Exact Lagrangian Fillings of Legendrian torus links
Abstract.
For a Legendrian torus knot or link with maximal ThurstonBennequin number, Ekholm, Honda, and Kálmán [EHK] constructed exact Lagrangian fillings, where is the th Catalan number. We show that these exact Lagrangian fillings are pairwise nonisotopic through exact Lagrangian isotopy. To do that, we compute the augmentations induced by the exact Lagrangian fillings to and distinguish the resulting augmentations.
1. Introduction
A Legendrian submanifold in the standard contact manifold , where , is a dimensional closed manifold such that everywhere. An exact Lagrangian filling of in the symplectization manifold is a dimensional surface that is cylindrical over when is sufficiently large. See Section 2.2 for the detailed definition and Figure 1.2 for a picture.
In this paper, we study oriented exact Lagrangian fillings of the Legendrian torus links with maximal ThurstonBennequin number (). When is even, we also require the link to have the right Maslov potential such that Reeb chords in Figure 1.2 are in degree (see Section 2.1 for detailed definitions). Ekholm, Honda, and Kálmán [EHK] gave an algorithm to construct exact Lagrangian fillings of the Legendrian torus link as follows. Starting with a Lagrangian projection of as shown in Figure 1.2, we can successively resolve crossings in any order through pinch moves (see Figure 1.3), which correspond to saddle cobordisms. As a result, we get two Legendrian unknots, which admit minimum cobordisms as shown in Figure 1.3. Concatenating the saddle cobordisms with these two minimum cobordisms, we get an exact Lagrangian filling of .
Different orders of resolving crossings may give different exact Lagrangian fillings of up to exact Lagrangian isotopy. Given a permutation of , write for the exact Lagrangian filling achieved by doing successive pinch moves at , , …, , respectively, and then concatenating with the two minimum cobordisms. Observe that two permutations may give isotopic exact Lagrangian fillings. For instance, let be the Legendrian torus knot and consider the exact Lagrangian fillings of that correspond to permutations and , respectively. Since the saddles corresponding to the pinch moves at and are disjoint when projected to , one can use a Hamiltonian vector field in the direction to exchange the heights of these two saddles. Therefore, the two fillings and are Hamiltonian isotopic and thus are exact Lagrangian isotopic. In general, for the Legendrian torus link , given any numbers such that , two permutations and , where only and are interchanged, give the same exact Lagrangian fillings of up to exact Lagrangian isotopy. Taking all the permutations of modded out by this relation, we obtain exact Lagrangian fillings of , where
is the th Catalan number. In this paper, we prove the following theorem.
Theorem 1.1 (see Theorem LABEL:knot and Corollary LABEL:link).
The exact Lagrangian fillings that come from the algorithm in [EHK] are all of different exact Lagrangian isotopy classes. In other words, the Legendrian torus link has at least exact Lagrangian fillings up to exact Lagrangian isotopy.
Shende, Treumann, Williams and Zaslow [STWZ] have also constructed exact Lagrangian fillings of the Legendrian torus knot using cluster varieties and shown that they are distinct up to Hamiltonian isotopy. They remarked that these are presumably the same as fillings obtained by [EHK]. But we do not resolve this issue here.
Remark 1.2.
We will see from Corollary LABEL:link that the conclusion of Theorem 1.1 for the case even can be derived from the result for the case when is odd. Therefore, for most of the paper, we focus on the case when is odd, which means is a knot.
Inspired by [EHK], we use augmentations to distinguish the exact Lagrangian fillings of the Legendrian torus knot . In order to talk about augmentations, we first introduce the ChekanovEliashberg differential graded algebra (DGA) of a Legendrian knot , which is a chain complex . This is an invariant of Legendrian submanifolds introduced by Chekanov [Che] and Eliashberg [Eli] in the spirit of symplectic field theory [EGH]. The underlying algebra of the ChekanovEliashberg DGA is freely generated by Reeb chords of over a commutative ring , where Reeb chords of correspond to double points of the Lagrangian projection of . The differential is defined by a count of rigid holomorphic disks with boundary on , taken with coefficients in . In general, the ChekanovEliashberg DGA of is defined with coefficients. For our purpose, it suffices to consider the DGA with coefficients, which means ignoring the orientations of moduli spaces of holomorphic disks. An augmentation of to a commutative ring is a DGA map . As shown in [EHK], an exact Lagrangian filling of gives an augmentation of by counting rigid holomorphic disks with boundary on . Moreover, by [EHK, Theorem 1.3], exact Lagrangian isotopic fillings give homotopic augmentations. Therefore, in order to distinguish two fillings, we only need to show their induced augmentations are not chain homotopic.
