Constructions of Lagrangian cobordisms
Sarah Blackwell, Noémie Legout, Caitlin Leverson, Maÿlis Limouzineau, Ziva Myer, Yu Pan, Samantha Pezzimenti, Lara Simone Suárez, Lisa Traynor
CCONSTRUCTIONS OF LAGRANGIAN COBORDISMS
SARAH BLACKWELL, NO´EMIE LEGOUT, CAITLIN LEVERSON,MA ¨YLIS LIMOUZINEAU, ZIVA MYER, YU PAN, SAMANTHA PEZZIMENTI,LARA SIMONE SU ´AREZ, AND LISA TRAYNOR
Abstract.
Lagrangian cobordisms between Legendrian knots arise in Sym-plectic Field Theory and impose an interesting and not well-understoodrelation on Legendrian knots. There are some known “elementary” build-ing blocks for Lagrangian cobordisms that are smoothly the attachmentof 0- and 1-handles. An important question is whether every pair of non-empty Legendrians that are related by a connected Lagrangian cobordismcan be related by a ribbon Lagrangian cobordism, in particular one thatis “decomposable” into a composition of these elementary building blocks.We will describe these and other combinatorial building blocks as well assome geometric methods, involving the theory of satellites, to constructLagrangian cobordisms. We will then survey some known results, derivedthrough Heegaard Floer Homology and contact surgery, that may providea pathway to proving the existence of nondecomposable (nonribbon) La-grangian cobordisms. Introduction
A contact manifold is an odd-dimensional manifold Y n +1 together with amaximally non-integrable hyperplane distribution ξ . In a contact manifold, Legendrian submanifolds play a central role. These are the maximal integralsubmanifolds of ξ : Λ n such that T p Λ ⊂ ξ , for all p ∈ Λ. In general, Legendriansubmanifolds are plentiful and easy to construct. In this article we will restrictour attention to the contact manifold R with its standard contact structure ξ = ker α , where α = dz − ydx . In this setting, every smooth knot or linkhas an infinite number of non-equivalent Legendrian representatives. Morebackground on Legendrian knots is given in Section 2.The even-dimensional siblings of contact manifolds are symplectic mani-folds. These are even-dimensional manifolds M n equipped with a closed,non-degenerate 2-form ω . In symplectic manifolds, Lagrangian submanifolds play a central role. Lagrangian submanifolds are the maximal dimensionalsubmanifolds where ω vanishes on the tangent spaces: L n such that ω | L = 0.When the symplectic manifold is exact, ω = dλ , it is important to understandthe more restrictive subset of exact Lagrangians: these are submanifolds where λ | L is an exact 1-form. Geometrically, L exact means that for any closed curve γ ⊂ L , (cid:82) γ λ = 0. In this article, we will restrict our attention to a symplecticmanifold that is symplectomorphic to R with its standard symplectic struc-ture ω = (cid:80) dx i ∧ dy i . In contrast to Legendrians, Lagrangians are scarce. For a r X i v : . [ m a t h . S G ] D ec CONSTRUCTIONS OF LAGRANGIAN COBORDISMS example, in R with its standard symplectic structure, the torus is the onlyclosed surface that will admit a Lagrangian embedding into R . A famoustheorem of Gromov [Gro85] states that there are no closed, exact Lagrangiansubmanifolds of R .There has been a great deal of recent interest in a certain class of non-closed,exact Lagrangian submanifolds, known as Lagrangian cobordisms . These La-grangian submanifolds live in the symplectization of a contact manifold andhave cylindrical ends over Legendrians. In this article, we will focus on exact,orientable Lagrangian cobordisms from the Legendrian Λ − to the LegendrianΛ + that live in the symplectization of R ; this symplectization is R × R equipped with the exact symplectic form ω = d ( e t α ), where t is the coor-dinate on R and α = dz − ydx is the standard contact form on R . SeeFigure 5 for a schematic picture of a Lagrangian cobordism and Definition 1for a formal definition. Such Lagrangian cobordisms were first introducedin Symplectic Field Theory (SFT) [EGH00]: in relative SFT, we get a cat-egory whose objects are Legendrians and whose morphisms are Lagrangiancobordisms. Lagrangian fillings occur when Λ − = ∅ and are key objects in theFukaya category, which is an important invariant of symplectic four-manifolds.A Lagrangian cap occurs when Λ + = ∅ .A basic question tied to understanding the general existence and behav-ior of Lagrangian submanifolds is to understand the existence of Lagrangiancobordisms: Given two Legendrians Λ ± , when does there exist a Lagrangiancobordism from Λ − to Λ + ? There are known to be a number of obstructionsto this relation on Legendrian submanifolds coming from both classical andnon-classical invariants of the Legendrians Λ ± . Some of these obstructionsare described in Section 2.3. To complement the obstructions, there are someknown constructions. For example, it is well known [EG98, Cha10, EHK16]that there exists a Lagrangian cobordism between Legendrians Λ ± that dif-fer by Legendrian isotopy. In addition, by [EHK16, Cha12], it is known thatthere exists a Lagrangian cobordism from Λ − to Λ + if Λ − can be obtainedfrom Λ + by a “pinch” move or if Λ + = Λ − ∪ U , where U denotes a Legendrianunknot with maximal Thurston-Bennequin number of − − . Topologically, between these slices,the cobordism changes by a saddle move (1-handle) and the addition of a lo-cal minimum (0-handle); see Figure 1. It is important to notice that there is not an elementary move corresponding to a local maximum (2-handle) move.By stacking these individual cobordisms obtained from isotopy, saddles, andminimums, one obtains what is commonly referred to as a decomposable La-grangian cobordism. Through these moves, it is easy to construct Lagrangiancobordisms and fillings; see an example in Figure 7.Towards understanding the existence of Lagrangians, it is natural to ask:
Does there exist a Lagrangian cobordism from Λ − to Λ + if and only if thereexists a decomposable Lagrangian cobordism from Λ − to Λ + ? We know the an-swer to this question is “No”: by studying the “movies” of the not necessarily
ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 3 ( A ) ( B )Λ + Λ − Λ + Figure 1. (A) The pinch move on Λ + produces a Lagrangiansaddle. (B) Λ + obtained by introducing an unknotted compo-nent to Λ − corresponds to the Lagrangian cobordism having alocal min.Legendrian slices of a Lagrangian. Sauvaget, Murphy, and Lin [Sau04, Lin16]have shown that there exists a genus two Lagrangian cap of the Legendrianunknot with Thurston-Bennequin number equal to − Lagrangian diagram moves used by [Lin16] to construct a Lagrangian capare described in Section 3.3. The necessity of a local maximum when Λ + (cid:54) = ∅ is not currently understood.To formulate some precise motivating questions, we will use ribbon cobor-dism to denote a 2 n -dimensional manifold that can be built from k -handleswith k ≤ n . This idea of restricting the handle index is well known in symplec-tic topology: Eliashberg [CE12, Oan15] has shown that any 2 n -dimensionalStein manifold admits a handle decomposition with handles of dimension atmost n , and thus any 2 n -dimensional Stein cobordism between closed, (2 n − R coordinate on R × R , we see that all decompos-able 2-dimensional Lagrangian cobordisms between 1-dimensional Legendriansubmanifolds are ribbon cobordisms. We are led to the following natural ques-tions. Motivating Questions.
