moresuccinctly as follows: Ψ( b ) = e Ψ( b ) = σ ( e − v )Ψ( a (cid:96) ) = a (cid:96) − H (Ψ ∂ (cid:48) a (cid:96) ) . (4.1)This definition looks circular since Ψ occurs on the right hand side of the definition of Ψ( a (cid:96) ),but in fact the height ordering and Stokes’ Theorem imply that for any (cid:96) , ∂ (cid:48) a (cid:96) involves only b , b , a , . . . , a (cid:96) − and not a (cid:96) +1 , . . . , a r , and so (4.1) can be used to recursively define Ψ( a (cid:96) ).Note that the height ordering does not appear explicitly in (4.1); however, the existence ofthe height filtration means that the recursive definition (4.1) terminates and thus producesa well-defined result. Remark 4.2.
It follows from the definition of Ψ that the chain map p ◦ Ψ : A Λ (cid:48) → A Λ hasthe following simple form:( p ◦ Ψ)( b ) = 0 ( p ◦ Ψ)( b ) = − σv ( p ◦ Ψ)( a (cid:96) ) = a (cid:96) . (cid:3) This concludes our description of the DGA maps associated to isotopy cobordisms.4.2.
The cobordism map for a saddle cobordism.
We now address the cobordism mapassociated to a saddle cobordism. Let Λ + be a Legendrian link with a contractible Reebchord a of degree 0; contractible chords of even degree can be similarly treated with suitablemodification to the grading. In the xy projection, replacing the crossing a by its orientedresolution yields a Legendrian link Λ − , and we write L a for the saddle cobordism betweenΛ − and Λ + .Our goal in this subsection is to write down a combinatorial formula for the cobordism mapΦ L a : ( A Λ + , ∂ ) → ( A Λ − , ∂ ). In [EHK16], Ekholm–Honda–K´alm´an describe such a formulafor this map over Z , subject to the assumption that the Reeb chord a is what they call“simple”. Our goal here is to describe the EHK map over Z and for what we call “properchords”, which are a different (and more general) class of contractible Reeb chords thansimple chords. The proof that our map is indeed the geometric cobordism map Φ L a (statedas Proposition 4.8 below) is deferred to Appendix A.Recall from Section 3.1 that the Legendrian contact differential ∂ for a Legendrian Λ countsimmersed disks with a single + corner, where “immersed disk” in our terminology includes he condition that all punctures are mapped to single quadrants (i.e., all corners are convex).We will now need to consider more general disks, which we call immersed disks with concavecorners . These are immersed disks where each boundary puncture is again mapped to acrossing of the Lagrangian projection Π xy (Λ), but where we now allow a neighborhood ofeach boundary puncture to be mapped to either a single quadrant at the crossing (a convexcorner) or the union of three quadrants (a concave corner). As with convex corners, we canlabel each concave corner as positive or negative, depending on whether 2 of the 3 quadrantscovered by the corner are positive or negative, respectively. Definition 4.3.
A contractible Reeb chord a of Λ + is proper if the following condition holds.For any immersed disk ∆, possibly with concave corners, such that: • ∆ has a positive convex corner at some Reeb chord besides a , • ∆ has at least one positive convex corner at a , • all other convex corners of ∆ are negative, and • the only possible concave corners of ∆ are positive concave corners at a ,then it must be the case that the (closure of the) boundary of ∆ in Π xy (Λ + ) passes throughthe crossing Π xy ( a ) only once. That is, any immersed disk ∆ must have no concave corners,∆ must have exactly one positive convex corner at a , and the boundary of ∆ never passesthrough a except at that corner. (cid:3) Remark 4.4.
We remark that being proper and being simple in the sense of [EHK16] arenot the same; we refer to [EHK16, Definition 6.16] for the index condition that defines thelatter property. In particular, the contractible Reeb chord a in Figure 21 below is properbut not simple. The necessity of considering saddle moves at Reeb chords like a in thispaper is what motivated our definition of proper chords. (cid:3) All of the Reeb chords that we use in this paper to perform saddle cobordisms are contractibleand proper. This is a consequence of the following result.
Proposition 4.5. If β ∈ Br + N is an admissible braid and a is a crossing of β such that β \{ a } contains a half-twist, then as a Reeb chord of the ( − -closure Λ( β ) , a is contractible andproper.Proof. Contractibility has already been shown in Proposition 2.8; we need to show properness.Suppose that ∆ is a disk as in Definition 4.3. Because ∆ has a positive convex corner at a , it must be “thin” in the sense that it lies in the neighborhood of the Legendrian unknotthat contains the satellite Λ( β ). The presence of the half-twist, and the fact that ∆ hasno concave corners in the half-twist, prevents ∆ from passing through the half-twist. Thisforces ∆ to be embedded in the neighborhood of the unknot, and so its boundary only passesthrough a once. (cid:3) We will next present a formula for the map for a saddle cobordism at a Reeb chord a when a is contractible and proper. As in [EHK16], the key is to consider immersed disks with two+ corners, one of which is at a . We break these into two types.Let a i be a Reeb chord of Λ + not equal to a . Define ∆ → a ( a i ), respectively ∆ ← a ( a i ), to be theset of immersed disks for Λ + , such that: • all corners are convex, and there are exactly two positive corners, one at a i and oneat a ; • at the corner at a , the orientation of Λ points toward, respectively away from (for∆ ← a ( a i )), the disk. igure 20. A disk in ∆ → a ( a i ) (left) and a disk in ∆ ← a ( a i ) (right). For thedisk on the left, w (∆) = a a and w (∆) = a . For the disk on the right, w (∆) = 1 and w (∆) = a .See Figure 20. For any ∆ ∈ ∆ → a ( a i ) ∪ ∆ ← a ( a i ), we can define three quantities. One is the signsgn(∆) ∈ {± } , which is the product of the orientation signs over all corners of Λ, multipliedby the signs of any base points traversed by the boundary of the disk. The other are twowords w (∆) , w (∆) ∈ A Λ + , defined as follows: w (∆) is the product of the − corners andbase points that we encounter as we traverse the boundary of ∆ counterclockwise from a i to a , and w (∆) is the analogous product as we traverse the boundary counterclockwise from a to a i . See Figure 20 for an example. Definition 4.6.
The combinatorial cobordism map, denoted Φ comb L a : A Λ + → A Λ − , is thecomposition of three algebra maps:Φ comb L a := Φ ← ◦ Φ → ◦ Φ , where Φ : A Λ + → A Λ − is defined by Φ ( a ) = s and Φ ( a i ) = a i for any Reeb chord a i besides a , and Φ → , Φ ← : A Λ − → A Λ − are defined as follows. Let a i be a generator of A Λ − ,that is, a Reeb chord of Λ − , which is then also a Reeb chord of Λ + . Then,Φ → ( a i ) := a i + (cid:88) ∆ ∈ ∆ → a ( a i ) ( − | w (∆) | sgn(∆)Φ → ( w (∆)) s − w (∆)Φ ← ( a i ) := a i + (cid:88) ∆ ∈ ∆ ← a ( a i ) ( − | w (∆) | sgn(∆)Φ ← ( w (∆)) s − w (∆) . (cid:3) Remark 4.7.
As with the definition of the Reidemeister II cobordism map, equation (4.1)in Section 4.1, these definitions may appear circular but can be used to recursively defineΦ → and Φ ← . The reason is that if we order the Reeb chords a , . . . , a r in increasing orderof height, then all disks with positive punctures at a and a j can only have negative cornersat a , . . . , a j − and not at a j +1 , . . . , a i r : in particular, if ∆ ∈ ∆ → a ( a j ) ∪ ∆ ← a ( a j ) then w (∆)only involves a , . . . , a j − . (cid:3) The key result in this subsection is the relation between Φ comb L a , as defined above, and Φ L a .This is the content of the following result: Proposition 4.8. If a ∈ A Λ + is a proper contractible Reeb chord, then the cobordism map Φ L a : A Λ + → A Λ − is equal to the combinatorial map Φ comb L a , up to a link automorphism of Λ − . That is, there is a link automorphism Ω : A Λ − → A Λ − such that Φ L a = Ω ◦ Φ comb L a . Proposition 4.8 is proved in Appendix A below. Let us illustrate how to compute Φ comb L a inan explicit example, which will also appear as part of our later computations. Example 4.9.
Consider the configuration shown in Figure 21, this appears as part of ourcalculations for the (cid:101) D -Legendrian in Section 6.2. The first step in that calculation is a saddle igure 21. Calculating the cobordism map for a saddle cobordism at a .Left, the Legendrian link at the top of the cobordism; right, an immersed diskshowing that a is not simple.move at a , and we calculate the corresponding map Φ ← here. By inspection we see that∆ ← a ( a i ) = ∅ for 10 ≤ i ≤
13, while ∆ ← a ( a ) and ∆ ← a ( a ) each contain one disk apiece, withnegative corners at a , a and a , a respectively. For a , ∆ ← a ( a ) contains three disks,one with no negative corners, one with negative corners at a , a , and one with negativecorners at a , a . It follows from this that Φ ← ( a i ) = a i for 10 ≤ i ≤
13 andΦ ← ( a ) = a − Φ ← ( a a ) s − = a − a a s − Φ ← ( a ) = a − Φ ← (1) s − a a = a − s − a a Φ ← ( a ) = a − s − − s − a a − Φ ← ( a a ) s − = a − s − − s − a a − ( a − s − a a ) a s − . For the complete Legendrian that we study in Section 6.2, an inspection of Figure 26 showsthat ∆ → a ( a i ) = ∅ and thus Φ → ( a i ) = a i for 10 ≤ i ≤
16. It follows that the map Φ comb L a sends a to s and agrees with Φ ← for a i , 10 ≤ i ≤ a is contractible and proper but not simple, and thus evenover Z we cannot directly apply the combinatorial formula from [EHK16]. The reason a is not simple is the disk shown in Figure 21, which has 2 positive corners at a , 1 positivecorner at a , and a concave corner at a . (cid:3) Remark 4.10. If a is not just proper but also simple, then our definition of Φ comb L a can bestated in an easier way, to match [EHK16]. In this case, write ∆ a ( a i ) = ∆ → a ( a i ) ∪ ∆ ← a ( a i ).If ∆ is any disk in ∆ a ( a i ) and a j is a negative corner of ∆, then it must be the casethat ∆ a ( a j ) = ∅ ; otherwise the union of ∆ and a disk ∆ (cid:48) ∈ ∆ a ( a j ) is an immersed diskwith concave corner at a j and two positive (convex) corners at a , violating the simplicitycondition. Then we can drop the Φ → and Φ ← in Φ → ( w (∆)) and Φ ← ( w (∆)), and concludedirectly that for all a i ,Φ comb L a ( a i ) = a i + (cid:88) ∆ ∈ ∆ a ( a i ) sgn(∆) w (∆) s − w (∆) . If we set s = 1 and reduce mod 2, this recovers the formula for the cobordism map from[EHK16, Proposition 6.18]. (cid:3) Assembling elementary cobordism maps.
Having described the cobordism mapsfor elementary cobordisms, we can calculate the map associated to any decomposable cobor-dism by composing the maps for its elementary pieces, and indeed this is what we do inSections 5 and 6 below. There is a possible difficulty with this approach: we have onlycalculated the saddle cobordism map up to a link automorphism (see Proposition 4.8). How-ever, for a filling, the extra flexibility provided by the basepoint parameters gets rid of thisproblem, as we explain in this subsection.Let L be a connected decomposable genus- g filling of an m -component Legendrian link Λ.As in Section 3.5, we decorate L with arcs corresponding to base points t , . . . , t m , s , . . . , s (cid:96) .Divide L into elementary cobordisms L , . . . , L k , where: L j is a cobordism between Legendrians Λ j − and Λ j , with Λ = ∅ and Λ k = Λ; • L is a disjoint union of minimum cobordisms; • for j = 2 , . . . , k , L j is either an isotopy cobordism or a saddle cobordism.Note that this decomposition differs slightly from our simplified setup in Section 3.5, wherewe suppressed isotopy cobordisms.As in Section 3.5, let R be the ring R := ( Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ]) / ( w = · · · = w k = − ∼ = Z [ H ( L ) ⊕ Z m − ] , where w , . . . , w k are words coming from the minima of L . For each j , let ( A Λ j , ∂ comb )denote the DGA for Λ j over R , with A Λ = A Λ k . Then each elementary cobordism gives amap Φ L j : ( A Λ j , ∂ comb ) → ( A Λ j − , ∂ comb ), and their composition is a (2 g + 2 m − L = Φ L ◦ · · · ◦ Φ L k : ( A Λ , ∂ comb ) → ( R, . Now suppose that for j = 1 , . . . , k , Ω j : A Λ j − → A Λ j − is a link automorphism of Λ j − , anddefine (cid:101) Φ L j = Ω j ◦ Φ L j . Proposition 4.11.
The maps Φ L = Φ L ◦ · · · ◦ Φ L k and (cid:101) Φ L = (cid:101) Φ L ◦ · · · ◦ (cid:101) Φ L k are equivalentsystems of augmentations of Λ .Proof. We prove by induction that for j = 1 , . . . , k , Φ L ◦ · · · ◦ Φ L j and (cid:101) Φ L ◦ · · · ◦ (cid:101) Φ L j areequivalent as maps ( A Λ j , ∂ comb ) → ( R, j = 1 is true since Φ L , and thus (cid:101) Φ L , are both the zero map on Reeb chords of Λ .For the induction step, assume that Φ L ◦ · · · ◦ Φ L j and (cid:101) Φ L ◦ · · · ◦ (cid:101) Φ L j are equivalent, so thatthere is an automorphism ψ j of R such that (cid:101) Φ L ◦ · · · ◦ (cid:101) Φ L j = ψ j ◦ (Φ L ◦ · · · ◦ Φ L j ). Sincethe map Φ L ◦ · · · ◦ Φ L j agrees with the geometric system of augmentations for L ∪ · · · ∪ L j by Proposition 3.13, and the geometric system incorporates link automorphisms of Λ j , thelink automorphism Ω j of Λ j induces an automorphism ω j of R such that(Φ L ◦ · · · ◦ Φ L j ) ◦ Ω j = ω j ◦ (Φ L ◦ · · · ◦ Φ L j ) . (Note that Proposition 3.13 assumes that L ∪ · · · ∪ L j is connected; however, the argu-ment here extends to the disconnected case as well, since the system of augmentations of adisconnected filling annihilates any Reeb chord with endpoints on different components.)We conclude that the following diagram commutes: A Λ j +1 Φ Lj +1 (cid:123) (cid:123) (cid:101) Φ Lj +1 (cid:35) (cid:35) A Λ j Ω j ∼ = (cid:47) (cid:47) Φ L ◦···◦ Φ Lj (cid:125) (cid:125) A Λ j Φ L ◦···◦ Φ Lj (cid:123) (cid:123) (cid:101) Φ L ◦···◦ (cid:101) Φ Lj (cid:33) (cid:33) R ω j ∼ = (cid:47) (cid:47) R ψ j ∼ = (cid:47) (cid:47) R. It follows that Φ L ◦ · · · ◦ Φ L j ◦ Φ L j +1 and (cid:101) Φ L ◦ · · · ◦ (cid:101) Φ L j ◦ (cid:101) Φ L j +1 are equivalent since one isthe composition of the other with ψ j ◦ ω j , and this completes the induction. (cid:3) By Proposition 4.11, when we build systems of augmentations for fillings by composing ele-mentary cobordism maps, we can replace any elementary cobordism map by its compositionwith a link automorphism. In particular, Proposition 4.8 implies that we can use the com-binatorial saddle map Φ comb as the cobordism map for a saddle cobordism, and this is whatwe will do in subsequent sections. . Legendrian Contact DGA and Cobordism Maps for ( − -closures In this section we present an algebraically amenable description of the Legendrian contactDGA for the ( − ϑ -loops andsaddle cobordisms on the DGA.5.1. The DGA of the ( − -closure of an admissible braid. Let σ k · · · σ k r ∈ Br + N bean admissible positive braid. Henceforth we will write Λ( σ k · · · σ k r ) for the ( − We decorate the xy projection of Λ( σ k · · · σ k r )as follows; see Figure 22. Place a column of base points on the n strands of the braidbetween braid crossings, as well as on either end of the braid, and label these base points t (cid:96),i ,1 ≤ i ≤ n , 0 ≤ (cid:96) ≤ r . (In practice we may only need some small subset of these base points;in that case we formally set t (cid:96),i = 1 for all of the other base points and then remove them.)The Reeb chords for Λ( σ k · · · σ k r ) consist of: • a , . . . , a r , of degree 0, corresponding to the crossings of the braid, and labeled in theobvious way; • c ij , 1 ≤ i, j ≤ n , of degree 1, corresponding to the Reeb chord of the standardLegendrian unknot U .Recall from Section 3.1 that in order to calculate degrees of Reeb chords, we need to choosea base point on each component of the link; any subset of the t (cid:96),i will do and produces thedegrees given above. Figure 22.
The Lagrangian projection of the Legendrian link Λ( σ k · · · σ k r ),with crossings and base points labeled. The braid itself is in the blue box.Arrows represent the orientation of the link.The differential on the Legendrian contact DGA of Λ( σ k · · · σ k r ) can be expressed in acompact way using the path matrices of K´alm´an [K´al06]. For k = 1 , . . . , n −
1, define an n × n matrix P k ( a ) (as a function of an input a ) as follows:( P k ( a )) ij = i = j and i (cid:54) = k, k + 11 ( i, j ) = ( k, k + 1) or ( k + 1 , k ) a i = j = k + 10 otherwise; Note that this differs from the notation Λ( β ) in Section 2.2, but by Proposition 2.6, the two notationsrepresent links that are Legendrian isotopic. Note that we number our braid strands in increasing order from bottom to top, while K´alm´an numbersbraid strands from top to bottom. We also incorporate base points while K´alm´an does not. hat is, P k ( a ) is the identity matrix except for the 2 × k and k + 1, which is ( a ). (These are the path matrices considered in [K´al06], but note thatwe number our braid strands in increasing order from bottom to top, while K´alm´an numbersbraid strands from top to bottom.) Also define t (cid:96) = ( t (cid:96), , . . . , t (cid:96),n ) and write D ( t (cid:96) ) for thediagonal n × n matrix with t (cid:96), , . . . , t (cid:96),n along the diagonal. Definition 5.1.
