Lagrangian cobordisms and Legendrian invariants in knot Floer homology

TL;DR

This paper studies how Legendrian link invariants in knot Floer homology change under decomposable Lagrangian cobordisms, providing new tools to obstruct their existence.

Contribution

It establishes functorial properties of LOSS and GRID invariants under certain cobordisms, offering new computable obstructions.

Findings

LOSS and GRID invariants behave functorially under decomposable Lagrangian cobordisms

New obstructions to the existence of such cobordisms are derived

Results are effective for computational applications

Abstract

We prove that the LOSS and GRID invariants of Legendrian links in knot Floer homology behave in certain functorial ways with respect to decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on $\mathbb{R}^3$. Our results give new, computable, and effective obstructions to the existence of such cobordisms.

Tables

Table 1. Table 1. The settings in which various knot Floer obstructions to Lagrangian cobordisms have been established.

	[BS18a, BS18b]	[GJ19]	Present paper
For $L$ in symplectization of $({ℝ}^3,{\xi }_{\mathrm{std}})$	✓	✓	✓
For $L$ in any symplectization	✓	✓	✗
For $L$ in any Weinstein cobordism	✗	✓	✗
For any decomposable $L$	✓	✓	✓
For any regular $L$	✓	✓	✗
For any exact $L$	✓	✗	✗
For $g\left(L\right)=0$	✓	✓	✓
For $g\left(L\right)>0$	✗	✗	✓

Equations

$${\alpha }_{\mathrm{std}}=dz-ydx.$$

$${\Lambda }_{-},{\Lambda }_{+}\subset ({\mathbb{R}}^3,{\xi }_{\mathrm{std}}),$$

$$({\mathbb{R}}_t\times {\mathbb{R}}^3,d(e^t{\alpha }_{\mathrm{std}})).$$

$$tb({\Lambda }_{+})-tb({\Lambda }_{-})=-\chi (L)\text{ and }r({\Lambda }_{+})=r({\Lambda }_{-}).$$

$${\hat{\lambda }}^{+}(\Lambda ),{\hat{\lambda }}^{-}(\Lambda )\in \widehat{\mathrm{HFK}}(-S^3,\Lambda ),$$

$${\hat{\lambda }}^{+}(\Lambda )=0\text{ or }{\hat{\lambda }}^{-}(\Lambda )=0.$$

$$\hat{\mathfrak{L}}(\Lambda )\in \widehat{\mathrm{HFK}}(-S^3,\Lambda ).$$

$${\varphi }_{\pm }:\widehat{\mathrm{HFK}}(-S^3,\Lambda )\to \widehat{\mathrm{HFK}}(-S^3,\pm \Lambda )$$

$${\varphi }_{\pm }({\hat{\lambda }}^{\pm }(\Lambda ))=\hat{\mathfrak{L}}(\pm \Lambda ).$$

$$\widehat{\mathrm{GH}}(\mathbb{G})\cong \widehat{\mathrm{HFK}}(-S^3,\Lambda ).$$

$${\hat{\lambda }}^{+}(\mathbb{G}),{\hat{\lambda }}^{-}(\mathbb{G})\in \widehat{\mathrm{GH}}(\mathbb{G}).$$

$${\hat{\lambda }}^{\pm }(\Lambda ):={\hat{\lambda }}^{\pm }(\mathbb{G}).$$

$$\widetilde{\mathrm{GH}}(\mathbb{G})\cong \widehat{\mathrm{HFK}}(-S^3,\Lambda )\otimes V^{\otimes |\mathbb{G}|-|\Lambda |},$$

$$V={\mathbb{F}}_{0,0}\oplus {\mathbb{F}}_{-1,-1}$$

$${\tilde{\lambda }}^{+}(\mathbb{G}),{\tilde{\lambda }}^{-}(\mathbb{G})\in \widetilde{\mathrm{GH}}(\mathbb{G}),$$

$$\widehat{\mathrm{GH}}(\mathbb{G})\hookrightarrow \widetilde{\mathrm{GH}}(\mathbb{G})$$

$${\hat{\lambda }}^{\pm }(\Lambda )={\hat{\lambda }}^{\pm }(\mathbb{G})=0\text{ iff }{\tilde{\lambda }}^{\pm }(\mathbb{G})=0.$$