Suppose Λ + (cid:54) = ∅ and there exists a connected La-grangian cobordism L from Λ − to Λ + . Then: (1) Does there exist a decomposable Lagrangian cobordism from Λ − to Λ + ? (2) Does there exist a ribbon Lagrangian cobordism from Λ − to Λ + ? (3) Is L Lagrangian isotopic to a ribbon and/or decomposable Lagrangiancobordism?
There are some results known about Motivating Question (3) for the spe-cial case of the simplest Legendrian unknot. If U denotes the Legendrianunknot with Thurston-Bennequin number −
1, it is known that every (ex-act) Lagrangian filling is orientable [Rit09], and there is a unique (exact,orientable) Lagrangian filling of U up to compactly supported Hamiltonian CONSTRUCTIONS OF LAGRANGIAN COBORDISMS isotopy [EP96]. Moreover, any Lagrangian cobordism from U to U is La-grangian isotopic, via a compactly supported Hamiltonian isotopy, to one ina countable collection given by the trace of a Legendrian isotopy induced bya rotation [CDRGG].Motivating Questions (1) and (2) are closely related and have deep ties toimportant questions in topology. Observe that a “yes” answer to (1) implies a“yes” to (2): if the existence of a Lagrangian cobordism implies the existenceof a decomposable Lagrangian cobordism, then we also know the existence ofa ribbon cobordism. Also note that when Λ + is topologically a slice knot andΛ − = ∅ , (2) is a symplectic version of the topological Slice-Ribbon conjecture:is every Lagrangian slice disk a ribbon disk? Cornwell, Ng, and Sivek conjec-ture that the answer to Motivating Question (1) and (3) is “No”: using thetheory of satellites, we know that there is a Lagrangian concordance betweenΛ ± shown in Figure 2, and in [CNS16, Conjecture 3.3] it is conjectured thatthe concordance between the pair is not decomposable.Λ + Λ − Figure 2.
There is a Lagrangian concordance between theseLegendrian knots that is conjectured to be non-decomposable.Here Λ − is a Legendrian trefoil and Λ + is a Legendrian White-head double of m (9 ).Very recently, Roberta Guadagni has discovered additional combinatorialmoves that can be used to construct a “movie,” meaning a sequence of slicepictures, of a Lagrangian cobordism; Figure 9 illustrates one of these tanglemoves. With one of Guadagni’s moves, it is possible to construct a movie ofa Lagrangian cobordism between the Legendrians pictured in Figure 2; seeFigure 10. Guadagni’s moves are “geometric”: they are developed throughproofs similar to those used in the satellite procedure, and thus the handleattachments involved in the cobordism are not obvious. In particular, at this ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 5 point it is not known if Guadagni’s tangle moves are independent from thedecomposable moves.This survey article is organized as follows. In Section 2, we provide somebackground on Legendrians and Lagrangians, formally define Lagrangian cobor-disms, and summarize known obstructions to the existence of Lagrangiancobordisms. In Section 3, we describe three “combinatorial” ways to constructLagrangian cobordisms, and in Section 4, we describe more abstract “geo-metric” ways to construct Lagrangian concordances and cobordisms throughsatellites. Then in Section 5, we describe some potential pathways – throughthe theory of rulings, Heegaard-Floer homology, and contact surgery – to po-tentially show the existence of Legendrians that are Lagrangian cobordant butare not related by a decomposable Lagrangian cobordism.
Acknowledgements:
This project was initiated at the workshop Women inSymplectic and Contact Geometry and Topology (WiSCoN) that took placeat ICERM in July 2019. The authors thank the NSF-HRD 1500481 - AWMADVANCE grant for funding this workshop. Leverson was supported by NSFpostdoctoral fellowship DMS-1703356. We thank Emmy Murphy for suggest-ing and encouraging us to work on this project. In addition, we thank JohnEtnyre, Roberta Guadagni, Tye Lidman, Lenny Ng, Josh Sabloff, and B¨ulentTosun for useful conversations related to this project.2.
Background
Legendrian Knots and Links.
In this section, we give a very briefintroduction to Legendrian submanifolds in R and their invariants. Moredetails can be found, for example, in the survey paper [Etn05].In R , the standard contact structure ξ is a 2-dimensional plane fieldgiven by the kernel of the 1-form α = dz − ydx . In ( R , ξ = ker α ), a Leg-endrian knot is a knot in R that is tangent to ξ everywhere. A useful wayto visualize a Legendrian knot is to project it from R to R . There are twouseful projections: the Lagrangian projection π L : R → R ( x, y, z ) (cid:55)→ ( x, y ) , as well as the front projection π F : R → R ( x, y, z ) (cid:55)→ ( x, z ) . An example of a Legendrian trefoil is shown in Figure 3.Legendrian submanifolds are equivalent if they can be connected by a 1-parameter family of Legendrian submanifolds. In fact, for each topologicalknot type there are infinitely many different Legendrian knots. Indeed, wecan stabilize a Legendrian knot (as shown in Figure 4) to get another Legen-drian knot of the same topological knot type. We can see that these are notLegendrian equivalent using Legendrian invariants.
CONSTRUCTIONS OF LAGRANGIAN COBORDISMS xz xy
Figure 3.
The front projection (left) and the Lagrangian pro-jection (right) of a Legendrian trefoil.
Figure 4.