Let β = σ k · · · σ k r be an n -stranded braid decorated with base points, withcrossings and base points labeled as in Figure 22. The path matrix of β is the n × n matrix P β = D ( t ) P k ( a ) D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r ) . Colloquially, the ( i, j ) entry of the path matrix P ( β ) counts paths beginning at the left of β on strand i , ending at the right on strand j , and at each crossing the path encounters, eitherpassing straight through the crossing, or turning a corner if the path changes direction fromsoutheast to northeast at the corner. Each path produces a word by reading the base pointstraversed and corners turned in order, and the ( i, j ) entry of P ( β ) is the sum of these words. Proposition 5.2.
The differential on the DGA ( A Λ( σ k ··· σ kr ) , ∂ ) for Λ( σ k · · · σ k r ) is givenas follows: ∂ ( a (cid:96) ) = 0 , and if we assemble the c ij into an n × n matrix C = ( c ij ) and write for the n × n identity matrix, then: ∂ ( C ) = + P β . Proof.
Each degree-0 generator a (cid:96) has vanishing differential for degree reasons. For c ij , thereare two possible types of immersed disks (all of which are in fact embedded) with + cornerat c ij , depending on which + quadrant at c ij is covered by the disk. There is an embeddeddisk with + puncture at the right quadrant of c ij and no − puncture if i = j , and otherwisethere is no immersed disk with + puncture at this right quadrant. This produces the termin the formula. For embedded disks with + puncture at the left quadrant of c ij , we need tokeep track of ways that the boundary of this disk can enter the braid from the left on strand i and exit the braid to the right on strand j , with possible convex corners at some crossings a (cid:96) . The contribution of these disks to ∂ ( c ij ) is precisely the ( i, j ) entry of the path matrix P β . (cid:3) ϑ -monodromy action on the DGA. Consider a Legendrian link Λ = Λ( β, k ; γ ) ⊂ ( R , ξ st ) and its ϑ -loop, as defined in Section 2.4. Here we compute the morphism A ( ϑ ) : A Λ → A Λ induced by this Legendrian isotopy, which we call the ϑ -monodromy or purple box mon-odromy. To be precise, any Legendrian isotopy between Legendrian links induces a chainisomorphism between the (suitably stabilized) DGAs of the links, as described in [Che02,ENS02]. In the case of the isotopy given by the ϑ -loop, which consists entirely of Reide-meister III moves, it is not necessary to stabilize the DGAs, and as a result we obtain theaforementioned chain isomorphism A ( ϑ ), which we now compute explicitly.The Lagrangian projection of Λ( β, k ; γ ) is given in the right diagram in Figure 12. Let n denote the braid index of γ , and let N be the number of braid strands in Λ( β, k ; γ ), so that thebraid index of β is N − n + 1. As in Section 5.1, the Reeb chords of Λ( β, i ; γ ), which generatethe Legendrian contact DGA A Λ , come in two types: the degree 1 chords c ij , i, j ∈ [1 , N ],and the degree 0 chords in the braiding region. We can divide these Reeb chords into twotypes in another way. Call the sublink of Λ( β, i ; γ ) corresponding to the i -th strand of β (andcontaining the purple box γ ) the satellite sublink ; this is depicted in purple in Figure 12.We call crossings of Λ( β, i ; γ ) satellite crossings and non-satellite crossings depending onwhether or not they involve the satellite sublink. Note that the satellite crossings of degree1 are precisely c ij with i, j ∈ { k, . . . , k + n − } , while the satellite crossings of degree 0 comein groups of n , with each group coming from a single crossing of β . e allow for the placement of arbitrarily many base points on Λ( β, k ; γ ), subject to therestriction that any base points lying on the satellite sublink actually lie in the purple boxfor γ . (In practice, there will be one base point per strand of Λ( β, k ; γ ), and the base pointsin the purple box will lie on its right edge.) Let P γ denote the n × n path matrix for γ withits base points. Extend this to an N × N matrix (cid:101) P γ by (cid:101) P γ = P γ
00 0 where the central matrix P γ corresponds to rows and columns k, . . . , k + n − Proposition 5.3.
The purple-box monodromy map A ( ϑ ) : A Λ → A Λ is given on generatorsas follows. Assemble the degree generators c ij into an N × N matrix: then A ( ϑ )( C ) = (cid:101) P γ C (cid:101) P − γ . For degree generators, A ( ϑ ) fixes all non-satellite crossings, while its action on degree satellite crossings is as follows: A ( ϑ ) (cid:32) h ... h n (cid:33) = P γ (cid:32) h ... h n (cid:33) A ( ϑ ) (cid:32) h (cid:48) ... h (cid:48) n (cid:33) = (cid:0) P Tγ (cid:1) − (cid:32) h (cid:48) ... h (cid:48) n (cid:33) . Here h , . . . , h n is any group of satellite crossings coming from a crossing of β where the i -thstrand is the overcrossing, while h (cid:48) , . . . , h (cid:48) n is any group of satellite crossings coming from acrossing of β where the i -th strand is the undercrossing. See Figure 23. Figure 23.
A group of satellite crossings coming from an overcrossing (left)and an undercrossing (right).
Proof.
The ϑ -loop consists of a sequence of Reidemeister III moves that push the purple boxaround, and consequently the map A ( ϑ ) is the composition of a sequence of algebra isomor-phisms corresponding to these Reidemeister III moves, as given concretely in Section 4.1.2.In particular, any non-satellite crossing does not participate in any of the Reidemeister IIImoves and so it is fixed by A ( ϑ ).Next consider a group of degree 0 satellite crossings h , . . . , h n as in the statement of theproposition (the argument for h (cid:48) , . . . , h (cid:48) n is similar and will be omitted). The ϑ -loop pushesthe purple box containing γ through h , . . . , h n from right to left. Since the path matrix P γ is a product of path matrices for individual crossings and columns of base points, and we canfactor the action of A ( ϑ ) on h , . . . , h n by pushing each individual crossing and base pointcolumn across h , . . . , h n from right to left and composing the results, the key is to observewhat happens when we push a single crossing or base point column of γ across h , . . . , h n . igure 24. Pushing a crossing (left) or a column of base points (right) acrossa group of satellite crossings.If we push a crossing a from γ across h , . . . , h n by a single Reidemeister III move as shown inFigure 24 (left), then from Section 4.1.2, the associated isomorphism sends a (cid:55)→ a , h i (cid:55)→ h i +1 ,and h i +1 (cid:55)→ h i + ah i +1 . (Note that compared to Figure 18, the crossings h i and h i +1 haveswitched places after the Reidemeister III move.) This is precisely the matrix map (cid:18) h i h i +1 (cid:19) (cid:55)→ (cid:18) a (cid:19) (cid:18) h i h i +1 (cid:19) . Thus pushing the crossing a across h , . . . , h n acts on (cid:32) h ... h n (cid:33) by left multiplication by thepath matrix for a . If instead we push a column of base points across h , . . . , h n as shownin Figure 24 (right), then from Section 4.1.1, the associated isomorphism sends h i to t i h i for i ∈ [1 , n ], which corresponds to left multiplication by the diagonal matrix with diagonalentries t , . . . , t n . Composing the individual isomorphisms, we conclude that the purple-boxmonodromy indeed acts on h , . . . , h n by left multiplication by the path matrix P γ , as desired.Finally, we consider the degree 1 crossings. As we perform the ϑ -loop, the purple box passesthrough the region with the degree 1 crossings twice. By essentially the same argument asfor degree 0 satellite crossings, the first pass results in the map C (cid:55)→ (cid:101) P γ C , while the secondpass yields C (cid:55)→ C (cid:101) P − γ . Composing these gives C (cid:55)→ (cid:101) P γ C (cid:101) P − γ . (cid:3) The K´alm´an loop.
The techniques of this section can be applied to compute the mon-odromy of other loops of Legendrian links besides ϑ -loops. One case where it is particularlysimple to calculate the monodromy in our setting is the loop of Legendrian T ( p, q )-toruslinks originally studied by K´alm´an in [K´al05]. For concreteness we focus here on the mostbasic example of the K´alm´an loop, which involves the max-tb Legendrian right handed trefoil( p = 2 , q = 3). K´alm´an constructs a loop in the space of these Legendrian trefoils and provesthat the induced action on the degree-0 Legendrian contact homology has order 5. Here wereinterpret this result in our setting.In our language, the Legendrian trefoil is the ( − σ ; in other words, it is Λ( β, γ ) ⊂ ( R , ξ st ) where β ∈ Br +1 is the 1-stranded braid and γ = σ ∈ Br +2 . We label the crossings of γ a , . . . , a and place base points t , t to the rightof γ , as shown in Figure 25 (left). The ϑ -loop moves the entire braid σ around the standardunknot Λ( β ) = U until it returns to its starting point. We can factor this loop as the fifthpower of another loop δ , which moves the single leftmost crossing of σ around the unknotuntil it returns to γ as the rightmost crossing. Note that this move shifts the position ofthe base points t , t ; we then slide t , t along the knot until they return to their originalpositions. See Figure 25. The combination of the crossing move and the base point move orms a loop beginning and ending at Λ( β, γ ), which is the K´alm´an loop and which wedenote by δ . Figure 25.
The Legendrian trefoil is the ( − σ . Left, the braiding region with base points; middle, the result of movingthe leftmost crossing to the right; right, the result of sliding the base pointsback to their original position (with the slides shown in the middle diagram).The action A ( δ ) of the loop δ on the Legendrian contact DGA A (Λ( β, γ )) is easy todescribe. Moving a to the right simply permutes the a i : a (cid:55)→ a , . . . , a (cid:55)→ a , a (cid:55)→ a .From Section 4.1.1, sliding the base points as indicated in Figure 25 fixes a and sends a i to t − a i t for i = 3 , t − a i t for i = 2 ,
4. Thus A ( δ ) acts on the DGA as follows: a (cid:55)→ t − a t a (cid:55)→ t − a t a (cid:55)→ t − a t a (cid:55)→ t − a t a (cid:55)→ a . (The degree 1 generators are fixed by A ( δ ).) By inspection we see that A ( δ ) has order 5, inagreement with K´alm´an’s result: A ( δ ) = A ( ϑ ) is the identity map.This argument readily generalizes to ( p, q )-torus links for arbitrary positive p, q . In the generalcase, the Legendrian link is the ( − σ . . . σ p − ) p + q ∈ Br + p ,and the K´alm´an loop δ moves the leftmost p − δ p + q acts as the identity on the Legendrian contactDGA. Remark 5.4.
The original proof in [K´al05] that A ( δ ) has order p + q uses the Legendrianlink given by the resolution of the rainbow closure of the braid ( σ · · · σ p − ) q . The DGA forthis link has ( p − q generators in degree 0, and K´alm´an’s computation of the monodromyof δ on this DGA is rather nontrivial, both because Reidemeister II moves are involved andbecause the DGA differential itself is quite complicated due to the presence of non-embeddeddisks. K´alm´an then performs an intricate computation to show that this monodromy hasorder p + q . The mere fact that p + q appears here, e.g. instead of p or q , is rather mysteriousfrom the geometric viewpoint. By contrast, in our setup with ( − p − p + q ) generators in degree 0, the monodromy of δ simply cyclically permutes thesegenerators, and it is evident without computation that this action has order p + q . (cid:3) The saddle cobordism map for ( − -closures. So far we have discussed the ϑ -monodromy. We now turn to the other principal computational ingredient in our calculationsfor the upcoming Section 6, namely the calculation of saddle cobordism maps: we will consideraugmentations corresponding to specific decomposable fillings, and these augmentations arethe composition of a number of saddle maps.In Section 4.2, we defined a combinatorial cobordism map Φ comb : A Λ + → A Λ − associatedto a saddle cobordism at any proper contractible Reeb chord. This combinatorial formulaallows us in Section 6 to calculate the augmentations corresponding to particular fillings of( − comb looks like forsaddle cobordisms of ( − − comb is indeed a chain map in this case, without going through the general theory. The interested eader may want to compare our discussion here with [GSW20a, section 3.3], which presentsan independent but rather similar matrix treatment of saddle cobordism maps.Consider a saddle cobordism whose top end is a Legendrian ( − + = Λ( σ k · · · σ k r ),and whose bottom end is the Legendrian link Λ − obtained by resolving a contractible propercrossing of Λ + . For ease of notation, we will assume that the crossing is a , correspondingto the braid generator σ k , and so Λ − = Λ( σ k · · · σ k r ). (The case of a saddle resolving anarbitrary crossing a (cid:96) is easy to deduce from this; just perform the cyclic-permutation isotopysending Λ + = Λ( σ k · · · σ k r ) to Λ( σ k (cid:96) · · · σ k r σ k · · · σ k (cid:96) − ) and similarly for Λ − .)From Section 5.1 above, we can write down the differentials ∂ ± on Λ ± in matrix form.Specifically, as in Section 5.1, we place base points t (cid:96),i , 1 ≤ i ≤ n , 1 ≤ (cid:96) ≤ r , next to thecrossings of Λ + . Then Λ − inherits this same array of base points, along with two new basepoints in place of the crossing a , one on strand k + 1 labeled by s and one on strand k labeled by − s − . By Proposition 5.2, in the notation from Section 5.1, the differentials ∂ + and ∂ − for the DGAs of Λ + and Λ − are given by the matrix formulas: ∂ + ( C ) = + P k ( a ) D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r ) ∂ − ( C ) = + D ( t ) D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r )where t = (1 , . . . , , − s − , s , , . . . ,
1) (with − s − and s in the k and k + 1 componentsrespectively).Let Φ comb = Φ ← ◦ Φ → ◦ Φ : A Λ + → A Λ − be the cobordism map from Proposition 4.8. Wefirst note that the action of Φ comb on degree-1 Reeb chords c ij is easy to write down. Indeed,write T ← k ( s ) for the n × n matrix equal to the identity matrix except with ( k , k + 1) entrygiven by s − . Then we have(5.1) Φ comb ( C ) = T ← k ( s ) C ( T ← k ( s )) − . This can be seen directly from an inspection of Figure 22, using the fact that ∆ → a ( c ij ) = ∅ ,while the only possible disks in ∆ ← a ( c ij ) are thin disks heading left from their + corner at a ,following the figure eight, and ending in the region containing the c ij ’s. We omit the detailshere.The explicit nature of this algebraic model allows us to sketch a direct argument for whyΦ comb is a chain map. Note that this argument is mainly provided for context and is notneeded in the rest of the paper, and so we do not provide full details; see also [GSW20a,section 3.3] for a related discussion with more details. Proposition 5.5. Φ comb ◦ ∂ + = ∂ − ◦ Φ comb .Proof. In order to show that Φ comb is a chain map, it suffices to show that Φ comb ( ∂ + ( C )) = ∂ − (Φ comb ( C )). Note thatΦ comb ( P k ( a )) = P k ( s ) = T ← k ( s ) D ( t ) T → k ( s )where T → k ( s ) is the identity matrix except with ( k + 1 , k ) entry given by s − . Since Φacts on C by conjugation by T ← k ( s ), showing that Φ is a chain map reduces to verifying thefollowing:(5.2)Φ comb (cid:0) T → k ( s ) D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r ) T ← k ( s ) (cid:1) = D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r ) . Call a matrix lower-unipotent if it is of the form + N where N is a strictly lower triangularmatrix; that is, a lower-unipotent matrix is a lower triangular matrix with 1’s along the The computation in the proof of Proposition 5.5 does contribute to the implementation of the program[Ng], in the code calculating the augmentation associated to a filling of a ( − iagonal. Note in particular that T → k ( s ) is lower-unipotent. It is straightforward to checkthat if T is lower-unipotent then T (cid:48) = ( P k (cid:96) ( a (cid:96) + T k (cid:96) +1 ,k (cid:96) )) − T P k (cid:96) ( a (cid:96) )is again lower-unipotent. We can thus inductively define a sequence of lower-unipotent ma-trices T (cid:48) , T , T (cid:48) , T , . . . , T (cid:48) r , T r as follows: T (cid:48) = T → k ( s ) ,T (cid:96) = D ( t (cid:96) ) − T (cid:48) (cid:96) D ( t (cid:96) ) ,T (cid:48) (cid:96) = ( P k (cid:96) ( a (cid:96) + ( T (cid:96) − ) k (cid:96) +1 ,k (cid:96) )) − T P k (cid:96) ( a (cid:96) ) . Then we have T (cid:96) − P k (cid:96) ( a (cid:96) ) D ( t (cid:96) ) = P k (cid:96) ( a (cid:96) + ( T (cid:96) − ) k (cid:96) +1 ,k (cid:96) ) D ( t (cid:96) ) T (cid:96) . Write x (cid:96) := ( T (cid:96) − ) k (cid:96) +1 ,k (cid:96) for short; we now have T → k ( s ) D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r )= D ( t ) P k ( a + x ) D ( t ) P k ( a + x ) · · · P k r ( a r + x r ) D ( t r ) T r . The key fact now, whose proof (and precise statement) we omit here, is that the matrices T (cid:96) have geometric meaning: for i > j , the ( i, j ) entry in T (cid:96) counts embedded disks whose leftmostend is a positive corner at a and whose rightmost end is a vertical line segment connectingstrands i and j just to the right of crossing a (cid:96) . (In particular, T r = .) Furthermore, themap Φ → from Section 4.2 is constructed exactly to satisfyΦ → ( a (cid:96) + x (cid:96) ) = a (cid:96) for all (cid:96) = 2 , . . . , r . As a consequence, we haveΦ → (cid:0) T → k ( s ) D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r ) (cid:1) = D ( t ) P k ( a ) D ( t ) P k ( a ) · · · P k r ( a r ) D ( t r ) . Similarly, Φ ← satisfiesΦ ← (cid:0) D ( t ) P k ( a ) D ( t ) · · · P k r ( a r ) D ( t r ) T ← k ( s ) (cid:1) = D ( t ) P k ( a ) D ( t ) P k ( a ) · · · P k r ( a r ) D ( t r ) . Combining this equation and the previous equation now yields (5.2), whence Φ comb is a chainmap. (cid:3) Proof of Infinitely Many Fillings
In this section we prove Theorem 1.1. First, we describe the scheme of proof that we willuse for all the Legendrian links Λ ∈ H . The cases Λ( (cid:101) D ) , Λ , Λ and Λ( β ) , Λ( β ) , Λ( β )are then proven directly using this strategy. The general cases Λ( (cid:101) D n ) , Λ n are concluded fromProposition 6.5 and the proofs we give for the two cases Λ( (cid:101) D ) , Λ .6.1. The argument.