In [EHK], the authors distinguished all the exact Lagrangian fillings from the algorithm when by computing all the augmentations of the Legendrian torus knot to and finding that they are pairwise nonchain homotopic. However, when , a computation shows that the number of augmentations of the DGA to is much less than the Catalan number .
In this paper, for an exact Lagrangian filling of the Legendrian torus knot , we consider its induced augmentation of to , where is the singular homology of . Note that and thus it is natural to count the rigid holomorphic disks in with boundary on with coefficients. However, the computation of augmentations is not as easy as the case with coefficients. For each exact Lagrangian filling from the [EHK] algorithm, we give a combinatorial formula of the induced augmentation of to . Observing from the formula, we find a combinatorial invariant to show that the augmentations are pairwise nonchain homotopic. In this way, we distinguish all the exact Lagrangian fillings of the Legendrian torus knot up to exact Lagrangian isotopy.
Outline. In Section 2, we review the ChekanovEliashberg DGA of a Legendrian submanifold and the DGA maps induced by an exact Lagrangian cobordism. In Section 3, we compute all the augmentations of the Legendrian torus knot to induced by the exact Lagrangian fillings and prove that all the resulting augmentations are distinct up to chain homotopy. In the end, we prove Theorem 1.1 for the case even as a corollary.
Acknowledgement. The author would like to thank Lenhard Ng for introducing the problem and many enlightening discussions. This work was partially supported by NSF grants DMS0846346 and DMS1406371.
2. Preliminaries
In Section 2.1, we review the definition of ChekanovEliashberg DGA of Legendrian submanifolds in and its extension to the setting of multiple base points. For the purpose of computing augmentations in Section 3.1, the definition of DGA we use here is slightly different from the versions in [NgRSFT] and [NRSSZ], where the underlying algebra is completely noncommutative. In our definition, we allow elements in the coefficient ring to commute with the elements corresponding to Reeb chords. This is a generalization of the definition of ChekanovEliashberg DGA from [ENS]. See [EENS, Section 2.3.2] for further discussions. In Section 2.2, we review the DGA map induced by an exact Lagrangian cobordism and revise coefficients of this map for the purpose of computing augmentations in Section 3.1.
2.1. ChekanovEliashberg DGA
Let be a Legendrian submanifold in , where . There are two projection diagrams associated to via the Lagrangian projection and the front projection respectively. As an example, a front projection and a Lagrangian projection of the Legendrian trefoil are shown in Figure 2.1. Moreover, starting from a front projection of , Ng [Ngresolve] gave an algorithm to get a Lagrangian projection of by smoothing the cusps of the front projection in a way shown in Figure 2.2.
Let be an oriented Legendrian link with connected components. Now let us define the ChekanovEliashberg DGA of . To simplify the definition of grading, we assume throughout the paper that the rotation number of is 0. Note that all the Legendrian torus links we consider have rotation number .
The underlying algebra. The underlying algebra is a unital graded algebra freely generated by Reeb chords of over , where is any basis of . A Reeb chord of in is a vertical line segment ( direction) with both ends on endowed with an orientation in the positive direction. Reeb chords of are in correspondence to double points of , which by Ng’s algorithm correspond to the crossings and right cusps of .
To define the grading of Reeb chords, we work on the front projection . Write for the set of cusps of , which divides into strands (ignoring double points). The Maslov potential is a function that assigns an integer to each strand such that around each cusp, the Maslov potential of the lower strand is one less than that of the upper strand. This is well defined up to a global shift on each component of . When is even, we can choose a Maslov potential of the Legendrian torus link such that for any Reeb chord as labeled in Figure 1.2, the upper strand and the lower strand of have the same Maslov potential. Once the Maslov potential is fixed, the grading of a Reeb chord that corresponds to a crossing of can be defined by
where is the upper strand of the crossing and is the lower strand of the crossing. The grading of Reeb chords that correspond to right cusps of are defined to be . Extend the definition of grading to by setting for and using the relation .
Differential. The differential is defined by counting rigid holomorphic disks in with boundary on .
For any Reeb chords of , define to be the moduli space of holomorphic disks:
with the following properties:

is a dimensional unit disk with points removed from the boundary and the points are labeled in counterclockwise order.

and the neighborhood of in the image of covers exactly one positive quadrant of the crossing (see Figure 2.3).

, for , and the neighborhood of in the image of covers exactly one negative quadrant of the crossing (see Figure 2.3).
According to [Che], we have the following dimension formula:
When , the disk is called rigid. There are finitely many rigid holomorphic disks and hence we can count the number of rigid holomorphic disks.
In order to count with coefficients, we want to take the homology class of the boundary of rigid disks in . However, for any rigid holomorphic disk , the boundary is not closed. Therefore, we introduce capping paths first. Equip each connected component with a reference point , for . For each , pick a path in that goes from to . For each Reeb chord of from to , the capping path is defined by concatenating the following four paths:

a path on from to ,

the chosen path connecting to ,

the chosen path connecting to ,

a path on from to .
See Figure 2.4 for an example of a capping path.
After associating each Reeb chord with a capping path, for any rigid holomorphic disk , the curve
is a loop in . Notice that . Thus we can view the homology class as in .
Now we are ready to define the differential of the ChekanovEliashberg DGA of .
Definition 2.1.
For any Reeb chord of , the differential is defined by:
(2.1) 
The definition of differential can be extended to by setting for , and using Leibniz rule
According to [Che], the map is a differential in degree . Moreover, up to stable tame isomorphism, the ChekanovEliashberg DGA is an invariant of under Legendrian isotopy.
Remark 2.2.
In general, for any commutative ring and a ring homomorphism , we define the ChekanovEliashberg DGA as a tensor product of the DGA with the ring :
where the ring homomorphism gives the structure of a module over .
Now we give a combinatorial definition of the differential of . Assign an orientation and label each component , for , with a base point , which is different from the reference point and ends of Reeb chords. For a union of oriented curves in , we associate it with a monomial in
(2.2) 
where is the number of times goes through counted with sign. The sign is positive if goes through following the link orientation and is negative if goes through against the link orientation. In particular, for a rigid holomorphic disk , we have
where is short for Plugging it into the formula (2.1), we get a combinatorial definition of the differential. It seems to depend on the choice of capping paths. However, we have the following wellknown proposition.
Proposition 2.3.
Let be a Legendrian link and , be two families of capping paths of Reeb chords of . The corresponding DGAs and are isomorphic.
Proof.
For a Reeb chord of , we have
For each Reeb chord , concatenate with and get a closed curve, denoted by . It is not hard to check that the map
is a chain map and is an isomorphism. ∎
Note that for an oriented link with minimal base points (i.e. each component has exactly one base point), we can choose a family of capping paths such that none of them pass through any base point. Therefore, we only need to count intersections of the disk boundary and base points. Thanks to Proposition 2.3, we can define the ChekanovEliashberg DGA of to be a unital graded algebra over generated by Reeb chords of endowed with a differential given by
where is defined by formula (2.2). This DGA is denoted by as well.
Example 2.4.
For the Legendrian torus knot with a single base point as shown in Figure 2.5. The underlying algebra is generated by Reeb chords over . Reeb chords and are in degree and the rest of Reeb chords are in degree . The differential is given by:
The definition of DGA of Legendrian link can be generalized to the case where there are more than one base point on some components of the link. Let be an oriented Legendrian link and be a set of points on such that each component of has at least one point in the set and the set does not include any end of any Reeb chord of . For a union of paths , associate it with a monomial in , where is defined similar as above. The DGA is a unital graded algebra generated by Reeb chords of over endowed with a differential given by
2.2. The DGA map induced by exact Lagrangian cobordisms
According to [EHK], the ChekanovEliashberg DGA acts functorially on exact Lagrangian cobordisms. We first recall the definition of exact Lagrangian cobordisms.
Definition 2.5.
Let and be Legendrian submanifolds in , where . An exact Lagrangian cobordism from to is a dimensional surface in such that there exists such that is

cylindrical over on the positive end, i.e. ;

cylindrical over on the negative end, i.e. ;

compact in ,
and for some function . (See Figure 2.7.)
When is empty, the surface satisfying the conditions above is called an exact Lagrangian filling of .
By [EHK], an exact Lagrangian cobordism from to gives a DGA map from to with coefficients. Therefore, an exact Lagrangian filling of a Legendrian submanifold , which can be viewed as a cobordism from the empty set to , gives a DGA map from to the trivial DGA , which is an augmentation of to .
For the purpose of computing augmentations of the Legendrian torus knots in Section 3.1, we revise the coefficient ring of the DGA map induced by exact Lagrangian cobordisms from [EHK]. In stead of using coefficients, we will show the following proposition:
Proposition 2.6.
Let and be Legendrian submanifolds in and be a connected exact Lagrangian cobordism from to . Assume that is a connected exact Lagrangian cobordism from to some other Legendrian link and is the concatenation of and as shown in Figure 2.7. The exact Lagrangian cobordism induces a DGA map
with coefficients.
Note that when is an exact Lagrangian cylinder over , this map agrees with the DGA map introduced by [EHK]. The proof of Proposition 2.6 follows [EHK, Section 3]. Our revision of the coefficient ring is based on a different choice of capping paths of and . The capping paths of are chosen on in [EHK] while we choose capping paths of on and capping paths of on . For the rest of the section, we will describe this DGA map.
The inclusion map makes it natural to define the DGA . The underlying algebra
is generated by Reeb chords of over the ring . Given that is connected, we can choose a family of capping paths for on . Therefore, for any rigid holomorphic disk counted by , it is natural to take the homology class of in . Hence the differential coefficients of are in . In addition, the DGA does not depend on the choice of capping paths on for a similar reason as Proposition 2.3. The DGA is defined similarly.
The DGA map induced by is a composition of two maps. The first map
is induced by the inclusion map . It is not hard to show is a DGA map. The second map
is defined by counting rigid holomorphic disks in with boundary on .
Fix an almost complex structure on which is adjusted to the symplectic form (see [EHK, Section 3.2] for details). For a Reeb chord of and Reeb chords of , define to be the moduli space of holomorphic disks:
with the following properties:

is a dimensional unit disk with points removed. The points are arranged on the boundary of the disk counterclockwise.

The image of is asymptotic to a strip around .

The image of is asymptotic to a strip around for .
By [CEL], there is a corresponding dimension formula:
When , the holomorphic disk is called rigid. For each rigid holomorphic disk , concatenate the image of the disk boundary with the capping paths of corresponding Reeb chords on and get
which is a loop in . Hence we can take the homology class of in , denoted by . The map
is defined as follows. For any Reeb chord of , the map maps to
The map is identity on . By [EHK, Section 3.5], the map is a DGA map.
Therefore, the exact Lagrangian cobordism induces a DGA map
3. Main Results
We consider the exact Lagrangian fillings of the Legendrian torus knot contructed from the [EHK] algorithm. Each filling can be achieved by concatenating successive saddle cobordisms with two minimum cobordisms. In Section 3.1, we combine results in [EHK] and Proposition 2.6 to write down combinatorial formulas for the DGA maps induced by a pinch move and a minimum cobordism. Composing all the DGA maps induced by ordered pinch moves and the two minimum cobordisms, we obtain a combinatorial formula for augmentations of to induced by exact Lagrangian fillings . In Section LABEL:equi, we find a combinatorial invariant to distinguish these resulting augmentations and hence we show that the exact Lagrangian fillings are distinct up to exact Lagrangian isotopy. As a corollary, we extend the result to the case is even.
3.1. Computation of augmentations
Consider the Lagrangian projection of the Legendrian torus knot with a base point and label the crossings in degree from left to right by as shown in Figure 3.1.
For each permutation of , the corresponding exact Lagrangian filling of the Legendrian torus knot is achieved in the following way:

Start with an exact Lagrangian cylinder over , denoted by . Label as .

For , concatenate from the bottom with a saddle cobordism corresponding to the pinch move at crossing and get a new exact Lagrangian cobordism . Label the new Legendrian submanifold after pinch move as .

Finally, use two minimal cobordisms, denoted by , to close up from the bottom and get the exact Lagrangian filling . To be consistent, let be the empty set.
By Proposition 2.6, for , each exact Lagrangian cobordism induces a DGA map:
The map that is induced by minimum cobordisms is well understood while the maps for that correspond to pinch moves are not. We will first study and give a geometric description of the DGA map that corresponds to a pinch move. Combining with [EHK], we will write down an explicit combinatorial formula for each , for .
To describe easily, we chop off the cylindrical top of and view it as a surface with boundary , denoted by as well. By Poincaré duality, we have . In particular, for each oriented curve in with ends on , which is an element in , there exists an element such that for any oriented loop in , the intersection number of and is . Thus, in order to know the homology class of a loop in , we only need to count the intersection number of each generator curve of with .
We choose the set of generator curves of as follows. Use coordinate to slice into a movie of diagrams (some of them may not be Legendrian diagrams). We study the trace of points on the diagram when is decreasing. For , the saddle cobordism flows all the points directly downward except ends of the Reeb chord . According to [Lin], the ends of the Reeb chord merge to a point , and then split into two points, labeled as and respectively. Now for , consider the trace of in , which is a flow line from to the bottom of . Concatenating it with the inverse trace of in , we get a curve in as shown in Figure 3.2. In addition, denote the trace of the base point in by . In this way, we have that is a set of generator curves of .
For each curve , where , Poincaré duality gives an element . Denote its dual in by . Therefore, for any union of paths in , the monomial associated to in is
where is intersection number of and counted with signs.
For , the map induced by the inclusion map is injective. A similar argument shows that for a union of paths in , the monomial associated to in counts intersections of with . Notice that the curves do not intersect . Hence the monomial in agrees with in .
Choose a family of capping paths for on for . By Proposition 2.6, for , each exact Lagrangian cobordism gives a DGA map :
which maps any Reeb chord of to