Two ways to stabilize a Legendrian knot in front projection.Two useful classical invariants of Legendrian knots Λ are the Thurston-Bennequin number tb (Λ) and the rotation number r (Λ). They can be com-puted easily from front projections. Given the front projection of a Legendrianknot or link Λ, the Thurston-Bennequin number is tb (Λ) = writhe( π F (Λ)) − , where the writhe is the number of crossings counted with sign. Once theLegendrian knot is equipped with an orientation, the rotation number is r (Λ) = 12 (cid:16) − (cid:17) . One can use these two invariants to see that stabilizations change the Legen-drian knot type.In future sections, we will not assume that our Legendrians Λ ± come equippedwith an orientation. In our Motivating Questions described in Section 1, ourLagrangian cobordisms are always orientable, so the existence of a Lagrangiancobordism from Λ − to Λ + will induce orientations on Λ ± .There are many powerful non-classical invariants that can be assigned toa Legendrian knot. Although this will not be a focus of this paper, we willgive a brief description of some of these invariants. One important invariantstems from normal rulings , defined independently by Chekanov and Pushkar[PC05] and Fuchs [Fuc03]. A count of normal rulings leads to ruling poly-nomials [PC05]; more details will be discussed in Section 5.1. Through theclosely related theory of generating families, one can also associate invari-ant polynomials that record the dimensions of generating family homologygroups [Tra01, JT06, FR11, ST13]. In addition, through the theory of pseudo-holomoprhic curves, one can associate to a Legendrian Λ a differential gradedalgebra (DGA) , A (Λ) [Che02, Eli98]. An augmentation is a DGA map from A (Λ) to a field. The count of augmentations is closely related to the count ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 7 of ruling polynomials [Fuc03, NR13, NS06]. Augmentations can be used toconstruct finite-dimensional linearized contact homology groups [Che02], whichare often known to be isomorphic to the generating family homology groups[FR11]. In addition, there are invariants for Legendrian knots coming from
Heegaard Floer Homology [LOSS09] [OST08].2.2.
Lagrangian Cobordisms.
Lagrangian cobordisms between Legendriansubmanifolds always have “cylindrical ends” over the Legendrians, but otherconditions vary: sometimes it is specified that the Lagrangian is exact, isembedded (or immersed), is orientable, or has a fixed Maslov class. In thispaper, a Lagrangian cobordism is always exact, embedded, and orientable.
Definition 1.
Let Λ ± be two Legendrian knots or links in ( R , ξ = ker α ).A Lagrangian cobordism L from Λ − to Λ + is an embedded, orientableLagrangian surface in the symplectization ( R × R , d ( e t α )) such that for some N > L ∩ ([ − N, N ] × R ) is compact,(2) L ∩ (( N, ∞ ) × R ) = ( N, ∞ ) × Λ + ,(3) L ∩ (( −∞ , − N ) × R ) = ( −∞ , − N ) × Λ − , and(4) there exists a function f : L → R and constant numbers c ± such that e t α | T L = df , where f | ( −∞ , − N ) × Λ − = c − , and f | ( N, ∞ ) × Λ + = c + .A Lagrangian filling of Λ + is a Lagrangian cobordism with Λ − = ∅ ; a La-grangian cap of Λ − is a Lagrangian cobordism with Λ + = ∅ . A Lagrangianconcordance occurs when Λ ± are knots and L has genus 0.Figure 5 is a schematic representation of a Lagrangian cobordism. tN Λ + Λ − L − N Figure 5.
A Lagrangian cobordism from Λ − to Λ + . Remark . In condition (4) of Definition 1, the fact that Λ ± are Legendrianwill guarantee that f ± will be locally constant. Using this, it follows thatany genus zero Lagrangian surface that is cylindrical over Legendrian knotswill be exact. When Λ ± have multiple components, one needs to check thatthe constant does not vary: this condition guarantees the exactness of the CONSTRUCTIONS OF LAGRANGIAN COBORDISMS
Lagrangian cobordism obtained by “gluing” together Lagrangian cobordisms[Cha15a].
Remark . In contrast to topological cobordisms, Lagrangian cobordisms forma non-symmetric relationship on Legendrian knots [Cha15b]. In this articlewe will always denote the direction of increasing R t coordinate by an arrow.2.3. Obstructions to Lagrangian Cobordisms.
The focus of this paperis on constructing Lagrangian cobordisms between two given Legendrians Λ ± .In the smooth world, any two knots are related by a smooth cobordism, but inthis more restrictive Lagrangian world, there are a number of obstructions thatare important to keep in mind when trying to explicitly construct Lagrangiancobordisms. Here we mention a few that come from classical and non-classicalinvariants of the Legendrians Λ ± . Obstructions: (1) If there exists a Lagrangian cobordism of genus g between Λ − andΛ + , then there must exist a smooth cobordism of genus g between thesmooth knot types of Λ − and Λ + . Thus any obstruction of a smoothgenus g cobordism between Λ − and Λ + would obstruct a Lagrangiangenus g cobordism.(2) Since there are no closed, exact Lagrangian surfaces [Gro85], if thereexists a Lagrangian cap (respectively, filling) for Λ, then there cannotexist a Lagrangian filling (respectively, cap) of Λ.(3) As shown in [Cha10], if there exists a Lagrangian cobordism L fromΛ − to Λ + , then r (Λ − ) = r (Λ + ) and tb (Λ + ) − tb (Λ − ) = − χ ( L ) . In particular, if a Legendrian knot Λ admits a Lagrangian filling orcap, then r (Λ) = 0. Also, combining this equality on tb and theslice-Bennequin inequality [Rud97], we see that, when Λ is a singlecomponent knot, if there exists a Lagrangian cap L of Λ, then tb (Λ) ≤− g ( L ) ≥ − to Λ + , and Λ − has an augmentation, then(a) Aug (Λ + ; F ) ≥ Aug (Λ − ; F ), where F is the finite field of twoelements, and Aug (Λ; F ) denotes the number of augmentationsof Λ to F up to DGA homotopy [Pan17, CSLL + R Λ ± ( z ) (see Section 5.1 for definitions)satisfy R Λ − ( q / − q − / ) ≤ q − χ (Σ) / R Λ + ( q / − q − / ) , for any q that is a power of a prime number [Pan17].(5) If Λ admits a Maslov 0 Lagrangian filling L , and if (cid:15) L denotes theaugmentation of Λ induced by L , then LCH k(cid:15) L (Λ) ∼ = H n − k ( L ), whichis known as the Ekholm-Seidel isomorphism [Ekh12], and whose proofwas completed by Dimitroglou Rizell in [DR16]. More generally, if ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 9 there is a cobordism from Λ − to Λ + , and if Λ − admits an augmentation,then [CDRGG20] provides several long exact sequences relating thehomology of the cobordism and the Legendrian contact (co)homologiesof its Legendrian ends. A version of this isomorphism and these longexact sequences using generating families are given in [ST13].(6) If Λ admits an augmentation, Λ does not admit a Lagrangian cap, asthe augmentation implies the non-acyclicity of the DGA A (Λ) [EES09,Theorem 5.5], and from [DR15, Corollary 1.9] if a Legendrian admitsa Lagrangian cap then its DGA A (Λ) (with Z coefficients) is acyclic.There are additional obstructions, obtained through Heegaard Floer Theory,that can be used to obstruct Lagrangian concordances and cobordisms [BSar,GJ19, BLWar]. Some of these will be discussed more in Section 5.3. Remark . Observe that the obstructions in (4) and (6) assume that the bot-tom Λ − has an augmentation, and stabilized knots will never have an augmen-tation. It would be nice to have more obstructions when Λ − is a stabilizedknot. This might be possible using the theory of “satellites” described in Sec-tion 4.1: it is possible for the satellite of a stabilized Legendrian to admit anaugmentation. See Section 4.3 for more discussions in this direction.3. Combinatorial Constructions of Lagrangian Cobordisms
A convenient way of visualizing topological cobordisms is through “movies”:a sequence of pictures that represent slices of the Lagrangian. In this section,we describe three known combinatorial ways to construct Lagrangian cobor-disms through such an approach.3.1.