Let Λ ⊂ ( R , ξ st ) be a Legendrian link Λ = Λ( β, i ; γ ), β ∈ Br + N , γ ∈ Br + M , and consider its ϑ -loop, as introduced in Section 2.4. The general structure of ourproofs can be described in three steps, as follows:(i) First, choose an ordered sequence of crossings for β and γ such that resolving thesecrossings yields an orientable exact Lagrangian filling L ⊂ ( R , λ st ) of the Legendrianlink Λ.(ii) Second, compute the augmentation ε L : A Λ → Z [ H ( L ) ⊕ Z m − ] associated to theexact Lagrangian filling L (where m is the number of components of Λ) and the in-duced maps ϑ k : A Λ → A Λ , k ∈ N . We note that all crossings chosen in (i) will havethe property that their complement contains a half-twist, and consequently they arecontractible and proper by Proposition 4.5. Thus we may apply the combinatorial ormulas from Section 4.2 in this step.(iii) Third, fix a crossing a for the braid word β associated to the Legendrian link Λ,which we consider as one of the generators a ∈ A Λ of the Legendrian contact DGA.Consider the invariant E ( k, a ) := max η : R → Z | ( η ◦ ε L ◦ ϑ k )( a ) | , k ∈ N , where R = Z [ H ( L ) ⊕ Z m − ] and η : R → Z runs over all possible unital ringmorphisms. Note that the set of such morphisms is finite, as the first Betti number b ( L ) is finite, and thus E ( k, a ) is a well-defined maximum over a finite set of integers.Finally, show that E ( k, a ) is a strictly increasing function of k ∈ N .The different choices for the Lagrangian filling (and thus the augmentation ε L ) and crossing a ∈ A Λ influence the computation of the invariant E ( k, a ). Finding the maximum over aset whose cardinality grows exponentially in l ( β ) + l ( γ ) makes brute force computation adifficult (though not unfeasible) route. Thus, particular care must be devoted in choosingthe augmentation ε L and the crossing a ∈ A Λ : we will find crossings a ∈ A Λ and Lagrangianfillings whose augmentations satisfy that ( ε L ◦ ϑ k )( a ) is a positive Laurent polynomial in Z [ H ( L )], for all k ∈ N , making the invariant E ( k, a ) readily computable. Remark 6.1.
Executing the argument laid out here for specific Legendrian links, includingall of the ones that we consider in this section, is readily amenable to calculation by computer.A
Mathematica notebook that performs the calculations contained in the remainder of thissection, and is suitable for calculations for general ( − (cid:101) D in detail, without recourseto the computer program, in Section 6.2 below; we provide fewer details for subsequentcomputations and refer the reader to the program. (cid:3) Augmentations for Λ( (cid:101) D ) . We now turn to proving Theorem 1.1 for the Legendrianlink Λ( (cid:101) D n ), n ≥
4. In this subsection we present the argument for n = 4; the general n ≥ (cid:101) D ) ⊂ ( R , ξ st ) ⊂ ( S , ξ st ) is defined tobe the rainbow closure of the positive braid ( σ σ σ σ ) , which is also the ( − σ σ σ σ ) σ σ = ( σ σ σ σ ) ∆ . A Lagrangian projection for Λ( (cid:101) D ) is depictedin Figure 26. Let us prove the following result: Theorem 6.2 (The (cid:101) D –Legendrian) . Let ϑ : S → L (Λ( (cid:101) D )) be the purple-box Legendrianloop. Then there exists a Lagrangian filling L ⊂ D of Λ( (cid:101) D ) such that the ϑ -orbit of thesystem of augmentations ε L is entire. In order to prove Theorem 6.2, we set some notation and lay out the pieces that go into theproof. Let us label the crossings of the positive braid ( σ σ σ σ ) σ σ from left to right as a , . . . , a , a , a , a , a . Figure 26 shows the Lagrangian projection of Λ( (cid:101) D ) that we use for the proof, where thelabeled crossings are also depicted. These crossings are the degree-0 Reeb chords of a Leg-endrian front for Λ( (cid:101) D ). The ϑ -monodromy is obtained by carrying around the purple boxcontaining the two crossings a , a and the two base points t , t , as shown in Figure 26,cf. Figure 2.The filling L of Λ( (cid:101) D ) that we will consider is the decomposable filling constructed as follows.Resolve the following crossings of Λ( (cid:101) D ) in order: a , a , a , a , a , a , a , a . igure 26. Lagrangian projection for the Legendrian link Λ( (cid:101) D ), as used inthe proof of Theorem 6.2. The crossings a , a , in blue, are used to detect theinfinite order of the ϑ -monodromy. In this case, the ϑ -monodromy is obtainedby moving the crossings a , a around this projection.Note that at each step the remaining braid is admissible in the sense of Definition 2.5: thisfollows from Proposition 2.7 and the fact that the crossings a , a , a , a , a , a comprisea half-twist. Thus each step produces a legal Lagrangian projection of a Legendrian link,and each resolved crossing is contractible. The result of resolving these 8 crossings is the( − and is precisely the standard 4-component Legendrian unlink. We then fill in each of the 4 component unknots. This givesthe desired filling L of Λ( (cid:101) D ), expressed as 8 saddle cobordisms and 4 minimum cobordisms.Following the discussion in Section 3.5, we use Z [ t ± , . . . , t ± , s ± , . . . , s ± ] as the coeffi-cient ring for the DGA A (Λ( (cid:101) D )). We will need two maps on A (Λ( (cid:101) D )), induced by the ϑ -monodromy and the filling L . The former map is an automorphism ϑ : A (Λ( (cid:101) D )) →A (Λ( (cid:101) D )). For the latter, as in Section 3.5, L induces an augmentation ε L : A (Λ( (cid:101) D )) → R .Here R = Z [ t ± , . . . , t ± , s ± , . . . , s ± ] / ( w = w = w = w = − w , w , w , w are the product of the labels of the base points on each unknot in Λ . Since by inspection t , t , t , t appear on all distinct components of Λ , the quotient allows us to solve for the t i ’s, and we conclude that R ∼ = Z [ s ± , . . . , s ± ].Our aim is to pairwise distinguish the iterates ε L ◦ ϑ k : A (Λ( (cid:101) D )) → R , k ∈ N , even upto automorphisms of R . We will do this by computing the image of a under each of thesemaps. In order to perform this computation, we need to partially compute the maps ϑ and ε L .We first consider the monodromy automorphism ϑ , which we compute using Proposition 5.3.First note that ϑ fixes the variables a , a , t , t that appear inside the purple box. We willbe interested in what ϑ does to the two Reeb chords a , a , which are depicted in blue inFigure 26. The path matrix associated to the purple box is given by M = (cid:18) a (cid:19) · (cid:18) a (cid:19) · (cid:18) t t (cid:19) = (cid:18) t t a t a t (1 + a a ) (cid:19) . By Proposition 5.3, the effect of the DGA automorphism ϑ ∈ Aut( A (Λ( (cid:101) D ))) on the twocrossings a , a , which are depicted in blue in Figure 26, is (cid:18) a a (cid:19) (cid:55)−→ ϑ (cid:18) a a (cid:19) = M (cid:18) a a (cid:19) . Next consider the augmentation ε L , which we can explicitly compute using the formulas fromSection 4. We will only need the following partial computation: emma 6.3. We have ε L ( a ) = s , ε L ( a ) = s , and ε L ( t ) = − s s ,ε L ( t ) = − s s s s s s ,ε L ( a ) = s s − s s s s s ,ε L ( a ) = − s s s s s s + s s s s s − s s s s s s + s s s s s s s . Proof.
For i = 9 , . . . ,
16, let Φ i = Φ comb L ai denote the combinatorial cobordism map associ-ated to the saddle cobordism at a i , as described in Section 4.2; also let ε : A Λ → R ∼ = Z [ s ± , . . . , s ± ] denote the augmentation associated to the disk filling of Λ . We have ε L = ε ◦ Φ ◦ · · · ◦ Φ . We begin by computing ε . Note that all Reeb chords of Λ either have degree 1 (for the4 crossings on the right) or connect different components of Λ (for the crossings a i for1 ≤ i ≤ ≤ i ≤ consists of four disjoint disks, it followsthat ε sends all Reeb chords to 0. As for the t i parameters, an inspection of Figure 26 yieldsthat the unknot components of Λ containing t and t contain the following base points inorder: − s − , − s − , t and − s − , s , − s − , − s − , s , − s − , t , respectively. Setting each ofthe products of these base points equal to − t = − s s and t = − s s s s s s , andthese are the respective images of t and t under ε (and thus under ε L as well).We now proceed to compute ε L for a , a , a , a . The sequence of saddle moves has beenchosen to simplify the computation of ε L ( a ) and ε L ( a ): indeed, ε L ( a ) = Φ ( a ) = s ,while Φ and Φ fix a and so ε L ( a ) = Φ ( a ) = s . Figure 27.
Disks with positive corners at a , a (left) and a , a (right),contributing to Φ ( a ) and Φ ( a ) respectively. For these disks, the pos-itive corner at a has positive orientation sign, while the positive corner at a and a has negative orientation sign.For a , we keep track of disks with two positive punctures, one at a and one at the crossingbeing resolved. There are no such disks when we resolve a and a . When we resolve a ,there is one disk ∆ ∈ ∆ → a ( a ) passing through − s − (against the orientation) with nonegative corners; see Figure 27. From Definition 4.6, we read off Φ → ( a ) = a + s s − andΦ ← ( a ) = a , and so Φ ( a ) = a + s s − . As we successively resolve a , . . . , a , theonly additional relevant disk with two positive punctures comes when we resolve a and isshown in Figure 27; this gives Φ ( a ) = a − s s − s s s − s − . We conclude that ε L ( a ) = ε (Φ (Φ ( a )) = ε (cid:18) a + s s − s s s s s (cid:19) = s s − s s s s s . he computation of ε L ( a ) is similar but slightly more involved. We compute thatΦ ( a ) = a − t a s − t − Φ ( a ) = a − t a s − t − Φ ( a ) = a + t s − a t − Φ ( a ) = a + t s − a t − ;piecing these together, along with Φ i ( a i ) = s i and the values computed above for ε ( t ) and ε ( t ), gives the desired expression for ε L ( a ). (cid:3) We are now in position to prove Theorem 6.2.
Proof of Theorem 6.2.
Consider the following matrices with entries in Z [ s ± , . . . , s ± ]: M := ε L ( M ) , v := ε L (cid:18) a a (cid:19) = (cid:18) s s (cid:19) , N := (cid:18) s s (cid:19) , M := N − M N. For k ∈ N , the augmentation ε L ◦ ϑ k sends the column vector ( a a ) to ε L ( M ) k · ε L (cid:18) a a (cid:19) = M k v = N ( N − M N ) k · (cid:18) (cid:19) = N M k (cid:18) (cid:19) . We can explicitly write down M using Lemma 6.3. This leads to the following observation: ifwe replace s , s , s , s by their negatives − s , − s , − s , − s , the matrix M becomes( M ) | { s j →− s j ,j =11 , , , } = (cid:18) m m m m (cid:19) , where the entries are m = s s s s s + s s s s + s s s s + s s s s + s s m = s s s + s m = s s s s + s s s m = s s . Note that all the coefficients are positive
Laurent polynomials in the variables s , . . . , s :this is the algebraic reason for our choice of augmentation ε L , and the change of signs for thevariables s , s , s , s .Let us now finally conclude that the iterates ε L ◦ ϑ k are pairwise distinct. We do this bystudying the quantity E ( k, a ) := max η : R → Z | ( η ◦ ε L ◦ ϑ k )( a ) | , where η : R → Z runs over all possible 2 unital ring morphisms. This is an integer-valuedinvariant of an augmentation ε L : A Λ → R even up to post-composition of an automorphismof R . That is, if E ( k, a ) (cid:54) = E ( l, a ) then there exists no automorphism ϕ ∈ Aut( R ) suchthat ϕ ( ε L ◦ ϑ k ) = ε L ◦ ϑ l , and thus the k -th and l -th ϑ -iterates of ε L are distinct. In orderto compute E ( k, a ), we note that | ( ε L ◦ ϑ k )( a ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) s (cid:1) M k (cid:18) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:0) (cid:1) M k (cid:18) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) is the absolute value of the upper-left entry of M k . A unital ring morphism η : R → Z is uniquely determined by specifying the values s , . . . , s ∈ {± } and since the entries m , m , m , m are positive Laurent polynomials, the value | ( ε L ◦ ϑ k )( a ) | is maximized hen s i = − i = 11 , , ,
16 and s i = 1 for i = 9 , , ,
14. It follows that E ( k, a )is equal to the upper-left entry of ( ) k , which is a strictly increasing function of k . Thisproves that E ( k, a ) (cid:54) = E ( l, a ) if k (cid:54) = l , as required. (cid:3) Three Variations on the Affine D -braid. Let us next consider the following threeLegendrian links from the Introduction:Λ( β ) = Λ(( σ σ σ σ ) σ , σ ) , Λ( β ) = Λ(( σ σ σ σ ) σ , σ ) , Λ( β ) = Λ(( σ σ σ σ ) σ , σ ) . These are obtained from the (cid:101) D -braid by removing the crossing a , for β , the crossing a ,for β or the two crossings a , a , for the braid β . See Figure 26 for the notation on thecrossings, we denote the crossings of these three braids by the same labels as in Figure 26.In these three cases, we can use the template given by the proof of Theorem 6.2, again bystudying the crossings a , a . We will omit the details and just give the choice of Lagrangianfilling L , its corresponding augmentation ε L as computed from the formulas in Section 4.2,and the augmented matrices M . These computations are also contained in the Mathematica notebook [Ng].-
The link Λ( β ). The Lagrangian filling L is obtained by resolving the crossings a , a , a , a , a , a , a in order. The augmentation ε L sends t → − s s , t → − s s s s s s , a → s , a → s , a → s s + s s s s s ,a → − s s s s s s − s s s s s − s s s s s s . The augmented matrix M = N − M N satisfies M | { s j →− s j ,j =11 , , , } = (cid:32) s s s s s + s s s s + s s s s s + s s s s s s s (cid:33) , whose entries are all positive Laurent polynomials.- The link Λ( β ). The Lagrangian filling L is obtained by resolving the crossings a , a , a , a , a , a , a in order. This augmentation ε L sends t → s s s s , t → − s s , a → s , a → s , a → s s s s s s + s s s s − s s s s s + s s . The augmented matrix M = N − M N satisfies M | { s j →− s j ,j =11 , , } = (cid:32) s s s s s + s s s s + s s s s s s s s s + s s s s s (cid:33) , whose entries are all positive Laurent polynomials.- The link Λ( β ). The Lagrangian filling L is obtained by resolving the crossings a , a , a , a , a , a in order. This augmentation ε L sends t → s s s s , t → − s s , a → s , a → s , a → s s s s s s + s s s s + s s . That is, the crossing a i for the (cid:101) D -braid is still denoted a i for the braids β ij , 1 ≤ i, j ≤
2, where β isprecisely the (cid:101) D -braid. he augmented matrix M = N − M N satisfies M | { s j →− s j ,j =11 , , } = (cid:32) s s s s s + s s s s s s s s s s s (cid:33) , whose entries are all positive Laurent polynomials.This completes the proof of Theorem 1.1 for the Legendrian links Λ( β ) , Λ( β ) , Λ( β ). Weemphasize that these links all have a stabilized component (or two, in the case of Λ( β )).In particular, these Legendrian links are not the rainbow closure of a positive braid, and ourFloer-theoretic argument is presently the only known argument that shows the existence ofinfinitely many Lagrangian fillings for these Legendrian links. Figure 28.
Lagrangian projection for the Legendrian link Λ , as used in theproof of Theorem 6.4. In this proof, the crossings a , a , in blue, are used todetect the infinite order of the ϑ -monodromy. The ϑ -monodromy is obtainedby moving the purple box around this projection. To construct Λ instead,add an additional crossing labeled a to the purple box, between a and thebase points t , t .6.4. Monodromy for the Braids Λ and Λ . We next prove Theorem 1.1 for the Legen-drian links Λ and Λ from the Introduction. Figure 28 depicts a Lagrangian projection ofΛ ; Λ comes from adding one additional crossing to the purple box. Theorem 6.4 (The Λ – and Λ –Legendrians) . Let Λ and Λ be the ( − -closures of the -braids ( σ σ σ σ ) σ and ( σ σ σ σ ) σ , respectively. Let ϑ : S → L (Λ ) be the purple-boxLegendrian loop. Then for n = 1 , , there exists a Lagrangian filling L ⊂ D of Λ n such thatthe ϑ -orbit of the system of augmentations ε L is entire.Proof. As in Section 6.3, this follows the proof of Theorem 6.2 and we will simply specifythe fillings and describe the corresponding augmentations and augmented matrices. Thecomputation of the augmentations can be found in [Ng].We begin with Λ , whose Lagrangian projection is shown in Figure 28. We choose the filling L given by resolving the crossings a , a , a , a , a , a , a n order. (As usual, this produces an unlink, and we then fill in each of the 3 unknotcomponents to complete the construction of L .) The augmentation ε L sends t → s s s s s s , t → − s s s s , a → s , a → s ,a → s s − s s + s s + s s s s s s − s s s s s s + 1 s s s s s s . Define M = ε L (cid:18) t t a t (cid:19) , N = ε L (cid:18) a a (cid:19) , and M = N − M N ; then M | { s j →− s j ,j =5 , , , } = (cid:18) m m m m (cid:19) , and the entries m = s s s s s + s s s s + s s s s + s s s s + 1 s s m = s s s s s m = s s s s + s s s s s + s s s s s + s s s m = s s s s are all positive Laurent polynomials.For Λ , we start with the diagram for Λ in Figure 28, and add one more crossing labeled a directly to the right of a . Choose the filling L of Λ given by resolving the crossings a , a , a , a , a , a , a , a in order. The augmentation ε L sends t → − s s s s , t → − s s s s s s s s , a → s , a → s , a → s s − s s + s s − s s s s s s s ,a → s s s s s s s − s s s s s s s s + s s s s s s s s − s s s s s s s s . Define M = ε L (cid:18) t a t a t (1 + a a ) t (cid:19) , N = ε L (cid:18) a a (cid:19) , and M = N − M N ; then M | { s j →− s j ,j =5 , , , } = (cid:18) m m m m (cid:19) , and the entries m = s s s s s s s + 2 s s s s s s + s s s s s s s + s s s s s + s s s s s s + s s s s s s ++ s s s s + s s s s + s s s s + 1 s s ,m = s s s s s s + s s s + s s s s s + s s s s s ,m = s s s s s s + 2 s s s s s s + s s s s s s s + s s s s + s s s s s + s s s s s + s s s ,m = s s s s s + s s s s s s + s s s s . are all positive Laurent polynomials. (cid:3) igure 29. Lagrangian projection for the Legendrian link Λ n , n ≥ Figure 30.