$${\Phi }_L:\widetilde{\mathrm{GH}}({\mathbb{G}}_{+})\to \widetilde{\mathrm{GH}}({\mathbb{G}}_{-})$$

$${\Phi }_L({\tilde{\lambda }}^{\pm }({\mathbb{G}}_{+}))={\tilde{\lambda }}^{\pm }({\mathbb{G}}_{-}).$$

$$(\chi (L),\frac{1}{2}(\chi (L)+|{\Lambda }_{-}|-|{\Lambda }_{+}|)),$$

$$({\mathcal{A}}_{{\Lambda }_{+}},{\partial }_{{\Lambda }_{+}})\to ({\mathcal{A}}_{{\Lambda }_{-}},{\partial }_{{\Lambda }_{-}}).$$

$$\widehat{\mathrm{HFK}}(-Y_{+},{\Lambda }_{+})\to \widehat{\mathrm{HFK}}(-Y_{-},{\Lambda }_{-})$$

$$(W,L):(Y_{-},{\Lambda }_{-})\to (Y_{+},{\Lambda }_{+}),$$

$$\{\text{decomposable}\}\subseteq \{\text{regular}\}\subseteq \{\text{exact}\},$$

$${\alpha }_{\mathrm{std}}=dz-ydx,$$

$$T_p\Lambda \subset ({\xi }_{\mathrm{std}}{)}_p\text{ for all }p\in \Lambda .$$

$$tb=\mathrm{wr}-\frac{1}{2}(c_{+}+c_{-})\text{ and }r=\frac{1}{2}(c_{-}-c_{+}),$$

$$tb(S_{\pm }(L))=tb(L)-1\text{ and }r(S_{\pm }(L))=r(L)\pm 1.$$

$$({\mathbb{R}}_t\times {\mathbb{R}}^3,d(e^t{\alpha }_{\mathrm{std}})),$$

$$d(e^t{\alpha }_{\mathrm{std}}){|}_L\equiv 0.$$

Full text

JAB was partially supported by NSF CAREER Grant DMS-1454865.

TL was partially supported by NSF DMS-1709702 and a Sloan Fellowship.

1. Introduction

Let $\xi_{\mathrm{std}}$ be the standard contact structure on $\mathbb{R}^{3}$, given by the kernel of the $1$-form

[TABLE]

A difficult problem in contact and symplectic geometry, which has attracted a great deal of attention in recent years, is to decide, given two Legendrian links

[TABLE]

whether there exists an exact Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$ in the symplectization

[TABLE]

In the smooth category, any two links are cobordant in $\mathbb{R}\times\mathbb{R}^{3}$, and the challenge is to determine the minimum genus among such cobordisms. The opposite is true in the Lagrangian setting, where the existence of an exact Lagrangian cobordism is constrained but its genus is completely determined by the classical Thurston–Bennequin and rotation numbers of the Legendrian links at the ends. Indeed, Chantraine showed in [Cha10] that if $L$ is an exact Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$, then

[TABLE]

An important goal, therefore, is to develop obstructions to the existence of exact Lagrangian cobordisms that are effective, meaning that they can obstruct such cobordisms where smooth topology and the classical invariants do not.

In this article, we restrict our attention to decomposable Lagrangian cobordisms, which are those that can be obtained as compositions of elementary cobordisms associated to Legendrian isotopies, pinches, and births, as shown in Figure 2. Decomposable cobordisms are exact, and constitute most known examples of exact Lagrangian cobordisms. It is open whether all exact Lagrangian cobordisms are decomposable.

Our main result, described in Sections 1.1-1.3, is that knot Floer homology provides effective obstructions to decomposable Lagrangian cobordisms. Symplectic Field Theory also furnishes various obstructions to exact Lagrangian cobordisms; see [EHK16, CNS16, Pan17, CDGG15, ST13]. One advantage of our knot Floer obstructions is that they are generally much easier to compute than those coming from SFT. Moreover, we show that knot Floer homology obstructs decomposable cobordisms in cases where the SFT invariants do not (and vice versa).

As discussed in Section 1.4, there is an existing body of work [BS18a, BS18b, GJ19] showing that knot Floer homology effectively obstructs Lagrangian cobordisms of genus-zero in various settings. Ours is the first result that shows that knot Floer homology can effectively obstruct Lagrangian cobordisms of positive genus.