Decomposable Moves.
It is well known that if Λ − and Λ + are Leg-endrian isotopic, then there exists a Lagrangian cobordism from Λ − to Λ + ;see, for example, [EG98, Cha10, EHK16]. Isotopy, together with two types ofhandle moves, form the basis for decomposable Lagrangian cobordisms. Theorem 2 ([EHK16, BST15]) . If the front diagrams of two Legendrian links Λ − and Λ + are related by any of the following moves, there is a Lagrangiancobordism L from Λ − to Λ + . Isotopy:
There is a Legendrian isotopy between Λ − and Λ + ; see Fig-ures 6(a)-6(c) for Reidemeister Move I-III. -handle: The front diagram of Λ − can be obtained from the front di-agram of Λ + by “pinching” two oppositely-oriented strands; see Fig-ure 6(d). We will also refer to this move as a “Pinch Move.” -handle: The front diagram of Λ − can be obtained from the front dia-gram of Λ + by deleting a component of Λ + that is the front diagramof a standard Legendrian unknot U with maximal Thurston-Bennequinnumber of − as long as there exist disjoint disks D U , D U c ⊂ R xz containing the xz -projection of U and the other components of Λ + ,respectively. Such an “unknot filling” can be seen in Figure 6(e). ( a ) ( b ) ( c ) ( d ) ( e ) ∅ Figure 6.
Decomposable moves in terms of front projections.Arrows indicate the direction of increasing R t coordinate in thesymplectization. The move in ( b ) only shows the ReidemeisterII move in the left cusp case, but there is an analogous move forthe right cusp. Definition 3.
A Lagrangian cobordism L from Λ − to Λ + is called elemen-tary if it arises from isotopy, a single 0-handle, or a single 1-handle. A La-grangian cobordism L from Λ − to Λ + is decomposable if it is obtained bystacking elementary Lagrangian cobordisms.Observe that there is not an elementary move corresponding to a 2-handle(maximum). Also note that the elementary 1-handle (saddle) move can beused to connect two components or to split one component into two.Decomposable cobordisms are particularly convenient as they are easy todescribe in a combinatorial fashion, through a list of embedded Legendriancurves, Λ − = Λ → Λ → · · · → Λ n = Λ + , where the front projection of the Legendrian Λ i +1 is related to that of Λ i byisotopy or one of the 0-handle or 1-handle moves. Example 4.
One can construct a Lagrangian filling of a positive Legendriantrefoil with maximal Thurston-Bennequin number using the series of movesshown in Figure 7: a 0-handle, followed by three Reidemeister I moves, fol-lowed by two 1-handles (or pinch moves). This gives a genus 1 (orientable,exact) Lagrangian filling of this Legendrian trefoil. Since we are assumingthat Lagrangian fillings and caps are always exact, this implies that this tre-foil cannot admit a Lagrangian cap; see Section 2.3 Obstructions (2).
Example 5.
Using elementary moves, one can also construct a Lagrangianconcordance from the unknot with tb = − m (9 ), as shown on Figure 8.3.2. Guadagni Moves.
Very recently, Roberta Guadagni has discovered anew “tangle” move; see Figure 9. This is not a local move: there are some
ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 11
Figure 7.
A decomposable Lagrangian filling of a Legendrian trefoil.
Figure 8.
A decomposable Lagrangian cobordism from a Leg-endrian unknot to a Legendrian m (9 ).global requirements. In particular, this move cannot be applied if all compo-nents of the tangle are contained in the same component of Λ − : the componentof Λ − containing the blue strand must be different than the components con-taining the other strands of the tangle. Figure 9.
Under some global conditions, there exists a La-grangian cobordism between these tangles.
Example 6.
With Guadagni’s tangle move, it is possible to construct a La-grangian cobordism between the Legendrians pictured in Figure 2; see Fig-ure 10. However, at this point it is not known if Guadagni’s tangle move isindependent of the decomposable moves.3.3.
Lagrangian Diagram Moves.
As shown in Section 3.1, decomposablecobordisms are constructed from 0-handles and some 1-handles (saddles) butno 2-handles (caps). Based on the work of Sauvaget [Sau04], Lin [Lin16] con-structs a genus two cap of a twice stabilized unknot, and thus gives the firstexplicit example of a non-decomposable Lagrangian cobordism. The construc-tion describes time-slices of a Lagrangian cobordism through a list of moveson “decorated Lagrangian diagrams.”
Figure 10.