Lagrangian projection for the Legendrian link Λ( (cid:101) D n ), n ≥
4, asused in the proof of Theorem 6.2.6.5.
The general case: Λ( (cid:101) D n ) and Λ n . We now turn to the Legendrian links Λ( (cid:101) D n ), n ≥
5, and Λ n , n ≥
3, which are depicted in Figures 29 and 30. The action of the ϑ -loops onthe Legendrian contact DGA for Λ( (cid:101) D n ), respectively Λ n , can be studied directly thanks toour understanding of the ϑ -loops for the Legendrian braids Λ( (cid:101) D ), respectively Λ . The mainingredient that allows us to deduce the general cases from a particular case is the following: Proposition 6.5.
Let β ∈ Br + N be an admissible braid, let k , k ∈ N with ≤ k ≤ k , andlet L be an exact Lagrangian filling of Λ( β, σ k ) . Consider an exact Lagrangian cobordism Σ from Λ( β, σ k ) to Λ( β, σ k ) obtained by resolving any combination of ( k − k ) crossingsin the braid σ k which are not the initial nor the final crossings. Let Φ Σ : A (Λ( β, σ k )) → A (Λ( β, σ k )) denote the induced map between the Legendrian contact DGAs. Let a denote any Reeb chordof Λ( β, σ k ) not in σ k , as well as the corresponding Reeb chord of Λ( β, σ k ) . Then forall m ∈ N , we have ( ε L ◦ ϑ m )( a ) = ( ε L ◦ Φ Σ ◦ ϑ m )( a ) , where ϑ i denotes the ϑ -loop of Λ( β, σ k i ) , i = 1 , .Consequently, if the ϑ -orbit of the augmentation ε L of Λ( β, σ k ) is entire, then the ϑ -orbitof the augmentation ε L ◦ Σ = ε L ◦ Φ Σ of Λ( β, σ k ) is entire.Proof. We begin by noting that f Σ fixes any Reeb chord of Λ( β, σ k ) outside of the braid σ k . This is because f Σ consists of a composition of saddle cobordism maps that count diskswith two positive corners, and the only such disks with a positive corner at one of the resolvedcrossings must have its other positive corner at a crossing in the braid, by our assumptionthat we do not resolve the two extreme crossings of σ k . here are two types of crossings in Λ( β, σ k ) besides the crossings in σ k : the ones thatcome from crossings of Λ( β ) involving the satellited strand of β , and the ones that do not. If a is of the latter type, then ϑ and ϑ both fix a . Since f Σ ( a ) = a , we are done in this case.Now assume that a is of the former type, and note that crossings of this type come in pairscorresponding to the two strands of σ k . Recall that the action of the ϑ -monodromy on sucha pair of crossings is completely determined by the path matrix of the braid. If we write P and P for the path matrices for the braids σ k and σ k respectively, then it suffices to showthat P = f Σ ( P ) . Note that these path matrices incorporate all base points in the braid region; in particular,the braid σ k includes base points in its interior, coming from the resolved crossings of σ k . Figure 31.
Resolving a crossing in σ to produce σ .To prove P = f Σ ( P ), by functoriality we may assume that Σ consists of a single saddlecobordism. Furthermore, since f Σ fixes any crossing besides the two crossings adjacent tothe saddle, it suffices to check the equality when k = 3 and k = 2; see Figure 31. In thiscase, if a , a , a denote the crossings in σ k as shown in Figure 31, we have f Σ ( a ) = a − s − , f Σ ( a ) = s , f Σ ( a ) = a − s − , and we compute: f Σ ( P ) = f Σ (cid:18)(cid:18) a (cid:19) (cid:18) a (cid:19) (cid:18) a (cid:19)(cid:19) = (cid:18) a (cid:19) (cid:18) − s − s (cid:19) (cid:18) a (cid:19) = P , as desired.The final sentence of the proposition follows from the fact that Φ Σ is surjective; see the proofof Proposition 7.5 below. (cid:3) From Proposition 6.5, and the fact that Λ( (cid:101) D ) and Λ have fillings for which the ϑ -orbit ofthe associated augmentation is entire, it follows that the same is true for Λ( (cid:101) D n ), n ≥
5, andΛ n , n ≥
3. This completes the proof of Theorem 1.1.
Remark 6.6.
A consequence is that the Legendrian links Λ( (cid:101) D n ), n ≥
5, and Λ n , n ≥ (cid:101) D ) or Λ to these links, using Proposition 7.5 below. However, Theorem 1.1 isstronger: we have actually constructed an infinite family of fillings of each of these links thatare all provably distinct from each other. (cid:3) Proof of Corollaries and Concluding Remarks
In this section we discuss some of the applications stated in the Introduction. First, weshow that the smooth isotopy type of the Lagrangian fillings we construct is independent ofthe iteration of the ϑ -loop. Then, we precisely state the notion of aug-infinite Legendrians(which implies the existence of infinitely many fillings) and prove some of its propertiesunder exact Lagrangian cobordisms. We also conclude Proposition 7.7, providing a gamutof small smooth knots with a max-tb Legendrian representative that admits infinitely manyLagrangian fillings. Finally, we prove Corollaries 1.6 and 1.7 regarding closed Lagrangianssurfaces in certain Weinstein 4-manifolds. .1. Smooth isotopy class of Lagrangian fillings.
Let L ⊂ ( D , λ st ) be an exact La-grangian filling of Λ ⊂ ( S , ξ st ) and ϑ : S → L (Λ) a Legendrian loop. The isotopy cobor-dism gr ( ϑ ) ⊂ S × [0 ,
1] associated to the Legendrian loop ϑ is an exact Lagrangian self-concordance of Λ, which we can concatenate with L . This yields another exact Lagrangianfilling L ϑ = L gr ( ϑ ) of Λ. Theorem 1.1 shows that, for certain ϑ , L ϑ may not be Hamil-tonian isotopic to L . For the Legendrian ϑ -loops we use in this article, let us prove that L ϑ is always smoothly isotopic to L . The argument is the same as in (the updated version of)[CG20]; we reproduce it here for convenience: Proposition 7.1.
Let Λ ⊂ ( S , ξ st ) be a Legendrian link of the form Λ = Λ( β, i ; γ ) , where β ∈ Br + N , γ ∈ Br + M . Let L π ⊂ ( D , λ st ) be an exact Lagrangian filling obtained by a pinchingsequence π ∈ S | β | , and ϑ : S → L (Λ) a Legendrian ϑ -loop. Then, the exact Lagrangianfillings L and L ϑ are smoothly isotopic relative to their boundary Λ .Proof. From the perspective of a positive braid representative β ∈ Br + N of Λ = Λ( β ), aLegendrian ϑ -loop consists of two moves: Reidemeister III moves and conjugations. Let usdenote the ordered crossings of β by ( a j ), j ∈ [1 , | β | ], with a π ( k ) being the k -th crossing to beresolved. First, any pinching (resolution) sequence π ∈ S | β | yields a surface which is smoothlyisotopic to L . From the smooth perspective, resolving a crossing corresponds to an elementarysurface cobordism of index 1. Two consecutive such cobordisms can be performed in eitherorder. Since any two different pinching sequences differ by a composition of transpositions,the smooth isotopy class of L π is equal for any pinching sequence π ∈ S | β | . It thus suffices toconsider the case of the identity permutation π = e , and show that L e ∪ Λ gr ( ϑ ) is smoothlyisotopic to L e .Consider a Reidemeister III move for three (consecutive) crossings a i − , a i , a i +1 , which leadsto a i +1 , a i , a i − . For the Lagrangian filling L e , these three crossings are resolved left to right:starting at a i − , then a i and a i +1 , in this order. Starting at a i +1 , a i , a i − we can describe two smooth cobordisms, both local to this piece of the braid (constant relative to its endpoints):(i) Apply a Reidemeister III move down to a i − , a i , a i +1 and then resolve according to π = e . Namely, first a i − , then a i and finally a i +1 .(ii) Directly resolve the three crossings a i +1 , a i , a i − , using the transposition π = ( i, i +2).That is, we resolve a i +1 first, then a i and lastly a i − .Both tangle cobordisms start at the tangle a i +1 , a i , a i − and end up in the trivial 3-strandedtangle. Since the crossings a i − and a i +1 are interchanged in a Reidemeister III move ( a i − before being geometrically the same as a i +1 ), as are a i +1 , a i − , the two tangle cobordisms aresmoothly isotopic. Hence, concatenating with the graph of an isotopy given by a sequenceof Reidemeister III moves, from Λ to itself, does not affect the smooth isotopy type of aLagrangian filling L .The same occurs for conjugation of the given positive braid β = ( a , a , . . . , a | β |− , a | β | ).Indeed, there are two smooth concordances starting with ( a | β | , a , a , . . . , a | β |− ):(i) Apply the cyclic shift from ( a | β | , a , a , . . . , a | β |− ) down to ( a , a , . . . , a | β |− , a | β | ) andthen resolve the crossings starting at a | β | and then left to right, that is, continuingwith a , a and resolving through a | β |− .(ii) Directly resolve the crossings of ( a | β | , a , a , . . . , a | β |− ) according to π = e : startingat a | β | , then a , a through a | β |− .These two concordances yield smoothly isotopic surfaces. In conclusion, starting with theLagrangian filling L e ⊂ ( D , λ st ), the concatenation L ϑ = L ∪ Λ gr ( ϑ ) yields a Lagrangianfilling of the form L π , for a permutation π ∈ S | β | . Since L π and L e are smoothly isotopic,the required statement follows. (cid:3) .2. Aug-infinite Legendrian links and cobordisms.
Here we describe a method forstarting with one Legendrian link known to have infinitely many fillings and producing others.First we need to define a condition that implies having infinitely many fillings and is in turnimplied in our examples by the ϑ -orbit being entire.Suppose that Λ is a Legendrian link with a (connected, orientable, exact Lagrangian) filling L of Maslov number 0. As discussed in Section 3.3, L induces a (2 g + 2 m − ε L : A Λ → Z [ s ± , . . . , s ± g +2 m − ], where g is the genus of L and m is the numberof components of Λ. Furthermore, up to equivalence (automorphism of Z [ s ± , . . . , s ± g +2 m − ]),this system is well-defined, independent of choices, and invariant under Hamiltonian isotopyof L . Here, as in Section 3.1, we have assumed in defining the DGA ( A Λ , ∂ ) that there is onebase point on each component of Λ.There are 2 g +2 m − ring morphisms from Z [ s ± , . . . , s ± g +2 m − ] to Z , each sending each s i to ±
1. By composing ε L with these homomorphisms, we obtain 2 g +2 m − augmentationsfrom A Λ to Z . In this way, the filling L of Λ induces finitely many Z -valued augmentations( A Λ , ∂ ) → ( Z , L is not connected: the augmenta-tions induced by a disconnected filling of Λ necessarily annihilate any Reeb chord of Λ whoseendpoints lie on different components of the filling, and each component of the filling inducesfinitely many augmentations of the sublink of Λ given by the boundary of the component. Definition 7.2.
A Legendrian link Λ is aug-infinite if the collection of all Z -valued aug-mentations ( A Λ , ∂ ) → ( Z ,
0) induced by orientable exact Lagrangian fillings of Λ of Maslovnumber 0, ranging over all possible such fillings, is infinite.Note that the aug-infinite condition is independent of the choices made along the way, in-cluding spin structure, capping paths and operators, and placement of base points. Addingextra base points also does not affect the condition; cf. the proof of Proposition 7.5 below.The following is an immediate consequence of the fact that each filling induces finitely many Z -valued augmentations. Proposition 7.3. If Λ is aug-infinite then it has infinitely many fillings. In the conclusion of Proposition 7.5, infinitely many Lagrangian fillings refers to the fact thatthere are infinitely many Lagrangian fillings up to Hamiltonian isotopy. A prior, they mightnot be smoothly isotopic. Nevertheless, as proven in Proposition 7.1, this is the case for theLagrangian fillings we construct with ϑ -loops.Next we observe that our arguments from Section 6 actually prove that the LegendriansΛ( (cid:101) D n ), Λ n , and Λ( (cid:101) A , ) satisfy this strengthened version of having infinitely many fillings. Proposition 7.4.
The three classes of Legendrian links Λ( (cid:101) D n ) ( n ≥ , Λ n ( n ≥ , and Λ( β ) = Λ( (cid:101) A , ) , Λ( β ) , Λ( β ) are aug-infinite. Now we claim that for particular decomposable Lagrangian cobordisms, if the bottom of thecobordism is aug-infinite, then the top is as well. To be precise, we have the following.
Proposition 7.5.
Let Λ + and Λ − be Legendrian links with rotation number , and supposethat Λ − is an aug-infinite Legendrian link. Suppose that the following two properties hold: - The xy projection Π xy (Λ − ) is obtained from Π xy (Λ + ) by a sequence of saddle cobor-disms at proper contractible Reeb chords of Λ + of degree ; - All Reeb chords of Λ − (and thus of Λ + ) are in nonnegative degree.Then the Legendrian link Λ + is aug-infinite.Proof. It suffices to consider the case where Λ + and Λ − are related by a single saddle move ata Reeb chord a of Λ + . Suppose that Λ + has m components, and place a base point on each; hese base points trace down to Λ − . As in Section 3.5, we place a pair of base points on Λ − coming from the saddle at a . Then both A Λ + and A Λ − are DGAs over Z [ t ± , . . . , t ± m , s ± ],and the cobordism gives a map Φ : ( A Λ + , ∂ ) → ( A Λ − , ∂ ).Any filling of Λ − produces a system of augmentations for Λ − as in the discussion in Sec-tion 3.5; note that now one component of Λ − has more than one base point, but the construc-tion from Section 3.5 works just as well in this case. From Remark 3.1, adding each extrabase point has the effect on ( A Λ − , ∂ ) of replacing one generator t of the coefficient ring by twogenerators t (cid:48) , t (cid:48)(cid:48) and setting t = t (cid:48) t (cid:48)(cid:48) in the differential. It follows that there is a two-to-onecorrespondence between Z -valued augmentations of ( A Λ − , ∂ ) after and before the extra basepoint is added, and so adding extra base points does not affect the aug-infinite condition.We conclude that ( A Λ − , ∂ ) has infinitely many augmentations coming from fillings. Sincethere are finitely many choices for the images of t , . . . , t m , s under such an augmentation,there exist t , . . . , t m , s ∈ {± } such that ( A Λ − , ∂ ) has infinitely many augmentations fromfillings that send t i to t i and s to s .Write A Z Λ + and A Z Λ − for the DGAs over Z obtained by setting t i = t i and s = s . Thecobordism map Φ induces a map Φ Z : A Z Λ + → A Z Λ − satisfying Φ Z ( a ) = ± a i of Λ + , Φ Z ( a i ) = a i + f ( a i ) for some f ( a i ) ∈ A Λ − determined by theconstruction in Section 4.2. As observed in Section 4.2, this map respects the height filtration:for each i , f ( a i ) only involves Reeb chords of strictly smaller height than a i . We concludefrom this that Φ Z is surjective.Since A Z Λ − has infinitely many augmentations from fillings, there is some Reeb chord a i of Λ − that is sent to infinitely many values in Z under these augmentations. Now use thesurjectivity of Φ Z and suppose that x ∈ A Λ + satisfies Φ Z ( x ) = a i . Each augmentation of A Z Λ − from a filling of Λ − produces an augmentation of A Z Λ + from a filling of Λ + by compositionwith Φ Z , and x is sent to infinitely many values in Z under these augmentations. This showsthat A Z Λ + has infinitely many augmentations from fillings of Λ + , and consequently that Λ + is aug-infinite. (cid:3) Remark 7.6.
It is expected that Proposition 7.5 should hold whenever there is an exactLagrangian cobordism between Λ + and Λ − , without the restriction of being composed strictlyof saddle moves (and not isotopy cylinders) or even of being decomposable. One approachto proving the more general result is to show that exact cobordisms induce injective mapson the augmentation categories of Legendrian links (over Z ), in the spirit of previous workof Pan [Pan17a] for Legendrian knots and the upcoming paper [CSLL +
21] for links. (cid:3)
As the first application of Proposition 7.5, we have Corollary 1.4:
Proof of Corollary 1.4.