1.1. Obstructions

In [OSzT08], Ozsváth, Szabó, and Thurston used the combinatorial grid diagram formulation of knot Floer homology [MOS09] to define invariants of Legendrian links in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$. These so-called GRID invariants assign to such a Legendrian link $\Lambda$ two elements in the hat flavor of the knot Floer homology of $\Lambda\subset-S^{3}$,111The GRID invariant is defined for knots in $S^{3}$, but a Legendrian knot in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$ can be viewed naturally as a Legendrian in the standard contact structure on $S^{3}$. We follow the conventions of [OSzT08] and view these invariants as living in $\widehat{\mathrm{HFK}}(S^{3},m(\Lambda))$, which we identify with $\widehat{\mathrm{HFK}}(-S^{3},\Lambda)$.

[TABLE]

which depend only on the Legendrian isotopy class of $\Lambda$. These elements are effective invariants in that they can distinguish Legendrian links that are not isotopic but have the same classical invariants (see [NOT08], for example), and are combinatorially computable.

Remark 1.1**.**

The Maslov (or Alexander) gradings of the classes $\widehat{\lambda}^{\pm}(\Lambda)$ recover the Thurston–Bennequin and rotation numbers of $\Lambda$ (see Section 2.2).

We prove that the GRID invariants are well-behaved under decomposable Lagrangian cobordisms. As explained in Section 1.3, this provides effective obstructions to such cobordisms.

Theorem 1.2**.**

Suppose $\Lambda_{-},\Lambda_{+}$ are Legendrian links in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$ such that either

• •

$\widehat{\lambda}^{+}(\Lambda_{+})=0$* and $\widehat{\lambda}^{+}(\Lambda_{-})\neq 0$, or*

• •

$\widehat{\lambda}^{-}(\Lambda_{+})=0$* and $\widehat{\lambda}^{-}(\Lambda_{-})\neq 0$.*

Then there is no decomposable Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$.

Theorem 1.2 has the following corollaries.

Corollary 1.3**.**

Suppose $\Lambda$ is a Legendrian link in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$ such that either

[TABLE]

Then there is no decomposable Lagrangian filling of $\Lambda$.

As explained in Section 3.3, this follows from the fact that the GRID invariants are nonzero for the $tb=-1$ Legendrian unknot.

Corollary 1.4**.**

Suppose $\Lambda_{-},\Lambda_{+}$ are Legendrian links in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$ such that either

• •

$\widehat{\lambda}^{+}(\Lambda_{-})\neq 0$* and $\Lambda_{+}$ is the positive stabilization of a Legendrian link, or*

• •

$\widehat{\lambda}^{-}(\Lambda_{-})\neq 0$* and $\Lambda_{+}$ is the negative stabilization of a Legendrian link.*

Then there is no decomposable Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$.

This follows immediately from the fact (see Proposition 2.2) that the elements $\widehat{\lambda}^{+}$ and $\widehat{\lambda}^{-}$ vanish for positively and negatively stabilized Legendrian links, respectively.

In [LOSSz09], Lisca, Ozsváth, Stipsicz, and Szabó used open book decompositions to define a knot Floer invariant of Legendrian knots in any closed contact 3-manifold. For a Legendrian knot $\Lambda\subset(\mathbb{R}^{3},\xi_{\mathrm{std}})$, their so-called LOSS invariant also takes the form of an element

[TABLE]

Although the LOSS invariant is not algorithmically computable, Baldwin, Vela-Vick, and Vértesi proved in [BVV13] that it agrees with the GRID invariants, for Legendrian knots in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$. More precisely, given such a knot $\Lambda$, there are isomorphisms

[TABLE]

such that

[TABLE]

This gives corresponding versions of Theorem 1.2 and its corollaries for the LOSS invariant.

1.2. Proof

Theorem 1.2 follows from a similar result for the tilde version of the GRID invariants. We explain this below after providing a bit of additional background on the construction of the GRID invariants (see Section 2.2 for details).