A movie, using a Guadagni move, of an (orientable,exact) Lagrangian cobordism from the trefoil to the Whiteheaddouble of m (9 ) in Figure 2.A decorated Lagrangian diagram is a curve in the xy -plane with thecompact regions decorated by a positive number, which is the area of theregion. Figure 11 shows some examples: in the illustration of the F move, U is a Lagrangian projection of the Legendrian unknot with maximal Thurston-Bennequin number; in the illustration of the C move, U m is a decorated La-grangian diagram, but is not the Lagrangian projection of a Legendrian knot. Theorem 7 ([Lin16]) . Let Λ ± be Legendrian links and D ± be their corre-sponding decorated Lagrangian projections. If one can create a sequence ofdecorated Lagrangian diagrams D − = D → D → · · · → D n = D + such that each diagram D i +1 can be obtained from D i by the following com-binatorial moves, then there is a compact Lagrangian submanifold in R × R with boundary Λ − ∪ Λ + , where Λ ± ⊂ {± N } × R , for some N > . (1) R : a planar isotopy that changes areas by the amount ± A , for A > .This operation can only be done in the direction specified. (2) R : a Reidemeister II move. One can either introduce or eliminate twocrossings assuming some area conditions are satisfied: it is possible tointroduce or remove two crossings as long as the area of the innerregion, denoted by in the diagram, is less than either the area δ orthe area η . One can also do this move with the lower strand passingunder the upper strand. (3) R : a Reidemeister III move. One can perform a Reidemeister IIImove as long as the area of the inner region, denoted by in the dia-gram, is less than either the area (cid:15) , the area δ or the area η . The fixedcenter crossing can be reversed. Additionally, the moving strand canalso occur as an overstrand. ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 13 ∅∅ + A + A − A − A δ η (cid:15)δ η H + H − F CR R R Ua a a aU m Figure 11.
The Lagrangian diagram moves. The labels in R move represent the change of area through the move, while otherlabels 0 , (cid:15), δ, η, a indicate the area of the corresponding regions;here 0 represents a positive area that is smaller than either thearea (cid:15) , the area δ or the area η .(4) H + : a handle attachment that creates a positive crossing in the dia-gram. (5) H − : a handle attachment that removes a negative crossing in the dia-gram. (6) F : a filling that creates the diagram U , which is the Lagrangian pro-jection of an unknot with maximal Thurston-Bennequin number. (7) C : a cap that eliminates the diagram U m , which is the topological mir-ror of U .These moves are called Lagrangian diagram moves . Moreover, the con-structed Lagrangian will be exact if, in addition, (E1)
Each move results in a diagram with all components having a totalsigned area equal to . The signed area of a region is determined bythe sum of the signed heights of its Reeb chords. (E2) If a handle attachment merges two components of a link, the compo-nents being merged must be vertically split, meaning that the images ofthe xy -projections of these components are contained in disjoint disks. Remark . (1) For condition (E2), the H − can never be applied to mergecomponents, and H + can only be applied if the components beingmerged are vertically split.(2) A main distinction between the Lagrangian diagram moves and thedecomposable moves is that each diagram D i in the middle of thesequence is not necessarily the Lagrangian projection of a Legendrianlink. They are just the xy -projection of some time t i -slice of the cobor-dism. Thus the Lagrangian diagram moves are more flexible than thedecomposable moves. However, keeping track of the areas is an addedcomplication. Example 8.
Figure 12 illustrates the construction of a Lagrangian torus usingthe Lagrangian diagram moves. This torus fails to be exact since condition(E1) is violated. Figure 13 gives another construction of a Lagrangian torus.This time, all components have signed area 0, but now condition (E2) isviolated. ∅ ∅
F H − H + Caa aa aa
Figure 12.
A (non-exact) Lagrangian torus constructed usingthe Lagrangian diagram moves. The middle figure violates (E1). ∅ ∅ Figure 13.
A (non-exact) Lagrangian torus constructed usingthe Lagrangian diagram moves. These figures satisfy (E1) but(E2) is violated in the step labelled by a red arrow.
ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 15 Geometrical Constructions of Lagrangian Cobordisms
An important general way to know of the existence of Lagrangian cobor-disms without using the constructions described in Section 3 comes throughthe satellite operation. In this section, we review the satellite construction andthen state results from [CNS16, GSY20] about the existence of a Lagrangianconcordance/cobordism from Λ − to Λ + implying the existence of a Lagrangianconcordance/cobordism between corresponding satellites.4.1. The Legendrian Satellite Construction.
We begin by reviewing theconstruction of a Legendrian satellite; see also [NT04, Appendix] and [CNS16,Section 2.2]. To construct a Legendrian satellite, begin by identifying the opensolid torus S × R with the 1-jet space of the circle, J S ∼ = T ∗ S × R , equippedwith the contact form α = dz − ydx , where x, y are the coordinates in T ∗ S and z is the coordinate in R . Similar to the situation for R ∼ = J R , we canrecover a Legendrian knot in J S from its front projection in S x × R z , which istypically drawn by representing S as an interval with its endpoints identified.Given an oriented Legendrian companion knot Λ ⊂ R and a orientedLegendrian pattern knot P ⊂ J ( S ), the Legendrian neighborhood theoremsays that Λ has a standard neighborhood N (Λ) such that there is a contacto-morphism κ : J ( S ) → N (Λ). The Legendrian satellite , S (Λ , P ), is thenthe image κ ( P ). The front projection of S (Λ , P ) is as shown in Figure 14. Inparticular, suppose that the front projection of the pattern P intersects thevertical line at the boundary of the S interval n times. We then make an n -copy of Λ by using n -disjoint copies of Λ that all differ by small translationsin the z -direction. Take a point on the front projection of Λ that is orientedfrom left to right, cut the front of the n -copy open along the n -copy at thatpoint, and insert the front diagram of P . The orientation on the satellite S (Λ , P ) is induced by the orientation on P . Λ P S (Λ , P ) Figure 14.
A example of Legendrian satellite.
Remark . The satellite operation often makes Legendrian knots “nicer”; forexample, in Figure 14, the companion Λ is stabilized and does not admit anaugmentation or a normal ruling. However, the satellite S (Λ , P ) does admita normal ruling and augmentation. Lagrangian Cobordisms for Satellites.
In [CNS16, Theorem 2.4],Cornwell, Ng, and Sivek, show that Lagrangian concordance is preserved bythe Legendrian satellite operation.