Observe that there is a decomposable Lagrangian cobordism to theLegendrian (4 ,
4) torus link Λ(4 , − σ σ σ ) =( σ σ σ σ ) σ σ , from the link Λ( (cid:101) D ) = Λ(( σ σ σ σ ) σ σ , consisting of two saddle cobor-disms at proper contractible degree 0 Reeb chords. Since Λ( (cid:101) D ) is aug-infinite, it follows thatΛ(4 ,
4) is aug-infinite as well. In addition, there is an another such cobordism from Λ(4 ,
4) tothe Legendrian ( n, m ) torus link Λ( n, m ) for any n, m ≥
4, and so by Proposition 7.5, Λ( n, m )is aug-infinite for any n, m ≥
4. Similarly, we can deduce that the Legendrian (3 ,
6) toruslink Λ(3 , − σ σ ) = ( σ σ σ ) σ , is aug-infinitebecause there is a cobordism to Λ(3 ,
6) from Λ ; it then also follows that the (3 , m )-toruslink Λ(3 , m ) is aug-infinite for all m ≥
6. The proof is complete. (cid:3)
We can also apply Proposition 7.5 to show that various other single-component Legendrianknots have infinitely many fillings. roposition 7.7. The Legendrian knots given by the ( − -closures of the following positivebraids have infinitely many fillings: (i) ( σ σ σ σ ) σ σ σ , which has smooth type m (10 ) , Thurston–Bennequin number and genus fillings, (ii) ( σ σ σ σ ) σ σ σ σ σ σ ∈ B which has smooth type , Thurston–Bennequinnumber and genus fillings, (iii) σ σ σ σ σ σ σ σ ∈ B , which has smooth type m (10 ) , Thurston–Bennequinnumber and genus fillings, (iv) σ σ σ σ σ σ σ σ ∈ B , of smooth type , Thurston–Bennequin number andgenus fillings, (v) σ σ σ σ σ σ σ σ σ σ ∈ B , of smooth type m (10 ) , Thurston–Bennequin number and genus fillings.Proof. The m (10 ) and 10 knots have a cobordism from the link Λ( β ), which is the( − σ σ σ σ ) σ σ . The other three knots have cobordisms from the link Λ ,which is the ( − σ σ σ σ σ σ σ σ . (cid:3) In light of Proposition 7.5, given two Legendrian links Λ + , Λ − with infinitely many fillings,we might consider Λ − to be “simpler” than Λ + if there is a saddle cobordism from Λ − to Λ + .Since such a cobordism increases Thurston–Bennequin number as we go from bottom to top,a rough measure of the simplicity of a Legendrian link with infinitely many fillings is givenby its Thurston–Bennequin number: the lower the tb , the simpler the link. (Alternatively,we could use 2 g + m where g is the genus of a connected filling and m is the number ofcomponents of the link, since tb = 2 g + m − m (10 ) ( tb = 3) isthe simplest knot that is known to us to have infinitely many fillings, while Λ( (cid:101) A , ) ( tb = 2)is the simplest known link. Remark 7.8.
We presently do not know of any Legendrian knots with infinitely many genus1 fillings, or of any Legendrian links with infinitely many planar (genus 0) fillings. From theperspective of cluster algebras, the existence of the former would be somewhat unexpectedif we restrict to the class of ( − (cid:3) Lagrangian surfaces in Weinstein 4-manifolds.
Here we prove Corollaries 1.6 and1.7. Let Λ ⊂ ( S , ξ st ) be a Legendrian link with m := | π (Λ) | components, and W (Λ) theWeinstein 4-manifold obtained by attaching m Weinstein handles to ( D , λ st ), one alongeach component of the Legendrian Λ ⊂ ( S , ξ st ) ∼ = ( ∂ D , ker( λ st | ∂ D )). Given an embeddedexact Lagrangian filling L ⊂ ( D , λ st ), we denote by L ⊂ W (Λ) the closed embedded exactLagrangian surface in W (Λ) given by the set-theoretic union L := L ∪ L cap , where L cap isthe (disjoint) union of the Lagrangian cores of the m Weinstein handles.The augmentations ε L : A Λ → Z [ H ( L )] of the Legendrian contact DGA used in this manu-script employ the system of coefficients Z [ H ( L )], geometrically keeping track of local systemsin a Lagrangian filling L ⊂ ( D , λ st ). In the transition from L to L ⊂ W (Λ), we must compare Z [ H ( L )] and Z [ H ( L )], which are not isomorphic unless Λ has a single component. Thismotivates the following definition. Definition 7.9.
Let L ⊂ ( D , λ st ) be a filling of a Legendrian link Λ ⊂ ( S , ξ st ), inducing thesystem of augmentations ε L : A Λ → Z [ H ( L )], where Λ is equipped with the null-cobordantspin structure. The restricted system of augmentations associated to L is the composition ε L : A Λ ε L −→ Z [ H ( L )] −→ Z [ H ( L )] , here the second map is induced by the quotient map H ( L ) → H ( L ). (cid:3) If we place a single base point t i on each component Λ i of Λ, then t i represents the homologyclass of Λ i in both H (Λ) and H ( L ), and the quotient map in Definition 7.9 sends each t i to1 since Λ i is null-homologous in L . For practical purposes, if L is a connected decomposablefilling of an m -component link Λ, we can compute the restricted system of augmentations ε L associated to L as follows.Let us write ( A Λ , ∂ ) for the DGA of Λ with the Lie group spin structure, which is a DGAover Z [ t ± , . . . , t ± m ]. Recall from Sections 3.5 and 3.6 the construction of the system ofaugmentations ε L : A Λ → R where R = ( Z [ t ± , . . . , t ± m , s ± , . . . , s ± (cid:96) ]) / ( w = · · · = w k = − R by the relations t = · · · = t m = − R := R/ ( t = · · · = t m = − H ( L ) to H ( L ), where the − sign comes from the fact that we are using the Lie group, rather than the null-cobordantspin structure, on L . From Remark 3.15, t · · · t m = ( − m in R , and this new quotientimposes m − R ∼ = Z [ H ( L ) ⊕ Z m − ] and R ∼ = Z [ H ( L ) ⊕ Z m − ];imposing the conditions t = · · · = t m = − L producesthe restricted system of augmentations ε L for the Lagrangian filling L , enhanced by linkautomorphisms in each case. Remark 7.10.
When Λ is a single-component Legendrian knot, there is no difference betweenthe system and the restricted system of augmentations for a filling L . This comes from theresult of Leverson [Lev16] that any augmentation in this case must necessarily send the unique t variable to −
1; geometrically, this correlates with the fact that Λ is already null-homologousin L before we pass to L . (cid:3) The purpose of restricted systems of augmentations for L is that they correspond to localsystems that extend to local systems for the closed exact Lagangian surface L . Let L , L ⊂ ( D , λ st ) be Lagrangian fillings of a Legendrian link Λ ⊂ ( S , ξ st ). If L , L are Hamiltonianisotopic, then their associated augmentations are DGA homotopic; see Theorem 3.6. Asnoted in Remark 3.7, for the Legendrian links studied in this paper, we can replace “DGAhomotopic” by a simpler notion. Following Definition 3.9, we define two restricted systemsof augmentations ε L : A Λ → Z [ H ( L )] , ε L : A Λ → Z [ H ( L )]to be equivalent if there exists an isomorphism ψ : Z [ H ( L )] → Z [ H ( L )] such that ε L = ψ ◦ ε L . Then for Legendrian links such that the entire DGA A Λ is concentrated in nonnegativedegree, as is the case for all of the examples in this paper, DGA homotopic (restricted)systems of augmentations are necessarily equivalent (restricted) systems of augmentations.Now, both Corollaries 1.6 and 1.7 will be proven by using the following Proposition 7.11,which is not essentially new and uses the recent articles [EL19, Ekh19, GPS19]. We thankT. Ekholm, S. Ganatra, and Y. Lekili for illuminating discussions regarding the proof ofProposition 7.11. Proposition 7.11.
Let Λ ⊂ ( S , ξ st ) be a Legendrian link and L , L ⊂ ( D , λ st ) two fillingsof Λ . Suppose that the two restricted systems of augmentations ε L : A Λ → Z [ H ( L )] , ε L : A Λ → Z [ H ( L )] are not DGA homotopic. Then the exact Lagrangian surfaces L , L ⊂ W (Λ) are not Hamil-tonian isotopic in the Weinstein 4-manifold W (Λ) .Proof. Let W ( W (Λ)) be the wrapped Fukaya category of the Weinstein 4-manifold W (Λ), C the union of the m co-cores of the Weinstein handles of W (Λ), and CW( C ) the endomorphism ing of C as an object in W ( W (Λ)). By [CDGG17, Theorem 1.1], C generates W ( W (Λ)), seealso [GPS19, Theorem 1.10], and thus we consider the category W ( W (Λ)) through its Yonedaembedding Hom( C, − ) := CW( C, − ). The Lagrangian surfaces L , L are exact and hencerepresent objects in W ( W (Λ)), equally denoted L , L . Under the Yoneda embedding, thesetwo objects become Hom( C, L i ) := CW( C, L i ), i = 1 ,
2. We will now argue that L , L ∈ Ob( W ( W (Λ))) are distinct objects, which proves that the exact Lagrangian surfaces L , L ⊂ W (Λ) are not Hamiltonian isotopic. It suffices to show that CW( C, L ) and CW( C, L ) aredistinct as CW( C )-modules.Let L ⊂ ( D , λ st ) be a filling of Λ and ε L : A Λ → Z [ H ( L )] its associated augmentation. Theholomorphic disks that define ε L are explained in detail in [EHK16], see also [EN18, Theorem6.8] and Sections 3 and 4 above. In short, a Reeb chord a ∈ A Λ is sent to the contributionsfrom rigid holomorphic disks u : ( D , ∂ D ) → ( D , λ st ) with a positive puncture at the Reebchord a , and each disk contribution is weighted by the homology class [ ∂u ] ∈ Z [ H ( L )], where ∂u ⊂ L is appropriately capped in L . The claim is that the holomorphic disks that definethe CW( C )-module structure of CW( C, L ), namely the composition A ∞ -map η L : CW( C ) ⊗ CW(
C, L ) → CW(
C, L ) , or equivalently CW( C ) → End(CW(
C, L )), are in bijection with those contributing to therestricted augmentation ε L . Indeed, we first observe that C ∩ L , which generates CW( C, L ),consists of precisely a point per each component of C . The disks contributing to η L have: apositive puncture at a generator of CW( C ), which is either a minimum of a Morse functionon C or a Reeb chord of its Legendrian boundary ∂C ⊂ ∂W (Λ); a positive puncture at agenerator of CW( C, L ); and a negative puncture at a generator of CW(
C, L ) (in fact, thetwo generators of CW(
C, L ) here must be the same). These disks are depicted in the rightdiagram of Figure 32. In our case, the contributions of these disks are weighted by theirboundary homology classes, where we only keep track of the piece of the boundary thatbelongs to the closed Lagrangian surface L . These contributions yield coefficients in theground ring Z [ H ( L )]. Figure 32.
On the left, the holomorphic disks contributing to the aug-mentation ε L (bottom) and the surgery isomorphism CW(C) ∼ = A Λ (top).On the right, the holomorphic disks contributing to the module structureCW( C ) → End(CW(
C, L )). The notation g (CW( C )) and g ( A Λ ) stands forgenerators of the algebras CW( C ) and A Λ : g (CW( C )) are Reeb chords of ∂C or the minimum of C , and g ( A Λ ) are Reeb chords of Λ.Now, the decomposition L = L ∪ Λ L cap of the Lagrangian surface L ⊂ W (Λ) into a Lagrangianfilling L and the cores L cap is compatible with neck-stretching along the contact hypersurface ∂ D , ξ st ) containing Λ, where the Weinstein handles are attached. That is, the Weinstein4-manifold decomposes as W (Λ) ∼ = ( D , λ st ) ∪ O p (Λ) (cid:32) l (cid:91) i =1 ( T ∗ D , λ st ) (cid:33) , and performing a neck-stretching procedure to the holomorphic disks contributing to η L breaks them into two pieces. See [Abb14, Chapter 3], [BEH +
03, Section 3], or [CDGG20,Section 5] for the neck-stretching technique along such a contact hypersurface, in this case astandard contact level set of the symplectization of ( S , ξ st ). For a sufficiently large stretching,see e.g. [EHK16, Corollary 3.10] or [BEH +
03, Section 11.3], there is a one-to-one correspon-dence between the rigid holomorphic disks contributing to η L and two-level broken disks.The first level consists of holomorphic disks in the moduli space M co ( c ), following the no-tation in [EL19], where c := c Λ z v cz w , c Λ is a product of Reeb chords in Λ, z v , z w areintersections in C ∩ L cap , and c is a generator of CW( C ). The boundaries of these holomor-phic disks start belonging in L cap , at the left of the leftmost positive puncture in c Λ , thencontinue to belong to L cap as the Reeb chords in c Λ are visited, and switch to belonging to C , when z v is reached; then, the boundary (away from the punctures) belongs to C as thechords in c are visited and we reach z w , where the boundary switches back to L cap . Thecurves in this first level are depicted in the top of the left diagram of 32, where z v , z w arethe two points marked by C ∩ L cap , and these moduli were studied in detail in [EL19] byusing the properties proved in [Ekh19]. In particular, [EL19, Theorem 2] shows that the A ∞ -map { Φ i } i ∈ N : CW( C ) ⊗ i → A Λ defined by counting rigid contributions of the modulispaces M co ( c ) (for i = 1; for i >
1, we have multiple positive punctures at generators ofCW( C )) is an A ∞ -quasi-isomorphism, see [EL19, Theorem 72] for details.The second level consists of holomorphic disks with a positive puncture at the Reeb chordsof Λ and boundary in L . These are the same rigid holomorphic disks as those contributing tothe augmentation map ε L : A Λ → Z [ H ( L )]. However, we note that the weights are countedwith coefficients in Z [ H ( L )]; that is, the count of holomorphic disks contributing to thesecond level of η L is precisely given by the restricted augmentation ε L : A Λ → Z [ H ( L )].In conclusion, the moduli space of disks contributing to the CW( C )-module structure η L splits into M co ( c ), which yields the A ∞ -quasi-isomorphism CW( C ) ∼ = A Λ , and the modulispace of holomorphic disks contributing to the restricted augmentation ε L , associated to theclosed Lagrangian L . Thus, under the surgery isomorphism, the CW( C )-module structure η L is precisely given by the augmentation ε L on A Λ .Now if L , L are Hamiltonian isotopic fillings of Λ, then L , L are (exact) Lagrangianisotopic in W (Λ), even relative to the co-cores. It follows that their associated restrictedaugmentations ε L and ε L are DGA homotopic, and the result follows. (cid:3) For the following two corollaries, we emphasize, and implicitly use, that the DGA A Λ as-sociated to the Legendrian braids Λ = Λ( β ) ⊂ ( S , ξ st ), β ∈ Br + N , are concentrated innonnegative degree, and thus DGA homotopic (restricted systems of) augmentations are thesame as equivalent (restricted systems of) augmentations. Proof of Corollary 1.6.
For g = 2, we consider the Legendrian knot Λ = Λ( β ) ⊂ ( S , ξ st )given by the positive braid β = ( σ σ σ σ ) σ σ σ . By Proposition 7.7, Λ( β ) admits infin-itely many genus 2 exact Lagrangian fillings { L i } i ∈ N , distinguished by their augmentations ε L i : A Λ → Z [ H ( L i )]. Consider the Weinstein 4-manifold W := W (Λ), which is homotopicto a 2-sphere S because Λ( β ) is a knot. For the same reason, all Lagrangian fillings of Λ arerestricted. Note that since Λ is a knot, the restricted augmentation ε L i is the same as ε L i for all i . By Proposition 7.11, it follows that the exact Lagrangian surfaces { L i } i ∈ N in W arenot Hamiltonian isotopic. This proves the assertion in the case of g = 2. or higher g ≥
2, it suffices to apply the same argument to the Legendrian knots associated tothe braids β g = ( σ σ σ σ ) σ σ σ σ g − . Since there exists an exact Lagrangian cobordismfrom Λ( β ) = Λ( β ) to Λ( β g ) for all g ≥
2, each knot Λ( β g ) admits infinitely many exactLagrangian fillings of genus g . Hence Proposition 7.11 implies that the Weinstein 4-manifold W g := W (Λ( β g )), homotopic to a 2-sphere S , also admits infinitely many exact Lagrangiansurfaces of genus g which are not Hamiltonian isotopic. In each case, W g does not admit anyembedded exact Lagrangian surface of genus h ≤ g − × (cid:0) tb (Λ( β g )) − (cid:1) . This concludes the proof. (cid:3) Proof of Corollary 1.7.
Consider the Legendrian link Λ = Λ( β ) and the Weinstein 4-manifold W (Λ( β )), which is homotopic to S ∨ S because Λ( β ) has two components. Theorem 1.1implies that this 2-component link admits infinitely many distinct exact Lagrangian fillings.In order to apply Proposition 7.11, we need to ensure that these infinitely many fillings aredistinguished by their restricted systems of augmentations. For that, let us study the aug-mentation ε L associated to the (initial) Lagrangian filling L in Subsection 6.3 and its ϑ -loopiterates. There are four homology variables t , t , t , t ; under ε L , these are augmented to t → s s s s , t → − s s , t → − s s , t → s s s s . Note that the ϑ -loop monodromy fixes each homology variable, and so the ϑ -loop iterates ε L ◦ ϑ k have the same effect as ε L on t , t , t , t for all k ∈ N .The first two variables t , t lie in one component of Λ and t , t lie in the other compo-nent. From the discussion following Definition 7.9, we can impose the additional conditions( t , t , t , t ) = (+1 , − , +1 , −
1) to obtain the restricted system of augmentations ε L (en-hanced by link automorphisms); this is because we first set t = t = 1 to reduce to a singlebase point on each component, and then set t = t = − H ( L ) to H ( L ). Interms of the s variables, there are 3 new conditions (the 4th is redundant): s s s = s , s s = 1 , s s = − . Now we note that ( s , s , s , s , s , s ) = (1 , , − , − , , −
1) in particular satisfy theseconditions. These values of the s i also produce the maximal value of | ( ε L ◦ ϑ k )( a ) | for all k ∈ N , from the computation in Section 6.3. It follows that the same argument that we usedthere, to show that the ϑ -orbit of the system of augmentations ε L is entire, also shows thatthe same is true of the restricted system of augmentations ε L . We can now apply Proposition7.11 to conclude that W (Λ( β )) admits infinitely many distinct exact Lagrangian tori, upto Hamiltonian isotopy. (cid:3) Appendix A. The Cobordism Map for an Elementary Saddle Cobordism
The goal of this section is to prove Proposition 4.8, the formula for the cobordism mapover Z for a saddle cobordism at a proper contractible Reeb chord. The proof will come inseveral steps. In Section A.1, we will first add what we call “mini-dips” on either side ofthe Reeb chord, which then propagate through the cobordism; this changes the cobordismby a Hamiltonian isotopy. The advantage of adding these mini-dips is that they localize thedisks that contribute to the cobordism map, so that the map mod 2 is quite simple andcan be written down very explicitly. The main technical result is lifting this map to Z andshowing that Proposition 4.8 holds for the cobordism with mini-dips; this is the contentof Proposition A.1 below. The proof of Proposition A.1 is somewhat indirect and involvesmaking the cobordism even more complicated, with the trade-off benefit being that thecobordism map becomes easier to handle. This is in the spirit of a well-known techniquein Legendrian knot theory called “dipping”, and occupies Sections A.2 and A.3. Finally, in ection A.4, we deduce Proposition 4.8 from its mini-dipped special case, Proposition A.1,by tracing the effect of mini-dips on the cobordism map.A.1. Formula for the saddle cobordism map.