A Legendrian link $\Lambda\subset(\mathbb{R}^{3},\xi_{\mathrm{std}})$ can be represented by a grid diagram $\mathbb{G}$. This grid diagram determines a combinatorially computable, bigraded chain complex whose grid homology agrees with the knot Floer homology of $\Lambda\subset-S^{3}$,

[TABLE]

There are two canonical cycles in this grid chain complex, representing elements

[TABLE]

The hat version of the GRID invariants discussed previously are defined by

[TABLE]

A specialization of this chain complex gives rise to the tilde version of grid homology, which agrees with the tilde flavor of knot Floer homology, and is related to the hat flavor by

[TABLE]

where

[TABLE]

is the two-dimensional vector space supported in the Maslov–Alexander bigradings indicated by the subscripts; $|\mathbb{G}|$ is the grid number of $\mathbb{G}$; and $|\Lambda|$ is the number of components of $\Lambda$. There are two canonical elements in this version of grid homology as well,

[TABLE]

which we refer to as the tilde version of the GRID invariants. Moreover, there is an injection

[TABLE]

that sends $\widehat{\lambda}^{\pm}(\mathbb{G})$ to $\widetilde{\lambda}^{\pm}(\mathbb{G})$ [NOT08]. In particular,

[TABLE]

Theorem 1.2 therefore follows immediately from our main technical result below, which states that the tilde versions of the GRID invariants satisfy a weak functoriality under decomposable Lagrangian cobordisms.

Theorem 1.5**.**

Suppose $\Lambda_{-},\Lambda_{+}$ are Legendrian links in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$ with grid representatives $\mathbb{G}_{-},\mathbb{G}_{+}$, respectively. Suppose there exists a decomposable Lagrangian cobordism $L$ from $\Lambda_{-}$ to $\Lambda_{+}$. Then there is a homomorphism

[TABLE]

such that

[TABLE]

This map has Maslov–Alexander bidegree

[TABLE]

where $|\Lambda_{\pm}|$ is the number of components of $\Lambda_{\pm}$.

Recall that a decomposable cobordism $L$ as in the theorem can be described as a composition of elementary cobordisms associated with Legendrian isotopies, pinches, and births. To prove Theorem 1.5, we define combinatorially computable maps on the tilde version of grid homology for each of these elementary cobordisms (the maps corresponding to Legendrian isotopies were defined in [OSzT08]), and show that these elementary maps preserve the tilde GRID invariant. We then define $\Phi_{L}$ to be the appropriate composition of these elementary maps.

In [Juh16, Zem19], Juhász and Zemke independently proved that decorated link cobordisms between pointed links induce well-defined maps on knot Floer homology. (They defined these maps differently, but showed in [JZ19] that their definitions agree for the tilde flavor of $\mathrm{HFK}$.) A grid diagram naturally specifies a pointed link, and the sequence of grid moves corresponding to a decomposition of a Lagrangian cobordism $L$ from $\Lambda_{-}$ to $\Lambda_{+}$ into elementary pieces specifies a decorated cobordism between pointed copies of $\Lambda_{\pm}$. We believe that the map $\Phi_{L}$ agrees with the functorial map of Juhász–Zemke associated to this decorated cobordism, but do not prove this here.

1.3. Effectiveness

In Section 4, we give several examples that show that Theorem 1.2 can be used to obstruct decomposable Lagrangian cobordisms where the classical invariants and smooth topology do not. In particular, we prove the following in Section 4.3.

Theorem 1.6**.**

For each $g\in\mathbb{Z}_{\geq 0}$, there are Legendrian knots $\Lambda_{-},\Lambda_{+}\subset(\mathbb{R}^{3},\xi_{\mathrm{std}})$ such that

• •

there is a smooth cobordism of genus $g$ in $\mathbb{R}\times\mathbb{R}^{3}$ between $\Lambda_{-}$ and $\Lambda_{+}$,

• •

$\mathit{tb}(\Lambda_{+})-\mathit{tb}(\Lambda_{-})=2g$* and $\mathit{r}(\Lambda_{+})=\mathit{r}(\Lambda_{-})$,*

• •

$\widehat{\lambda}^{+}(\Lambda_{+})=0$* and $\widehat{\lambda}^{+}(\Lambda_{-})\neq 0$.*

The last item implies that there is no decomposable Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$.