Theorem 9 ([CNS16]) . Suppose P ⊂ J S is a Legendrian knot. If thereexists a Lagrangian concordance L from a Lengendrian knot Λ − to a Lengen-drian knot Λ + , then there exists a Lagrangian concordance L P from S (Λ − , P ) to S (Λ + , P ) . In particular, as shown in Figure 8, there is a Lagrangian concordancefrom Λ − , which is the Legendrian unknot with tb = −
1, to Λ + , which isthe Legendrian m (9 ) with maximal tb = −
1. Using the Legendrian “clasp”tangle P as shown in Figure 14 – which produces the Legendrian Whiteheaddouble – we can conclude that there exists a Lagrangian concordance from S (Λ − , P ) to S (Λ + , P ). In fact, S (Λ − , P ) is the positive trefoil with tb = 1.Thus Theorem 9 implies that there exists a Lagrangian concordance betweenthe Legendrian knots in Figure 2. Conjecture 10 ([CNS16, Conjecture 3.3]) . The Lagrangian concordance from S (Λ − , P ) to S (Λ + , P ) built through the satellite construction is not decompos-able. Theorem 9 has been extended to higher genus cobordisms by Guadagni,Sabloff, and Yacavone in [GSY20]. To state their theorem, we need to firstintroduce the notion of “twisting” and then closing a tangle T ⊂ J [0 , T ⊂ J [0 , T is the tangle obtained by addingthe tangle T and the full twist tangle ∆, which is illustrated in Figure 15;the tangle ∆ t T can be thought of as T followed by t full twists. Given aLegendrian tangle T ⊂ J [0 , T ⊂ J ( S ) will denote the associated closureto a Legendrian link. ... Figure 15.
For an n -stranded tangle, repeating this basic tan-gle n times produces a full twist. Theorem 11 ([GSY20]) . Suppose T ⊂ J [0 , is a Legendrian tangle whoseclosure T ⊂ J ( S ) is a Legendrian knot. If there exists a Lagrangian cobor-dism L from Λ − to Λ + of genus g ( L ) , then there exists a Lagrangian cobordism L T from S (Λ − , ∆ g ( L )+1 T ) to S (Λ + , ∆ T ) . In fact, Theorem 11 can be generalized to use the closure of different tangles T − and T + that are Lagrangian cobordant; for details, see [GSY20]. ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 17
Remark . It is natural to wonder if, along the lines of Conjecture 10, thishigher genus satellite procedure can create additional candidates for Legen-drians that can be connected by a Lagrangian cobordism but not by a de-composable Lagrangian cobordism. In [GSY20, Theorem 1.5], it is shownthat if the cobordism L from Λ − to Λ + is decomposable and the handles inthe decomposition satisfy conditions known as “Property A”, then the cor-responding satellites S (Λ − , ∆ g ( L )+1 P ) and S (Λ + , ∆ P ) will also be connectedby a decomposable Lagrangian cobordism. In particular, if there exists a de-composable cobordism L that does not satisfy Property A and is not isotopicto a cobordism that satisfies Property A, then the satellite construction wouldlead to a higher genus candidate that generalizes Conjecture 10.4.3. Obstructions to Cobordisms through Satellites.
In Section 2.3,some known obstructions to the existence of a Lagrangian cobordism werementioned. As mentioned in Remark 3, a number of these obstructions re-quire Λ − to admit an augmentation, and thus in particular Λ − must be non-stabilized. However, as mentioned in Remark 5, it is possible for the satelliteof a Legendrian Λ to admit an augmentation even if Λ does not. So thecontrapositive of Theorem 9 provides a potential strategy for further obstruc-tions to the existence of a Lagrangian cobordism when Λ − does not admitan augmentation. For example, motivated by Obstruction (4) in Section 2.3,one can ask: Can a count of augmentations give an obstruction to the ex-istence of a Lagrangian concordance from S (Λ − , P ) to S (Λ + , P ) and therebyobstruct the existence of a Lagrangian concordance from Λ − to Λ + ? In fact,this augmentation count will not likely provide a further obstruction: a simplecomputation shows that when Λ is stabilized enough, the number of augmen-tations of S (Λ , P ) only depends on the Legendrian pattern P . If trying topursue this path to obtain further obstructions to Lagrangian cobordisms, itis useful to keep in mind the following result of Ng that shows the DGA of thesatellite of a Legendrian Λ might only remember the underlying knot type ofΛ. Theorem 12 ([Ng01]) . Suppose Λ and Λ are stabilized Legendrian knots thatare of the same topological knot type and have the same Thurston-Bennequinand rotation numbers. For a Legendrian pattern P whose front intersects avertical line by two points, the DGAs of S (Λ , P ) and S (Λ , P ) are equivalent. Candidates for Non-Decomposable Lagrangian Cobordisms
Now that we have developed some ways to construct a Lagrangian cobor-dism through combinatorial moves and satellites, we state some theorems thatshow if a Lagrangian cobordism does exist, then it cannot be decomposable:this addresses Motivating Question (1). While we discuss these theorems, itis useful to keep in mind the known obstructions to Lagrangian cobordismsthat were mentioned in Section 2.3. Candidates for Non-decomposable Lagrangian Cobordisms fromNormal Rulings.
One simple way to show that two Legendrians Λ ± cannotbe connected by a decomposable Lagrangian cobordism comes from a countof “combinatorial” rulings. Roughly, a normal ruling of a Legendrian Λ is a“decomposition” of the front projection into pairs of paths from left cusps toright cusps such that(1) each pair of paths starts from a common left cusp and ends at a com-mon right cusp, has no further intersections, and bounds a topologicaldisk whose boundary is smooth everywhere other than at the cuspsand certain crossings called switches , and(2) near a switch, the pair of paths must be arranged as in one of thediagrams in Figure 16; observe that near the switch, vertical slices ofthe associated disks are either disjoint or the slices of one are containedin the slices of the other.Formal definitions of normal rulings can be found in, for example, [PC05] and[Fuc03]. Figure 16.
Normal rulings near a switch.As an illustration, all normal rulings of a particular Legendrian trefoil areshown in Figure 17.
Figure 17.
All normal rulings of this max tb positive Legen-drian trefoil.For each normal ruling R , let s ( R ) and d ( R ) be the number of switches andnumber of disks, respectively. By [PC05], the ruling polynomial is R Λ ( z ) = (cid:88) R z s ( R ) − d ( R ) , where the sum is over all the normal rulings, is an invariant of Λ under Leg-endrian isotopy. Normal rulings and augmentations are closely related eventhough they are defined in very different ways [Fuc03, FI04, NS06, Sab05]. ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 19
We have the following obstruction to decomposable cobordisms in terms ofnormal rulings.