Figure 33.
A saddle cobordism (left) and the mini-dipped version of thiscobordism (right).As in Section 4.2, we consider a saddle cobordism L a between Legendrian links Λ − and Λ + ,where Λ − is obtained from Λ + by replacing a contractible Reeb chord a of Λ + by the orientedresolution of the crossing. To simplify the cobordism map, we perturb Λ ± by a Legendrianisotopy (and consequently the saddle cobordism by a Hamiltonian isotopy) as follows: usetwo Reidemeister II moves to push the understrand of a over the overstrand on either sideof a , as shown in Figure 33. We call these moves “mini-dips” of a . Note that the crossing a is situated differently in Figure 33 than the similar-looking Figure 18 from [EHK16], andconsequently our mini-dip is different from the dip considered there. Also note that thecrossing data for the mini-dips (with the understrand of a passing over the overstrand inthe minidips) is forced by the condition that we want the resulting diagrams to representLagrangian projections of Legendrian links—apply Stokes’ Theorem to a bigon whose twocorners are the contractible chord a and an adjacent crossing in either of the mini-dips.For the next few subsections, we will assume that Λ + and Λ − contain the mini-dips shownin Figure 33; we will return to the general case without mini-dips in Section A.4. Over Z ,the mini-dips force the cobordism map Φ L a : A Λ + → A Λ − to have the following simple form:Φ L a ( a ) = s Φ L a ( a ) = a + s − Φ L a ( a ) = a + s − Φ L a ( a i ) = a i where the final equation holds for all Reeb chords a i of Λ + besides a, a , a . This followsdirectly from the work of [EHK16] (cf. Section 4.2) because there are only two disks with apositive puncture at a , the bigon between a and a and the bigon between a and a .Over the course of Sections A.2 and A.3, we will prove that the cobordism map Φ L a withsigns is given as follows. Figure 34.
A saddle cobordism with mini-dips. Crossings and base pointsare labeled, and quadrants with negative orientation sign are shaded. These are independently introduced in [GSW20a], where they are called “double dipping” and are usedfor the same purpose of simplifying cobordism maps. roposition A.1. Suppose that Λ + and Λ − are related by a saddle cobordism as in Figure 34:the Lagrangian projection of Λ + has a contractible crossing a flanked by mini-dips with thecrossings on either side of a labeled by a and a , and Λ − is the result of resolving the crossing a and placing base points labeled s and − s − on either side of the resolved crossing. Then,the cobordism map Φ L a : A Λ + → A Λ − over Z is given, up to a link automorphism of Λ − , by: Φ( a ) = s Φ( a ) = a − s − Φ( a ) = a − s − Φ( a i ) = a i where the final equation holds for all Reeb chords a i of Λ + besides a, a , a . More precisely,there is a link automorphism Ω : A Λ − → A Λ − such that the Φ L a : A Λ + → A Λ − is chainhomotopy equivalent to Ω ◦ Φ with Φ as defined above. Remark A.2.
We believe that the auxiliary data needed to define signs (capping operators,etc.) can be chosen so that the combinatorial formula for Φ in Proposition A.1 is preciselythe geometric map Φ L a , without composing with a link automorphism of Λ − . However, wewill not need the stronger statement for our purposes. (cid:3) A.2.
Splashes and diagonal automorphisms.
Our strategy for proving Proposition A.1is as follows: the signs for the formula for Φ L a given there are essentially forced, up to a linkautomorphism of Λ − , by the algebraic requirement that Φ L a needs to be a chain map over Z . This forcing is not true in full generality, but we will see that it is true if we isotop Λ ± viaReidemeister II moves so that their differentials consist of many terms, each of which is easyto handle. This sort of strategy is familiar in the subject through the technique of dipping;see, e.g., [FR11, Sab05, Siv11]. We will present a variant of this technique in this subsection,and then return to the proof of Proposition A.1 in Section A.3 below.Let Λ be a Legendrian link. By applying planar isotopy and Reidemeister II moves, we canisotop Λ so that its Lagrangian projection Π xy (Λ) satisfies the following properties: • all vertical tangencies (parallel to the y axis) lie on two lines x = c and x = c , andthere are at least 2 vertical tangencies on each of these lines; • no crossings in either the Lagrangian or front projections occur at the same x coor-dinate.Note in particular that Π xy (Λ) is the plat closure of some braid between x = c and x = c where the braid strands go from left to right. Furthermore, in the front projection Π xz (Λ),all left cusps lie on the line x = c and all right cusps lie on x = c .Subdivide the interval [ c , c ] by choosing x < x < · · · < x p with x = c , x p = c suchthat: • there are no crossings in either Π xy (Λ) or Π xz (Λ) in the intervals x ∈ [ x , x ] and[ x p − , x p ]; • for i = 1 , . . . , p −
2, in the interval x ∈ [ x i , x i +1 ] there is exactly one crossing in eitherΠ xy (Λ) or Π xz (Λ), and no crossing in the other.Now in a neighborhood of the x = x i slices for i = 1 , . . . , p −
1, introduce a collection of“splashes” as shown in Figure 35. This is a C -small perturbation in the front projection, This terminology is inspired by [FR11], though our splashes are slightly different from theirs and moreresemble what [EHK16] call “dips”. igure 35. A set of splashes, in the front projection (left) and correspondingLagrangian projection (right).while in the Lagrangian projection, each strand is pushed through the other strands. Fordefiniteness, we order the collection of splashes at x = x i from left to right in increasingorder of the y -coordinate of the splashed strand; in the Lagrangian projection, the crossinginformation for the new crossings is determined by the relative z coordinates of the strands at x = x i . Let Λ (cid:48) denote the resulting Legendrian link, and note that Π xy (Λ (cid:48) ) is obtained fromΠ xy (Λ) by a (large) number of Reidemeister II moves. See Figure 36 for a sample illustrationof Λ (cid:48) . Figure 36.
A full example of splashing. Top to bottom: the front andLagrangian projections of Λ, with strands joining x = c and x = c labeled1 , , ,
4, and the Lagrangian projection of Λ (cid:48) with some crossings labeled.Write the Chekanov–Eliashberg DGA of Λ (cid:48) as ( A Λ (cid:48) , ∂ ). Say that an automorphism Ψ of thealgebra A Λ (cid:48) is diagonal if it is of the following form: if a i denote the Reeb chords of Λ (cid:48) , thenthere is a collection of (invertible) scalars λ i such that Ψ( a i ) = λ i a i for all i . roposition A.3. Suppose that Ψ is a diagonal automorphism of A Λ (cid:48) that is also a chainmap: Ψ ◦ ∂ = ∂ ◦ Ψ . Then Ψ is a link automorphism of Λ (cid:48) . In order to prove Proposition A.3, we need some more notation. Let s denote the number ofvertical tangencies in the Lagrangian projection of Λ at each of x = c and x = c , so thatthe Lagrangian projection is the plat closure of a 2 s -stranded braid. Number these strands1 , , . . . , s so that in [ x , x ], the strands are numbered in increasing order of y coordinate;keep the numbering of braid strands consistent throughout the braid, and that in general thestrands will not remain numbered in increasing order beyond x = x .Now suppose that Ψ satisfies the hypotheses of Proposition A.3. We will construct units u , . . . , u s such that the following condition holds for all Reeb chords a of Λ (cid:48) :( ∗ ) Ψ( a ) = u r ( a ) u − c ( a ) a. Here we use r ( a ) and c ( a ) to denote the labels of the strands that are the endpoint andbeginning point of a , respectively.The following lemma is a useful tool for propagating condition ( ∗ ). Say that an embeddedbigon with boundary on Π xy (Λ (cid:48) ) and two convex corners at Reeb chords of Λ (cid:48) is a standardbigon if one corner is + and one is − ; similarly say that an embedded triangle is a standardtriangle if one corner is + and the other two are − . Lemma A.4. If a , a are Reeb chords of Λ (cid:48) such that there is a (unique) standard bigonwith corners at a , a , then ( ∗ ) holds for a if and only if it holds for a . If a , a , a areReeb chords such that there is a (unique) standard triangle with corners at a , a , a , then if ( ∗ ) holds for two of a , a , a , then it holds for the third as well.Proof. A bigon with + corner at a and − corner at a contributes a term a to ∂ ( a ); since r ( a ) = r ( a ) and c ( a ) = c ( a ) and Ψ ∂ = ∂ Ψ, it follows that if ( ∗ ) holds for one of a , a ,then it holds for the other. Similarly, a triangle with + corner at a and − corners at a , a contributes a term a a to ∂ ( a ); now use the fact that r ( a ) = r ( a ), c ( a ) = r ( a ), and c ( a ) = c ( a ) to conclude the desired result. See Figure 37. (cid:3) Figure 37.
Left two diagrams: a standard bigon and a standard triangle.Right diagram: a chain of bigons joining a q, ij , a q, ij , a q, ji , a q, ji . (If strand i isinstead above strand j , then these are still standard bigons but all + and − labels are interchanged.)We next label the crossings of Λ (cid:48) in the splashes as follows. Consider the splashed portionof strand i at x = x q . For any j (cid:54) = i , this splash crosses strand j twice; label these twocrossings a q, ij (left) and a q, ij (right). In this way we label all splashed crossings in Λ (cid:48) as a q,lij for 1 ≤ i, j ≤ s ( i (cid:54) = j ), 1 ≤ q ≤ p −
1, and 1 ≤ l ≤
2. See Figure 36 for an example.
Lemma A.5.
For fixed i, j, q , if one of the crossings a q, ij , a q, ij , a q, ji , a q, ji satisfies ( ∗ ) , then sodo the other three.Proof. In a neighborhood of x = x q , strand i lies either completely above or completely belowstrand j in the z coordinate. It follows that there is a chain of three standard bigons linking a q, ij , a q, ij , a q, ji , a q, ji ; see Figure 37. The result follows from Lemma A.4. (cid:3) emma A.6. If Ψ satisfies the hypotheses of Proposition A.3, then there are u , . . . , u s suchthat ( ∗ ) holds for all Reeb chords of Λ (cid:48) .Proof. We will prove that ( ∗ ) holds for all a = a q,lij by induction on q . In the course of theproof, we will also show that ( ∗ ) holds for all other Reeb chords of Λ (cid:48) , which correspondprecisely to the Reeb chords of Λ.We first establish the induction base case q = 1. Set u = 1. Then for j = 2 , . . . , s , theReeb chords a , j have one endpoint on strand 1 and one endpoint on strand j ; since eachΨ( a j ) , is an invertible scalar multiple of a , j , it follows that there are unique choices of u , . . . , u s so that ( ∗ ) holds for a = a , j for all j = 2 , . . . , s . Thus by Lemma A.5, ( ∗ ) alsoholds for a ,l j and a ,lj , j = 2 , . . . , s , l = 1 ,
2. Next suppose j > i ≥
2. Consider the twotriangles shown in Figure 38. Of the two corners at a , ij , one must be + and one must be − ,and similarly for the two corners at a , i . Of the corner at a , j and the corner at a , j , againone must be + and one must be − since the union of the two triangles is a standard bigon.Since no triangle can have three − corners by Stokes’ Theorem, it follows that one of the twotriangles in Figure 38 must be standard. Thus by Lemma A.4, ( ∗ ) holds for a , ij , whence itholds for a ,lij and a ,lji by Lemma A.5. This completes the base case q = 1. Figure 38.
Showing that a , ij satisfies ( ∗ ).Now suppose that ( ∗ ) holds for a = a q,lij for fixed q and all i, j, l ; we need to show that italso holds for a = a q +1 ,lij for all i, j, l . There are two cases depending on whether the crossingbetween x = x q and x = x q +1 is in Π xy (Λ) or in Π xz (Λ). First suppose that the crossing is inΠ xy (Λ), and let k , k denote the labels of the strands involved in the crossing. Choose anytwo indices i (cid:54) = j and assume without loss of generality that a q, ji is to the right of a q, ij . Aslong as { i, j } (cid:54) = { k , k } , there is a standard bigon joining a q, ji to a q +1 , ij , and it follows fromLemmas A.4 and A.5 and the induction hypothesis that a q +1 , ij , a q +1 , ij , a q +1 , ji , a q +1 , ji satisfy( ∗ ). If on the other hand { i, j } = { k , k } , then if we label the crossing between x = x q and x = x q +1 by a , there are standard bigons joining a q, ji to a and a to a q +1 , ij , and it follows asbefore that a q +1 , ij , a q +1 , ij , a q +1 , ji , a q +1 , ji , along with a itself, all satisfy ( ∗ ).It remains to treat the case where the crossing between x = x q and x = x q +1 is in Π xz (Λ).Say that this crossing is between strands k and k , where we choose the labels so that strand k has larger y coordinate than strand k between x = x q and x = x q +1 . The only differencebetween the splashes at x = x q and x = x q +1 is that strand k lies above k at x q while k lies above k at x q +1 , or vice versa. It follows that for any two indices i (cid:54) = j , as long as { i, j } (cid:54) = { k , k } , there is a standard bigon joining a q, ji to a q +1 , ij (or a q, ij to a q +1 , ji ) as in theprevious case, and we conclude as before that a q +1 , ij , a q +1 , ij , a q +1 , ji , a q +1 , ji satisfy ( ∗ ). inally suppose { i, j } = { k , k } . We will show that a q +1 , k k satisfies ( ∗ ), whence by Lemma A.5all four crossings of the form a q +1 ,lij for { i, j } = { k , k } and l = 1 , ∗ ), and theinduction step will be complete. Since Λ has at least 4 strands joining left and right, thereis some other strand labeled k with k (cid:54) = k , k . There are three cases depending on theposition of the y coordinate of strand k relative to strands k and k in [ x q , x q +1 ].If k lies above both k and k in the y direction, then consider the two triangles shownin Figure 39. For both of these triangles, one corner is at a q +1 , k k and the other two cornerssatisfy ( ∗ ). Since these triangles split in two a standard bigon with corners at a q, k k and a q +1 , k k , as in the q = 1 case one of the triangles must be standard. It follows from Lemma A.4that ( ∗ ) holds for a q +1 , k k , as desired. If k lies between k and k , or k lies below both k and k , entirely similar arguments using the triangles shown in Figure 40 again show that a q +1 , k k satisfies ( ∗ ), and we are done. (cid:3) Figure 39.
Showing that a q +1 , k k satisfies ( ∗ ). Figure 40.
Two more cases to show that a q +1 , k k satisfies ( ∗ ).We can now finally prove Proposition A.3. Proof of Proposition A.3.
Suppose Ψ : A Λ (cid:48) → A Λ (cid:48) is a chain map and an isomorphism. ByLemma A.6, we have u , . . . , u s so that ( ∗ ) holds for all Reeb chords of Λ (cid:48) . Now the strands1 , . . . , s are joined in pairs at the left end of Λ (cid:48) , and joined in pairs again at the right end. Onthe left end, for k = 1 , . . . , s , strands 2 k − k are connected, and this yields an embeddeddisk with a single corner at a , k − , k , which must be a + corner by Stokes. This contributes aconstant (1) term to δ ( a , k − , k ). Since Ψ is a chain map and Ψ( a , k − , k ) = u k u − k − a , k − , k by ( ∗ ), it follows that u k − = u k .More generally, the same argument shows that if strands i and j are joined at either end ofΛ (cid:48) , then u i = u j . It follows that u i = u j whenever i and j are part of the same connected omponent of Λ (cid:48) . Thus we may remove duplicates and rename u , . . . , u s as u , . . . , u m ,where m is the number of components of Λ (cid:48) . Then ( ∗ ) becomes precisely the condition forΨ to be a link automorphism of Λ (cid:48) , and we are done. (cid:3) A.3.
Proof of Proposition A.1.
With the auxiliary result Proposition A.3 in hand, wenext prove Proposition A.1. Suppose that Λ + and Λ − are related by a saddle cobordism ata contractible crossing flanked by mini-dips, as in the statement of Proposition A.3 or theright hand side of Figure 33. We first show that the desired map Φ is indeed a chain map,and then proceed to the main proof. Lemma A.7.
The map
Φ : A Λ + → A Λ − defined in Proposition A.1 is a chain map: Φ ◦ ∂ + = ∂ − ◦ Φ .Proof. We show that Φ ◦ ∂ + and ∂ − ◦ Φ agree on all Reeb chords of Λ + . Note that ∂ + ( a ) = 0, soΦ( ∂ + ( a )) = 0 = ∂ − ( s ) = ∂ − (Φ( a )). Also if we denote the mini-dip crossing next to a by a ,then ∂ + ( a ) = ∂ − ( a ) = − a , so Φ( ∂ + ( a )) = − Φ( a ) = − a = ∂ − ( a − s − ) = ∂ − (Φ( a ));similarly Φ( ∂ + ( a )) = ∂ − (Φ( a )). Figure 41.