As alluded to above, Symplectic Field Theory [EGH00] also provides effective obstructions to Lagrangian cobordisms. The most-studied such SFT obstruction comes from the Chekanov–Eliashberg DGA [Che02, Eli98], which assigns to a Legendrian knot $\Lambda\subset(\mathbb{R}^{3},\xi_{\mathrm{std}})$ a differential graded algebra $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$, which is an invariant of the Legendrian isotopy class of $\Lambda$, up to stable tame isomorphism. This DGA is said to be trivial if it is stable tame isomorphic to a DGA in which the unit is a boundary. Ekholm, Honda, and Kálmán proved in [EHK16] that an exact Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$ induces a DGA morphism

[TABLE]

Therefore, if the first DGA is trivial and the second is nontrivial then there cannot exist such a cobordism. It can be difficult to determine whether these DGAs are trivial, meaning that this obstruction can be hard to apply in practice. By contrast, there is a simple algorithm to decide whether the GRID invariants vanish and apply Theorem 1.2.

Another advantage of the GRID invariants is that the elements $\widehat{\lambda}^{+}$ and $\widehat{\lambda}^{-}$ are preserved by negative and positive Legendrian stabilization, respectively. This implies, for example, that for any pair $\Lambda_{-},\Lambda_{+}$ of Legendrian knots as in Theorem 1.6, the GRID invariants also obstruct the existence of a decomposable Lagrangian cobordism from any negative stabilization of $\Lambda_{-}$ to any negative stabilization of $\Lambda_{+}$. By contrast, the Chekanov–Eliashberg DGA is trivial for stabilized knots, and therefore cannot obstruct such cobordisms.

We should point out that there are also examples for which the DGA obstructs decomposable Lagrangian cobordisms where the GRID invariants do not (see Section 4.4).

1.4. Antecedents

As mentioned above, there are a few prior works that use knot Floer homology to obstruct genus zero Lagrangian cobordisms; such cobordisms are called Lagrangian concordances, and are automatically exact.

In [BS18a], for instance, Baldwin and Sivek defined an invariant of Legendrian knots in arbitrary closed contact 3-manifolds using monopole knot homology, and showed that it satisfies functoriality with respect to Lagrangian concordances in symplectizations of such manifolds. They then proved in [BS18b] that there is an isomorphism between monopole knot homology and knot Floer homology that identifies their Legendrian invariant with the LOSS invariant. This implies that the LOSS invariant is well-behaved with respect to Lagrangian concordances, and, in particular, reproduces Theorem 1.2 for concordances between knots in the symplectization of $(\mathbb{R}^{3},\xi_{\mathrm{std}})$, without the assumption of decomposability.

The other notable result in this area is due to Golla and Juhász, who proved in [GJ19] that the LOSS invariant satisfies functoriality with respect to regular Lagrangian concordances in Weinstein cobordisms between closed contact 3-manifolds (which include symplectizations). More precisely, they showed that the functorial map (of Juhász–Zemke)

[TABLE]

associated to a decorated regular Lagrangian concordance $L$ in a Weinstein cobordism $W$,

[TABLE]

sends $\widehat{\mathfrak{L}}(\Lambda_{+})$ to $\widehat{\mathfrak{L}}(\Lambda_{-})$, for decorations consisting of two parallel arcs that partition the cylinder into disks. We note that regular cobordisms are exact and, in the symplectization of $(\mathbb{R}^{3},\xi_{\mathrm{std}})$, include decomposable cobordisms [CET19]; in brief,

[TABLE]

and it is open whether any of these inclusions are proper. The Golla–Juhász result therefore recovers Theorem 1.2 for concordances between knots in the symplectization of $(\mathbb{R}^{3},\xi_{\mathrm{std}})$, with the potentially weaker assumption of regularity.

As noted previously, what most differentiates the results in this paper from those in previous works is that ours apply to positive genus Lagrangian cobordisms as well as to concordances. The table below summarizes the different settings in which the various knot Floer obstructions to Lagrangian cobordism are known to hold.

1.5. Organization

Section 2 consists of background material. In Section 3, we prove Theorem 1.5, which, as described above, implies Theorem 1.2. In Section 4, we show via examples that our obstructions are effective, proving Theorem 1.6.

1.6. Acknowledgements

The authors thank Lenny Ng for his insights on the Chekanov–Eliashberg DGA and, in particular, for computing this DGA for the knot $m(10_{145})$. The authors also thank Caitlin Leverson, Marco Marengon, Lenny Ng, Peter Ozsváth, Yu Pan, Josh Sabloff, Steven Sivek, Zoltán Szabó, and Shea Vela-Vick for helpful conversations. The third author thanks North Carolina State University for their hospitality.