Theorem 13. If Λ − has m normal rulings and Λ + has n normal rulings with m > n , then there is no decomposable Lagrangian cobordism from Λ − to Λ + .Proof. One can compare the number of normal rulings of the two ends for thedecomposable moves, as shown in Figure 18. Thus any normal ruling of Λ − induces a normal ruling of Λ + . Different normal rulings of Λ − induce differentnormal rulings of Λ + . Therefore the number of normal rulings of Λ + is biggerthan or equal to the number of normal rulings of Λ − . (cid:3) Figure 18.
Comparison of normal rulings for decomposable moves.Here is a strategy to show the existence of Legendrians that can be con-nected by a Lagrangian cobordism but not by one that is decomposable.
Strategy 1.
Choose Legendrians Λ ± such that: (1) Λ + has fewer graded normal rulings than Λ − , and (2) it is possible to construct, via a combination of the combinatorial con-structions from Section 3 or the satellite construction from Section 4,a Lagrangian cobordism from Λ − to Λ + .Remark . If Λ ± admit normal rulings, they will admit augmentations [FI04,Sab05]. From Section 2.3 obstructions (4)b, we then know that if there is aLagrangian cobordism from Λ − to Λ + , their ruling polynomials satisfy R Λ − ( q / − q − / ) ≤ q − χ (Σ) / R Λ + ( q / − q − / ) , for any q that is a power of a prime number. Satisfying condition (1) inStrategy 1 means that the polynomial on the right side of the inequality hasfewer terms than the polynomial on the left side of the inequality. If followingthis approach, it may be helpful to start by first finding a pair of positiveinteger coefficient polynomials that satisfy this inequality and condition (1) atthe same time. One can start with checking the ruling polynomials of smallcrossing number Legendrian knots on [CN13].5.2. Candidates for Non-decomposable Lagrangian Concordances fromTopology.
Observe that any decomposable Lagrangian concordance will be asmooth ribbon concordance. Thus it is potentially possible to use known ob-structions to ribbon concordances to find examples of smooth knots whose Leg-endrian representatives cannot be connected by a decomposable Lagrangian concordance: constructing a Lagrangian concordance between very stabilizedLegendrian representatives of these knot types, via the combinatorial tech-niques of Section 3 or geometric techniques of Section 4, will give an exampleof an exact Lagrangian concordance between knots that cannot be connectedby a decomposable Lagrangian concordance.For example, it is known [Gor81, Zem19, LZ19] that the only knot thatadmits a ribbon concordance to the unknot is the unknot itself. This has asa corollary the following obstruction to a decomposable Lagrangian concor-dance.
Theorem 14 ([CNS16, Theorem 3.2]) . If Λ − is topologically non-trivial and Λ + is topologically an unknot, then there is no decomposable Lagrangian con-cordance from Λ − to Λ + . Example 15.
To illustrate this theorem, here is a possible low crossing num-ber Legendrian knot to examine as Λ − . Consider the topological knot 6 whichis slice and ribbon. Its maximum tb Legendrian representative Λ (see Fig-ure 19) has tb = − r = 0. The DGA of this Legendrian A (Λ ) admitsan augmentation, and thus Λ does not admit a Lagrangian cap; see obstruc-tions (6) in Section 2.3. Since we are trying to construct a Legendrian Λ − thatcould be Lagrangian concordant to a stabilized unknot, which might have aLagrangian cap, we will add some stabilizations that will prevent augmenta-tions and thereby allow the possibility of a Lagrangian cap. If we now add apositive and a negative stabilization to Λ , we get a knot Λ ± with tb = − r = 0, which has no augmentation and is still topologically the knot 6 .If, by a sequence of moves in Section 3, one can construct a concordance fromΛ ± to the tb = − as many times as we wish resulting in tb (Λ − ) = t and r (Λ − ) = r and try,using the combinatorial constructions of Section 3, to construct a Lagrangianconcordance to Λ + , where Λ + is a Legendrian unknot with tb (Λ + ) = t and r (Λ + ) = r . If possible , such a construction would prove the existence of anon-decomposable Lagrangian concordance.
Figure 19.
Front diagram of Λ .There are additional results from topology that give obstructions to theexistence of ribbon concordances. For example, as shown by Gilmer [Gil84]and generalized by Friedl and Powell [FPar], if K − is ribbon concordant to ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 21
Figure 20.
Any Lagrangian concordance from the doubly sta-bilized Λ to the tb = − r = 0 Legendrian unknot wouldnecessarily be non-decomposable. K + , then the Alexander polynomial of K − divides the Alexander polynomialof K + . We can invoke these results in a strategy to show the existence ofnon-decomposable Lagrangian concordances. Strategy 2. (1)
Use results from smooth topology to find examples of smoothknots K ± such that K − is not ribbon concordant to K + . (2) For any pair of Legendrian representatives Λ ± of the knot type K ± ,even highly stabilized, use a combination of the combinatorial movesdescribed in Section 3 to construct a Lagrangian concordance from Λ − to Λ + . The example with the knot 6 given above is a concrete example to tryto apply this strategy with K − = 6 and K + being an unknot. A possibleexample when K + is non-trivial is the following. Example 16.
Let K − be the connect sum of the right- and left-handed tre-foils, K − = T r T l , and let K + be the connect sum of the figure 8 knot withitself, K + = F F . These knots are concordant but there is no ribbon concor-dance from K − to K + , as first shown by Gordon [Gor81]. Choose Legendrianrepresentatives Λ ± of K ± such that tb (Λ − ) = tb (Λ + ) and r (Λ − ) = r (Λ + ); notethat Λ ± can be very stabilized. If we can construct a Lagrangian concordancefrom Λ − to Λ + , via the combinatorial moves of Section 3, then we will haveshown the existence of a pair of Legendrians that are (exactly, orientably) La-grangian concordant but cannot be connected by a decomposable Lagrangianconcordance. Remark . Some known obstructions to ribbon concordance are, in fact, ob-structions to generalizations of ribbon concordance, namely strong homo-topy ribbon concordance and homotopy ribbon concordance . A stronghomotopy ribbon concordance is one whose complement is ribbon, i.e., can bebuilt with only 1-handles and 2-handles. A homotopy ribbon concordancefrom K − to K + is a concordance where the induced map on π of the comple-ment of K − (resp. K + ) injects (resp. surjects) into π of the complement ofthe concordance. Gordon [Gor81] showed thatribbon concordant = ⇒ strong homotopy ribbon concordant= ⇒ homotopy ribbon concordant. There have been a number of recent results obstructing (homotopy or stronghomotopy) ribbon concordances from Heegaard-Floer and Khovanov homol-ogy [Zem19, LZ19, MZer, GL20]; these results play an important role in Strat-egy 2.5.3.