Labeling the strands of Λ + (left) and Λ − (right) in the cobordismregion, and disks that pass through the cobordism region and contribute to ∂ + ( a i ) and ∂ − ( a i ).Now suppose that a i is a Reeb chord of Λ + besides a, a , a : we need to show that Φ( ∂ + ( a i )) = ∂ − ( a i ). The disks that make up the differentials ∂ + ( a i ) and ∂ − ( a i ) are exactly the sameexcept where they pass through the cobordism region encompassing a , a, a . Where thesedisks pass through the cobordism region, there is also a precise correspondence between thedisks for ∂ + ( a i ) and ∂ − ( a i ). The oriented boundary of such a disk enters the region on one ofthe strands on the left and exits on one of the strands on the right, or it enters on the rightand exits on the left. If we label the strands as shown in Figure 41, then for instance any diskcontributing to ∂ − ( a i ) that enters on the left on strand 1 and exits on the right on strand 2must pass s and turn a corner at a ; there are two corresponding disks contributing to ∂ + ( a i )with the same enter and exit data, one of which turns no corners in the cobordism regionand one of which turns corners at a and a . See Figure 41; the result replaces a monomial sa in ∂ − ( a i ) by 1 + aa in ∂ + ( a i ). In all, there are 8 ways to pass through the cobordismregion, with resulting contributions to ∂ ± ( a i ) as follows: ∂ + ( a i ) ∂ − ( a i ) ∂ + ( a i ) ∂ − ( a i )1 → a s ← − a − a − a aa s − − a sa → aa sa ← aa sa → a a a s ← a a a s → a + a + a aa − s − + a sa ← − a − s ow an inspection of this table shows that each entry in the ∂ − ( a i ) column is obtained fromthe corresponding entry in the ∂ + ( a i ) column by replacing a, a , a by s, a − s − , a − s − respectively. It immediately follows that Φ( ∂ + ( a i )) = ∂ − ( a i ). (cid:3) We now have a chain map Φ : A Λ + → A Λ − . In order to prove Proposition A.1, we wantto show that this is equal to the geometric cobordism map Φ L a up to a link automorphism.To do this, we will first localize the differentials of Λ ± by introducing splashes in the spiritof Section A.2. In what follows, we continue to refer to the small region of Λ ± containing a and a (and a for Λ + ) as the “cobordism region”, outside of which Λ + and Λ − coincide.We now change Λ − by a sequence of Reidemeister II moves that avoid the cobordism region,first pulling all vertical tangencies of Π xy (Λ − ) left or right so that they line up vertically,then adding splashes to separate any crossings in Π xy (Λ − ) or Π xz (Λ − ) outside the cobordismregion. From this we obtain a link Λ (cid:48)− , Legendrian isotopic to Λ − , for which there are x < x < · · · < x p such that: • all vertical tangencies lie on x = x or x = x p , and the number of vertical tangencieson each of these lines is at least 2; • there is a collection of splashes in a neighborhood of x = x i for i = 1 , . . . , p − • there is one i ∈ { , . . . , p − } such that [ x i , x i +1 ] contains the cobordism region, andin that interval [ x i , x i +1 ] the only crossings in either Π xy (Λ − ) or Π xz (Λ − ) are betweenthe two strands involved in the cobordism region; • for every other i = 1 , . . . , p −
2, in the interval [ x i , x i +1 ] there is exactly one crossingin either Π xy (Λ − ) or Π xz (Λ − ), and no crossing in the other; • [ x , x ] and [ x p − , x p ] contain no crossings in Π xy (Λ − ) or Π xz (Λ − ).In short, we follow the prescription from Section A.2, except that we do not separate thecrossings in the cobordism region from each other.If we follow the same sequence of Reidemeister II moves going from Λ − to Λ (cid:48)− , but start withΛ + , then we obtain a Legendrian link Λ (cid:48) + that differs from Λ (cid:48)− only in the cobordism region.We summarize the picture as follows: Λ + (cid:47) (cid:47) (cid:15) (cid:15) Λ (cid:48) + (cid:15) (cid:15) Λ − (cid:47) (cid:47) Λ (cid:48)− , where the horizontal arrows are Legendrian isotopies given by (the same) Reidemeister IImoves, and the vertical arrows are (identical) elementary saddle cobordisms. Note that thesaddle cobordism between Λ + and Λ − is Hamiltonian isotopic to the concatenation of thethree cobordisms specified by the other three sides of the square: from top to bottom, theisotopy from Λ + to Λ (cid:48) + , followed by the saddle cobordism between Λ (cid:48) + and Λ (cid:48)− , followedby the isotopy from Λ (cid:48)− to Λ − . By [EHK16, Kar20], the cobordism map Φ L a : A Λ + →A Λ − is chain homotopy equivalent to the composition of the cobordism maps given by thethree cobordisms. We will show that this composition is the map Φ from the statement ofProposition A.1.We first consider the cobordism map Φ (cid:48) : A Λ (cid:48) + → A Λ (cid:48)− . By Lemma A.7, we know of anotherchain map Φ : A Λ (cid:48) + → A Λ (cid:48)− : this is defined by Φ ( a ) = s , Φ ( a ) = a − s − , Φ ( a ) = a − s − , and Φ ( a i ) = a i for all other Reeb chords a i . Since [EHK16] gives a formula forgeometric cobordism maps mod 2 and this formula is especially simple in our case, we knowthat the geometric map Φ (cid:48) agrees with Φ up to signs. By replacing s by − s if necessary, wecan assume that Φ (cid:48) ( a ) = s . Lemma A.8.
There is a link automorphism
Ω : A Λ (cid:48)− → A Λ (cid:48)− such that Φ (cid:48) = Ω ◦ Φ . roof. Write ∂ (cid:48) + and ∂ (cid:48)− for the differentials on A Λ (cid:48) + and A Λ (cid:48)− respectively.Since the terms in Φ (cid:48) agree with the terms in Φ up to sign, there are signs σ i ∈ {± } such that Φ (cid:48) ( a ) = σ a ± s − , Φ (cid:48) ( a ) = σ a ± s − , and Φ (cid:48) ( a i ) = σ i a i for all other i . Infact, because Φ (cid:48) is a chain map, we must more specifically have Φ (cid:48) ( a ) = σ a − s − andΦ (cid:48) ( a ) = σ a − s − . To see this for a (with a similar argument for a ), we use the factthat Λ (cid:48)± have more than 2 strands joining left and right in the x direction, as stipulatedin their construction. In particular, there is a strand of Λ (cid:48)± that lies either above or belowthe cobordism region in the xy projection. Assume this strand lies above (the argumentfor below is very similar). The splashes from this strand on either side of the cobordismregion intersect the strands from the cobordism region in a number of crossings, two of whichare labeled a and a in Figure 42. In Π xy (Λ (cid:48) + ), there is a standard bigon with corners at a and a , contributing either a to ∂ (cid:48) + ( a ) or a to ∂ (cid:48) + ( a ). For definiteness assume theformer (the argument is same for the latter). An inspection of Figure 42 shows that ∂ (cid:48) + ( a )contains the terms ± (1 + a a ) a while ∂ (cid:48)− ( a ) contains ± a sa , and furthermore that theseare the only terms in ∂ (cid:48)± ( a ) that involve a . Since ∂ − Φ (cid:48) ( a ) = Φ (cid:48) ∂ + ( a ), we must have ± a sa = Φ (cid:48) ((1 + a a ) a ) = ± (1 + ( σ a ± s − ) s ) a , which implies that the ± sign is − asclaimed. Figure 42.
Splashes on either side of the cobordism region in Λ (cid:48) + (left) andΛ (cid:48)− (right), with relevant crossings labeled.Now let Ω to be the algebra automorphism of A Λ (cid:48)− defined by Ω( a i ) = σ i a i for all i ; then byour expression for Φ (cid:48) , we have Φ (cid:48) = Ω ◦ Φ . It follows from the fact that Φ (cid:48) and Φ are bothchain maps that Ω is also a chain map. Indeed, for any i we have ∂ (cid:48)− Ω a i = ∂ (cid:48)− Φ (cid:48) a i = Φ (cid:48) ∂ (cid:48) + a i = ΩΦ ∂ (cid:48) + a i = Ω ∂ (cid:48)− Φ a i = Ω ∂ (cid:48)− a i , where when i = 1 , ∂ (cid:48)− ( s − ) = 0.It remains to show that Ω is a link automorphism of Λ (cid:48)− . To do this, we use the fact that Ω is adiagonal automorphism of A Λ (cid:48)− and a chain map, and appeal to a variant of Proposition A.3.We cannot use Proposition A.3 directly because Λ (cid:48)− does not have a splash between a and a .However, we can still follow the inductive proof of Proposition A.3 in this setting. The onlything we need to check is the inductive step where we are given that a satisfies the condition( ∗ ) from the proof and need to conclude that a also satisfies this condition. To do this, let a and a be the crossings depicted in Figure 42, and note that there is a standard bigon inΠ xy (Λ (cid:48)− ) with corners at a and a . If the positive corner of this bigon is at a , then ∂ (cid:48)− ( a )contains the terms ( a sa − s − ) a , while if the positive corner is at a , then ∂ (cid:48)− ( a ) containsthe terms a ( a sa − s − ). In either case, since Ω is a chain map, Ω( a ) s Ω( a ) − s − must beequal to ± ( a sa − s − ). Since a satisfies( ∗ ), Ω( a ) = u r ( a ) u − c ( a ) a ; but this implies thatΩ( a ) = ( u r ( a ) u − c ( a ) ) − a = u r ( a ) u − c ( a ) a and so a satisfies ( ∗ ), as desired. This completesthe proof of Lemma A.8. (cid:3) e next examine the maps given by the Legendrian isotopies between Λ + and Λ (cid:48) + , andbetween Λ − and Λ (cid:48)− . Suppose that Λ (cid:48)− is obtained from Λ − by N Reidemeister II moves.Then we can follow [Che02, ENS02] to construct a DGA isomorphism Ψ − between A Λ (cid:48)− and the DGA S N ( A Λ − ) given by stabilizing A Λ − N times (adding 2 N generators in theprocess). This isomorphism comes from N applications of the isomorphism coming from asingle Reidemeister II move, as already described in Section 4.1. By that construction, ifwe start with Λ − and add the Reidemeister II moves one by one, we see that the nontrivialparts of Ψ − come from disks with two positive punctures, one of which is at a crossing inthe Reidemeister II move. By inspection, there is no point at which there is such a diskwhere the other positive puncture is at either a or a , and it follows that Ψ − ( a ) = a andΨ − ( a ) = a .Similarly, since Λ (cid:48) + is obtained from Λ + by the same Reidemeister II moves, we have a DGAisomorphism Ψ + between A Λ (cid:48) + and S N ( A Λ + ), and Ψ + ( a ) = a , Ψ + ( a ) = a , Ψ + ( a ) = a .Indeed, we can say more about the relation between Ψ + and Ψ − . The key point is thatthere is a precise correspondence between the twice-positive-punctured disks that determineΨ + and the twice-positive-punctured disks that determine Ψ − : algebraically, one obtains thelatter from the former by replacing a, a , a by s, a − s − , a − s − just as in the proof ofLemma A.7. Consequently, for any Reeb chord a i of Λ (cid:48) + (and thus of Λ (cid:48)− ) besides a, a , a ,Ψ − ( a i ) is obtained from Ψ + ( a i ) by this algebraic replacement.Put another way, let Φ be as above, and similarly define Φ : S N ( A Λ + ) → S N ( A Λ − ) byΦ ( a ) = s , Φ ( a ) = a − s − , Φ ( a ) = a − s − , and Φ is the identity on all othergenerators of S N ( A Λ + ). Note that by Lemma A.7, Φ and Φ are both chain maps. By theabove discussion, we conclude that the following diagram commutes: S N ( A Λ + ) Φ (cid:15) (cid:15) A Λ (cid:48) + Ψ + ∼ = (cid:111) (cid:111) Φ (cid:15) (cid:15) S N ( A Λ − ) A Λ (cid:48)− . Ψ − ∼ = (cid:111) (cid:111) The cobordism map A Λ + → A Λ (cid:48) + is simply the composition of the inclusion map i : A Λ + → S N ( A Λ + ) and the inverse of Ψ + , and the cobordism map A Λ (cid:48)− → A Λ − is the composition ofΨ − and the projection map p : S N ( A Λ − ) → A Λ − .We can now finally turn to the geometric cobordism map Φ L a : A Λ + → A Λ − . To completethe proof of Proposition A.1, we want to show that Φ L a = Ω ◦ Φ for some link automorphismΩ of Λ − .At this point we have broken down Φ L a into a composition of three cobordism maps: Ψ − ◦ i : A Λ + → A Λ (cid:48) + , Φ (cid:48) : A Λ (cid:48) + → A Λ (cid:48)− , and p ◦ Ψ − : A Λ (cid:48)− → A Λ − . That is, Φ L a is chain homotopyequivalent to the composition p ◦ Ψ − ◦ Φ (cid:48) ◦ Ψ − ◦ i of the five maps going around the sides ofthe following rectangle: A Λ + i (cid:47) (cid:47) Φ La (cid:15) (cid:15) S N ( A Λ + ) Φ (cid:15) (cid:15) A Λ (cid:48) + Ψ + ∼ = (cid:111) (cid:111) Φ (cid:48) (cid:15) (cid:15) A Λ − S N ( A Λ − ) p (cid:111) (cid:111) A Λ (cid:48)− . Ψ − ∼ = (cid:111) (cid:111) From Lemma A.8, there is a link automorphism Ω of Λ (cid:48)− such that Φ (cid:48) = Ω ◦ Φ . Since Λ (cid:48)− and Λ − are Legendrian isotopic, Ω induces a link automorphism of Λ − , which we also call Ω,so that Ω commutes with the chain map p ◦ Ψ − : A Λ (cid:48)− → A Λ − induced by the isotopy. ThusΦ L a (cid:39) p ◦ Ψ − ◦ Φ (cid:48) ◦ Ψ − ◦ i = p ◦ Ψ − ◦ Ω ◦ Φ ◦ Ψ − ◦ i = Ω ◦ p ◦ Ψ − ◦ Φ ◦ Ψ − ◦ i = Ω ◦ p ◦ Φ ◦ i. ut p ◦ Φ ◦ i is exactly equal to Φ as defined in the statement of Proposition A.1, and weare done with the proof.A.4. Proof of Proposition 4.8.
The remainder of this section is devoted to the proof ofProposition 4.8. At this point, by Proposition A.1, we know the saddle cobordism map fora saddle flanked by mini-dips; to prove Proposition 4.8, we just need to compose this mapwith maps corresponding to the Reidemeister II moves of adding and removing mini-dips.This is similar to the proof of Proposition A.1 in the previous subsection, except that it willnow be important to calculate these Reidemeister II maps in more detail.Suppose that, as in the statement of Proposition 4.8, we have a saddle cobordism betweenΛ + and Λ − , where the cobordism is given by resolving a proper contractible Reeb chord a ofΛ + . Let Λ (cid:48) + be the result of adding a mini-dip to Λ + just after a following the orientation ofΛ + , and let Λ (cid:48)(cid:48) + be result of further adding a mini-dip to Λ (cid:48) + on the other side of a . Similarlydefine Λ (cid:48)− and Λ (cid:48)(cid:48)− . Then Λ (cid:48)± are obtained from Λ ± by a single Reidemeister II move, Λ (cid:48)(cid:48)± areobtained from Λ (cid:48)± by another Reidemeister II move, and Λ (cid:48)(cid:48) + and Λ (cid:48)(cid:48)− are related by a saddlemove of the precise form that we considered in Proposition A.1. See Figure 43. Figure 43.
Adding mini-dips to Λ ± .The properness condition for a translates into the following result. Lemma A.9.
Given that the Reeb chord a is proper: • if a q is any Reeb chord of Λ (cid:48) + , and ∂ (cid:48) denotes the differential on Λ (cid:48) + , then any termin ∂ (cid:48) ( a q ) that contains a must contain a exactly once and cannot contain a ; • if a q is any Reeb chord of Λ (cid:48)(cid:48) + , and ∂ (cid:48)(cid:48) denotes the differential on Λ (cid:48)(cid:48) + , then any termin ∂ (cid:48) ( a q ) that contains a must contain a exactly once and cannot contain any of a , a , a . Figure 44.
Turning an immersed disk ∆ (cid:48) for Λ (cid:48) + into an immersed disk ∆for Λ + . roof. We will establish the statement for Λ (cid:48) + ; the proof of the statement for Λ (cid:48)(cid:48) + is similar.If a q is any of a, a , a , then the statement is trivially true: by action considerations, the onlyterm in ∂ (cid:48) ( a q ) that could contain a is just the term a itself in ∂ (cid:48) ( a ). Now assume a q is not a, a , a . Consider any word in ∂ (cid:48) ( a q ), corresponding to an immersed disk ∆ (cid:48) in Λ (cid:48) + with sole+ corner at a q and a − corner at a . Then ∆ (cid:48) in turn produces an immersed disk ∆ in Λ + ,now possibly with concave corners at a : see Figure 44. If ∆ (cid:48) contained multiple corners at a , or corners at both a and a , then the boundary of ∆ would pass through Π xy ( a ) morethan once, violating the properness condition from Definition 4.3. (cid:3) We will now piece together the five maps A Λ + → A Λ (cid:48) + → A Λ (cid:48)(cid:48) + → A Λ (cid:48)(cid:48)− → A Λ (cid:48)− → A Λ − toget the desired cobordism map. The central map A Λ (cid:48)(cid:48) + → A Λ (cid:48)(cid:48)− has already been computed,while the remaining maps come from Reidemeister II isotopies.We will focus for now on the map A Λ + → A Λ (cid:48) + , which we call Ψ → + . This is the chain mapinduced by adding a Legendrian Reidemeister II move, as derived in [Che02, ENS02] andsummarized in Section 4.1 above, and we describe it explicitly now. Label the Reeb chordsof Λ + besides a as a , . . . , a r , so that we can write A Λ + = A ( a, a , . . . , a r ) and A Λ (cid:48) + = A ( a, a , a , a , . . . , a r ). We stabilize A Λ + by adding two new generators e , e with | e | = 0, | e | = − ∂ ( e ) = e , ∂ ( e ) = 0, to produce a new DGA S ( A Λ + ) = A ( a, a , . . . , a r , e , e ).As described in Section 4.1 and specifically defined in (4.1), there is a chain isomorphismΨ : A Λ (cid:48) + → S ( A Λ + ), which in our case is defined by Ψ( a ) = e , Ψ( a ) = − e , Ψ( a ) = a ,and for (cid:96) ≥
5, Ψ( a (cid:96) ) = a (cid:96) − H Ψ ∂ (cid:48) a (cid:96) where ∂ (cid:48) is the differential on Λ (cid:48) + . Then Ψ → + is defined to be equal to Ψ − ◦ i .We now claim that Ψ → + satisfies the following formula, which can be compared to the definitionof Φ → from Section 4.2. Lemma A.10.