Candidates for Non-decomposable Lagrangian Cobordisms fromGRID Invariants.
Some candidates for non-decomposable Lagrangian cobor-disms of higher genus come from knot Floer homology. Using the grid formu-lation of knot Floer homology [OST08], Ozsv´ath, Szab´o, and Thurston definedLegendrian invariants of a Legendrian link Λ ⊂ R , called GRID invariants,which are elements in the hat flavor of knot Floer homology of Λ ⊂ − S : (cid:98) λ + (Λ) , (cid:98) λ − (Λ) ∈ (cid:92) HF K ( − S , Λ) . For more background, see [OST08, MOS09].Baldwin, Lidman, and Wong [BLWar] have shown that these GRID in-variants can be used to obstruct the existence of decomposable Lagrangiancobordisms.
Theorem 17 ([BLWar, Theorem 1.2] ) . Suppose that Λ ± are Legendrian linksin R such that either (1) (cid:98) λ + (Λ + ) = 0 and (cid:98) λ + (Λ − ) (cid:54) = 0 , or (2) (cid:98) λ − (Λ + ) = 0 and (cid:98) λ − (Λ − ) (cid:54) = 0 .Then there is no decomposable Lagrangian cobordism from Λ − to Λ + .Remark . By [BVVV13], in the standard contact manifold R , the GRIDinvariants agree with the LOSS invariant [LOSS09]. The LOSS invariant isfunctorial on Lagrangian concordances by [BS18, BSar]. Thus Theorem 17would also obstruct the existence of general Lagrangian concordances andnot only the decomposable ones. To find non-decomposable cobordisms usingobstructions from [BLWar], we should focus on non-zero genus cobordisms.Using the facts that the GRID invariants are non-zero for the tb = − (cid:98) λ + (Λ + ) (resp. (cid:98) λ − (Λ + )) vanish for positively (nega-tively) stabilized Legendrian links, Theorem 17 gives the following corollary. Corollary 18 ([BLWar, Corollaries 1.3, 1.4]) . (1) If Λ ⊂ R is a Legen-drian link such that (cid:98) λ + (Λ) = 0 or (cid:98) λ − (Λ) = 0, then there is no decom-posable Lagrangian filling of Λ.(2) Suppose Λ ± are Legendrian links such that either(a) (cid:98) λ + (Λ − ) (cid:54) = 0 and Λ + is the positive stabilization of a Legendrianlink, or(b) (cid:98) λ − (Λ − ) (cid:54) = 0 and Λ + is the negative stabilization of a Legendrianlink.Then there is no decomposable Lagrangian cobordism from Λ − to Λ + .This provides another strategy to show the existence of Legendrians Λ ± that are Lagrangian cobordant but cannot be connected by a decomposableLagrangian cobordism. ONSTRUCTIONS OF LAGRANGIAN COBORDISMS 23
Strategy 3. (1)
Find Legendrians Λ ± satisfying the GRID invariants con-ditions of Corollary 18 and Theorem 17 such that there are no knownobstructions, as described in Section 2.3, to the existence of a La-grangian cobordism from Λ − to Λ + . (2) Use a combination of the combinatorial moves described in Section 3to construct a Lagrangian cobordism from Λ − to Λ + . Example 19.
Concrete examples mentioned in [BLWar, Section 4.1] can beused for Strategy 3. Let Λ , Λ be the Legendrian m (10 ) knots and Legen-drian m (12 n ) knots shown in [NOT08, Figures 2 and 3]. Modify them witha pattern shown in [BLWar, Figure 13] to get Λ (cid:48) and Λ (cid:48) , which are of knottype m (12 n ) and m (14 n ) (or its mirror), respectively. For i, j = 0 , tb (Λ (cid:48) i ) = tb (Λ i ) + 2 and r (Λ (cid:48) i ) = r (Λ i ). There is no decomposableLagrangian cobordism from(1) Λ to Λ (cid:48) , or(2) Λ to Λ (cid:48) . If we can construct, using the combinatorial techniques of Section 3, a La-grangian cobordism (necessarily of genus 1) from Λ to Λ (cid:48) or from Λ to Λ (cid:48) ,then we will have found a non-decomposable Lagrangian cobordisms. Example 20.
In [BLWar, Section 4.3], the authors provide an infinite fam-ily of pairs of Legendrian knots where there does not exist a decomposableLagrangian cobordism between them.
Remark . In Strategies 2 and 3, we emphasized the construction of La-grangian cobordisms using the combinatorial techniques of Section 3. It wouldbe interesting to know if the geometric constructions of Section 4 could alsobe used to show the existence of a Lagrangian concordance/cobordism fromthe theory of normal rulings, topology, or grid invariants, that are known tonot be decomposable.5.4.
Non-decomposable Candidates through Surgery.
An additionalstrategy to show the existence of a non-decomposable Lagrangian filling comesfrom understanding properties of the contact manifold that is obtained fromsurgery on the Legendrian knot. In particular, Conway, Etnyre, and Tosun[CETar] have detected a relationship between Lagrangian fillings of a Legen-drian and symplectic fillings of the contact manifold obtained by performinga particular type of surgery on the Legendrian.
Theorem 21 ([CETar, Theorem 1.1]) . There is a Lagrangian disk filling of Λ + if and only if the contact +1 -surgery on Λ + ⊂ R ⊂ S produces a contactmanifold that is strongly symplectically fillable. If Λ + has a decomposableLagrangian filling, then the filling can be taken to be Stein. In fact, [CETar] also shows that a filling will be a Stein filling if and onlyif Λ + bounds a regular Lagrangian disk: a Lagrangian disk is regular if thereis a Liouville vector field that is tangent to the disk. Any decomposableLagrangian filling is regular.
We now see another strategy to construct a non-decomposable Lagrangianfilling.
Strategy 4.
Find a Legendrian Λ such that the +1 -surgery on Λ produces acontact manifold that is strongly symplectically fillable but does not admit aStein filling. An issue with this approach is a lack of examples: there are very few man-ifolds which carry strongly fillable but not Stein fillable contact structures.The main examples are the 1 /n surgeries on the positive and negative trefoils;see works by Ghiggini [Ghi05] and Tosun [Tos20]. However it is not obvi-ous whether any of these contact structures are a contact +1 surgery on aLegendrian knot in S . 6. Conclusion
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