For all (cid:96) ≥ , we have (A.1) Ψ → + ( a (cid:96) ) = a (cid:96) − (cid:88) ∆ ∈ ∆ → a ( a (cid:96) ) ( − | w (∆) | sgn(∆)Ψ → + ( w (∆)) a w (∆) . Proof.
Assume without loss of generality that a, a , . . . , a r are ordered by height (note that a is contractible and thus has the shortest height). We first claim that for all q ≥
5, Ψ( a q ) − a q only includes terms that involve at least one e and no e : we abbreviate this condition byΨ( a q ) − a q = O ( e ). We prove this by induction on q , where the base case is actually a q = a (note Ψ( a ) − a = 0). For the induction step, note that Ψ( a q ) − a q = − H Ψ ∂ (cid:48) a q , and the righthand side only contains terms involving at least one e ; we need to show that H Ψ ∂ (cid:48) a q doesnot involve e .Consider any word w in ∂ (cid:48) a q . If a does not appear in w , then w involves only a, a , a , . . . , a q − ,and so by induction Ψ( w ) − w = O ( e ) and H Ψ( w ) = H ( w ) = 0. On the other hand, if w doesinvolve a , then by Lemma A.9, w = w a w where w , w involve only a, a , . . . , a q − ; thenby induction again, H Ψ( w ) = − H ((Ψ( w )) e (Ψ( w ))) = − H ( w e Ψ( w )) = ± w e Ψ( w )does not involve e . This completes the proof that Ψ( a q ) − a q = O ( e ) for all q ≥ (cid:96) . The base case is actually Ψ → + ( a ) = a ,which is (A.1) with a = a (cid:96) . For the induction step, we compute that:Ψ → + ( a (cid:96) ) = Ψ − ( a (cid:96) ) = a (cid:96) + Ψ − H Ψ ∂ (cid:48) a (cid:96) . Now suppose that w is a word in ∂ (cid:48) a (cid:96) , and again apply Lemma A.9. If w does not contain a , then H Ψ( w ) = 0. If w does contain a , then we write w = w a w and compute: H Ψ( w ) = H Ψ( w a w ) = − H (Ψ( w ) e Ψ( w )) = − H ( w e Ψ( w )) = ( − | w | w e Ψ( w ) nd thusΨ − H Ψ( w ) = ( − | w | Ψ − ( w e Ψ( w )) = ( − | w | Ψ − ( w ) a w = ( − | w | Ψ → + ( w ) a w , where we have used the fact that w does not involve a or a and thus Ψ − ( w ) = Ψ − i ( w ) =Ψ → + ( w ). Finally note that the disk for w in Λ (cid:48) + precisely corresponds to a disk ∆ in ∆ → a ( a (cid:96) )in Λ + , and that the sign for w in ∂ (cid:48) a (cid:96) is − sgn(∆) since ∆ replaces a corner at a withpositive orientation sign with a corner at a with negative orientation sign. Now the signedsum of Ψ − H Ψ( w ) = Ψ → + ( w ) a w over all disks in ∆ → a ( a (cid:96) ) gives (A.1), and this completesthe induction. (cid:3) In a similar way, we write Ψ ← + for the cobordism map from A Λ (cid:48) + = A ( a, a , a , a , . . . , a r ) to A Λ (cid:48)(cid:48) + = A ( a, a , a , a , a , a , . . . , a r ) induced by the Reidemeister II isotopy between Λ (cid:48) + andΛ (cid:48)(cid:48) + . Lemma A.11.
For all (cid:96) ≥ , we have Ψ ← + ( a (cid:96) ) = a (cid:96) − (cid:88) ∆ ∈ ∆ ← a ( a (cid:96) ) ( − | w (∆) | sgn(∆)Ψ → + ( w (∆)) a w (∆) . Proof.
This is essentially identical to the proof of Lemma A.10. Given our choice of orienta-tion signs, there are two sign differences here from the proof of Lemma A.10: Ψ( a ) is now e rather than − e , and the sign of a word contributing to ∂ (cid:48) ( a (cid:96) ) is now equal to + sgn(∆) ratherthan − sgn(∆) for the corresponding disk ∆ ∈ ∆ ← a ( a (cid:96) ). These two sign changes cancel out.One other subtle difference is that if we follow the proof of the previous lemma, then ∆ ← a ( a (cid:96) )in the statement of the present lemma should be for Λ (cid:48) + rather than for Λ + . However, by theproperness condition for a , there is a one-to-one correspondence between disks in ∆ ← a ( a (cid:96) ) forΛ (cid:48) + and Λ + , and so the desired formula holds for either form of ∆ ← a ( a (cid:96) ). (cid:3) We can now finally piece together our various subsidiary results to prove Proposition 4.8.To distinguish between the saddle cobordisms in the dipped and undipped settings, let L a be the cobordism between Λ + and Λ − as in the statement of Proposition 4.8, and let (cid:101) L a bethe cobordism between Λ (cid:48)(cid:48) + and Λ (cid:48)(cid:48)− . As shown in Figure 43, we can concatenate (cid:101) L a and fourLagrangians coming from Legendrian isotopies to create a five-story cobordism between Λ + and Λ − which is Hamiltonian isotopic to L a : from top to bottom, the five cobordisms gobetween Λ + , Λ (cid:48) + , Λ (cid:48)(cid:48) + , Λ (cid:48)(cid:48)− , Λ (cid:48)− , and Λ − .The chain map Φ L a : A Λ + → A Λ − is then chain homotopic to the composition of the chainmaps coming from the five cobordisms. We summarize this in the following diagram, whichcommutes up to chain homotopy: A Λ + Ψ → + (cid:47) (cid:47) Φ La (cid:15) (cid:15) A Λ (cid:48) + Ψ ← + (cid:47) (cid:47) A Λ (cid:48)(cid:48) + Φ (cid:101) La (cid:15) (cid:15) A Λ − A Λ (cid:48)− p (cid:111) (cid:111) A Λ (cid:48)(cid:48)− . p (cid:111) (cid:111) Here Ψ → + and Ψ ← + are the maps computed in Lemmas A.10 and A.11, while p and p arethe maps induced by the reverse Reidemeister II moves from Λ (cid:48)(cid:48)− to Λ (cid:48)− and from Λ (cid:48)− to Λ − .By Remark 4.2, these last two maps (which correspond to p ◦ Ψ in Remark 4.2) are givensimply by projection: p ( a ) = p ( a ) = p ( a ) = p ( a ) = 0 and p , p are the identity onall other generators.Now by Proposition A.1, Φ (cid:101) L a = Ω ◦ Φ where Ω is a link automorphism of Λ (cid:48)(cid:48)− and Φ is themap given in the statement of the proposition. Since Λ (cid:48)(cid:48)− and Λ − are isotopic, Ω induces a ink automorphism of Λ − which we also denote by Ω, and p ◦ p ◦ Ω = Ω ◦ p ◦ p . At thispoint we have: Φ L a (cid:39) p ◦ p ◦ Φ (cid:101) L a ◦ Ψ ← + ◦ Ψ → + = Ω ◦ p ◦ p ◦ Φ ◦ Ψ ← + ◦ Ψ → + . We will be done if we can show that the composition p ◦ p ◦ Φ ◦ Ψ ← + ◦ Ψ → + is equal to the mapΦ comb L a = Φ ← ◦ Φ → ◦ Φ from Proposition 4.8. But Φ comb L a is specifically designed so that thisis the case. Specifically, if a (cid:96) is any Reeb chord of Λ + besides a , then Φ → ( a (cid:96) ) and Φ ← ( a (cid:96) )are precisely the result of replacing a and a by − s − in the expressions for Ψ → + ( a (cid:96) ) andΨ ← + ( a (cid:96) ) from Lemmas A.10 and A.11. But by the definition of Φ, this replacement is exactlythe effect of composing with the map p ◦ p ◦ Φ, which sends a , a to − s − and sends a (cid:96) toitself for (cid:96) ≥
5. It follows thatΦ comb L a ( a (cid:96) ) = (Φ ← ◦ Φ → )( a (cid:96) ) = ( p ◦ p ◦ Φ)((Ψ ← + ◦ Ψ → + )( a (cid:96) )for all (cid:96) . Combined with the fact that Φ comb L a ( a ) = s = ( p ◦ p ◦ Φ ◦ Ψ ← + ◦ Ψ → + )( a ), thisestablishes that Φ comb L a = p ◦ p ◦ Φ ◦ Ψ ← + ◦ Ψ → + . The proof of Proposition 4.8 is complete. References [Abb14] Casim Abbas.
An introduction to compactness results in symplectic field theory . Springer, Heidel-berg, 2014.[Ad90] V. I. Arnol (cid:48) d. Singularities of caustics and wave fronts , volume 62 of
Mathematics and its Appli-cations (Soviet Series) . Kluwer Academic Publishers Group, Dordrecht, 1990.[BEH +
03] F. Bourgeois, Y. Eliashberg, H. Hofer, K. Wysocki, and E. Zehnder. Compactness results insymplectic field theory.
Geom. Topol. , 7:799–888, 2003.[Ben83] Daniel Bennequin. Entrelacements et ´equations de Pfaff. In
Third Schnepfenried geometry confer-ence, Vol. 1 (Schnepfenried, 1982) , volume 107 of
Ast´erisque , pages 87–161. Soc. Math. France,Paris, 1983.[BFZ05] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky. Cluster algebras. III. Upper boundsand double Bruhat cells.
Duke Math. J. , 126(1):1–52, 2005.[BLL18] Sebastian Baader, Lukas Lewark, and Livio Liechti. Checkerboard graph monodromies.
Enseign.Math. , 64(1-2):65–88, 2018.[Cas20] Roger Casals. Lagrangian skeleta and plane curve singularities. arXiv:2009.06737, 2020.[CDGG17] Baptiste Chantraine, Georgios Dimitroglou Rizell, Paolo Ghiggini, and Roman Golovko.Geometric generation of the wrapped Fukaya category of Weinstein manifolds and sectors.arXiv:1712.09126, 2017.[CDGG20] Baptiste Chantraine, Georgios Dimitroglou Rizell, Paolo Ghiggini, and Roman Golovko. Floertheory for Lagrangian cobordisms.
J. Differential Geom. , 114(3):393–465, 2020.[CG20] Roger Casals and Honghao Gao. Infinitely many Lagrangian fillings. arXiv:2001.01334, 2020.[CGGS20] Roger Casals, Eugene Gorsky, Mikhail Gorsky, and Jos´e Simental. Algebraic weaves and braidvarieties. arXiv:2012.06931, 2020.[Cha10] Baptiste Chantraine. Lagrangian concordance of Legendrian knots.
Algebr. Geom. Topol. ,10(1):63–85, 2010.[Che02] Yuri Chekanov. Differential algebra of Legendrian links.
Invent. Math. , 150(3):441–483, 2002.[CSLL +
21] Orsola Capovilla-Searle, No´emie Legout, Ma¨ylys Limouzineau, Emmy Murphy, Yu Pan, and LisaTraynor. Obstructions to exact Lagrangian cobordisms. In preparation, 2021.[CZ20] Roger Casals and Eric Zaslow. Legendrian weaves. arXiv:2007.04943, 2020.[EES05] Tobias Ekholm, John Etnyre, and Michael Sullivan. Orientations in Legendrian contact homologyand exact Lagrangian immersions.
Internat. J. Math. , 16(5):453–532, 2005.[EHK16] Tobias Ekholm, Ko Honda, and Tam´as K´alm´an. Legendrian knots and exact Lagrangian cobor-disms.
J. Eur. Math. Soc. (JEMS) , 18(11):2627–2689, 2016.[EK08] Tobias Ekholm and Tam´as K´alm´an. Isotopies of Legendrian 1-knots and Legendrian 2-tori.
J.Symplectic Geom. , 6(4):407–460, 2008.[Ekh19] Tobias Ekholm. Holomorphic curves for legendrian surgery. arXiv:1906.07228, 2019.[EL19] Tobias Ekholm and Yanki Lekili. Duality between lagrangian and legendrian invariants.arXiv:1701.01284, 2019.[EN18] John B. Etnyre and Lenhard L. Ng. Legendrian contact homology in R . arXiv:1811.10966, 2018.[ENS02] John B. Etnyre, Lenhard L. Ng, and Joshua M. Sabloff. Invariants of Legendrian knots andcoherent orientations. J. Symplectic Geom. , 1(2):321–367, 2002. EP96] Y. Eliashberg and L. Polterovich. Local Lagrangian 2-knots are trivial.
Ann. of Math. (2) ,144(1):61–76, 1996.[Etn03] John B. Etnyre. Introductory lectures on contact geometry. In
Topology and geometry of manifolds(Athens, GA, 2001) , volume 71 of
Proc. Sympos. Pure Math. , pages 81–107. Amer. Math. Soc.,Providence, RI, 2003.[EV18] John Etnyre and Vera V´ertesi. Legendrian satellites.
Int. Math. Res. Not. IMRN , (23):7241–7304,2018.[FOOO09] Kenji Fukaya, Yong-Geun Oh, Hiroshi Ohta, and Kaoru Ono.
Lagrangian intersection Floer the-ory: anomaly and obstruction. Part II , volume 46 of
AMS/IP Studies in Advanced Mathematics .American Mathematical Society, Providence, RI; International Press, Somerville, MA, 2009.[FR11] Dmitry Fuchs and Dan Rutherford. Generating families and Legendrian contact homology in thestandard contact space.
J. Topol. , 4(1):190–226, 2011.[Gei08] Hansj¨org Geiges.
An introduction to contact topology , volume 109 of
Cambridge Studies in Ad-vanced Mathematics . Cambridge University Press, Cambridge, 2008.[GPS19] Sheel Ganatra, John Pardon, and Vivek Shende. Sectorial descent for wrapped Fukaya categories.arXiv:1809.03427, 2019.[GSW20a] Honghao Gao, Linhui Shen, and Daping Weng. Augmentations, fillings, and clusters.arXiv:2008.10793, 2020.[GSW20b] Honghao Gao, Linhui Shen, and Daping Weng. Positive braid links with infinitely many fillings.arXiv:2009.00499, 2020.[K´al05] Tam´as K´alm´an. Contact homology and one parameter families of Legendrian knots.
Geom. Topol. ,9:2013–2078, 2005.[K´al06] Tam´as K´alm´an. Braid-positive Legendrian links.
Int. Math. Res. Not. , pages Art ID 14874, 29,2006.[Kar17] Cecilia Karlsson. To compute orientations of Morse flow trees in Legendrian contact homology.arXiv:1704.05156, 2017.[Kar20] Cecilia Karlsson. A note on coherent orientations for exact Lagrangian cobordisms.
QuantumTopol. , 11(1):1–54, 2020.[Lev16] C. Leverson. Augmentations and rulings of Legendrian knots.
J. Symplectic Geom. , 14(4):1089–1143, 2016.[Lev17] Caitlin Leverson. Augmentations and rulings of Legendrian links in k ( S × S ). Pacific J. Math. ,288(2):381–423, 2017.[Lin16] Francesco Lin. Exact Lagrangian caps of Legendrian knots.
J. Symplectic Geom. , 14(1):269–295,2016.[Mis03] K. Mishachev. The N -copy of a topologically trivial Legendrian knot. J. Symplectic Geom. ,1(4):659–682, 2003.[Mis17] Filip Misev. Cutting arcs for torus links and trees.
Bull. Soc. Math. France , 145(3):575–602, 2017.[Ng] Lenhard Ng.
Mathematica notebook fillings.nb . Available at https://math.duke.edu/~ng/math/ .[Ng03] Lenhard L. Ng. Computable Legendrian invariants.
Topology , 42(1):55–82, 2003.[Ng10] Lenhard Ng. Rational symplectic field theory for Legendrian knots.
Invent. Math. , 182(3):451–512,2010.[NR13] Lenhard Ng and Daniel Rutherford. Satellites of Legendrian knots and representations of theChekanov-Eliashberg algebra.
Algebr. Geom. Topol. , 13(5):3047–3097, 2013.[NRS +
20] Lenhard Ng, Dan Rutherford, Vivek Shende, Steven Sivek, and Eric Zaslow. Augmentations areSheaves.
Geom. Topol. , 24(5):2149–2286, 2020.[NT04] Lenhard Ng and Lisa Traynor. Legendrian solid-torus links.
J. Symplectic Geom. , 2(3):411–443,2004.[OS04] Burak Ozbagci and Andr´as I. Stipsicz.
Surgery on contact 3-manifolds and Stein surfaces , vol-ume 13 of
Bolyai Society Mathematical Studies . Springer-Verlag, Berlin, 2004.[Pan17a] Yu Pan. The augmentation category map induced by exact Lagrangian cobordisms.
Algebr. Geom.Topol. , 17(3):1813–1870, 2017.[Pan17b] Yu Pan. Exact Lagrangian fillings of Legendrian (2 , n ) torus links.
Pacific J. Math. , 289(2):417–441, 2017.[Rud92] Lee Rudolph. Quasipositive annuli. (Constructions of quasipositive knots and links. IV).
J. KnotTheory Ramifications , 1(4):451–466, 1992.[Sab05] Joshua M. Sabloff. Augmentations and rulings of Legendrian knots.
Int. Math. Res. Not. ,(19):1157–1180, 2005.[Sau04] Denis Sauvaget. Curiosit´es lagrangiennes en dimension 4.
Ann. Inst. Fourier (Grenoble) ,54(6):1997–2020 (2005), 2004.[Siv11] Steven Sivek. A bordered Chekanov-Eliashberg algebra.
J. Topol. , 4(1):73–104, 2011. niversity of California Davis, Dept. of Mathematics, Shields Avenue, Davis, CA 95616, USA Email address : [email protected] Duke University, Department of Mathematics, Durham, NC 27708, USA
Email address : [email protected]@math.duke.edu