2. Background

In this section, we provide some background on Legendrian knots, Lagrangian cobordisms, knot Floer homology, and the GRID invariants.

2.1. Legendrian knots and Lagrangian cobordisms

Let $\xi_{\mathrm{std}}=\ker(\alpha_{\mathrm{std}})$ be the standard contact structure on $\mathbb{R}^{3}$, where

[TABLE]

as in the introduction. Recall that a smooth link $\Lambda\subset(\mathbb{R}^{3},\xi_{\mathrm{std}})$ is called Legendrian if

[TABLE]

We will primarily study Legendrian links up to Legendrian isotopy (and will frequently blur the distinction between Legendrian links and Legendrian link types). Furthermore, our Legendrian links will generally be oriented but we will often suppress the orientation from the notation.

We will typically represent a Legendrian link by its front diagram, which is its projection to the $xz$-plane, as illustrated in Figure 1. Note that a Legendrian link is completely determined by its front diagram; in particular, the crossing information is encoded in the slopes of the strands in the diagram (strands with more negative slope pass over strands with less negative slope). Front diagrams for Legendrian isotopic links are related by a sequence of Legendrian planar isotopies and Legendrian Reidemeister moves, shown in the first three diagrams of Figure 2.

There are two classical Legendrian isotopy class invariants: the Thurston–Bennequin number $\mathit{tb}$ and rotation number $\mathit{r}$. These can be computed from a front diagram by

[TABLE]

where $\mathrm{wr}$ denotes the writhe of the diagram, and $c_{+}$ and $c_{-}$ denote the number of upward and downward pointing cusps in the oriented diagram. Two important operations on Legendrian isotopy classes are positive and negative Legendrian stabilization. These operations are defined locally in terms of front diagrams as in Figure 3. In particular, given a Legendrian link $L$, its positive and negative stabilizations $S_{+}(L)$ and $S_{-}(L)$ are obtained by adding downward and upward pointing cusps, respectively, as shown in the figure. Note that

[TABLE]

Recall that the symplectization of $(\mathbb{R}^{3},\xi_{\mathrm{std}})$ is the symplectic 4-manifold

[TABLE]

and that an embedded surface $L$ in the symplectization is called Lagrangian if

[TABLE]

Suppose $\Lambda_{-},\Lambda_{+}$ are two Legendrian links in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$. A Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$ is an embedded Lagrangian surface $L$ in the symplectization such that

[TABLE]

for some $T>0$. This Lagrangian is said to be exact if there exists a function $f:L\to\mathbb{R}$ that is constant on the cylindrical ends and satisfies

[TABLE]

A Lagrangian cobordism of genus zero is called a Lagrangian concordance, and is automatically exact. As mentioned in the introduction, Chaintraine proved in [Cha10] that the existence of a Lagrangian cobordism $L$ from $\Lambda_{-}$ to $\Lambda_{+}$ implies that

[TABLE]

In particular, Lagrangian cobordisms (even concordances [Cha15]) are directed.

By work of Bourgeois, Sabloff, and Traynor [BST15], Chantraine [Cha10], Dimitroglou Rizell [Dim16], and Ekholm, Honda, and Kálmán [EHK16], there exists an elementary exact Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$ whenever $\Lambda_{+}$ is obtained from $\Lambda_{-}$ via Legendrian isotopy, a pinch, or a birth, as illustrated in Figure 2. Note that for a pinch, it is $\Lambda_{-}$ that in fact looks as if it has been obtained from pinching $\Lambda_{+}$. Topologically, these elementary cobordisms are annuli, saddles, and cups, respectively. Any composition of elementary cobordisms yields an exact Lagrangian cobordism, and an exact Lagrangian cobordism is called decomposable if it is isotopic through exact Lagrangians to such a composition [Cha12]. As mentioned in the introduction, it is open whether every exact Lagrangian cobordism is decomposable.

2.2. Knot Floer homology and the GRID invariants

We begin by reviewing the grid diagram formulation of knot Floer homology, following the conventions in [OSSz15]. See also [MOS09, MOSzT07].

A grid diagram $\mathbb{G}$ is an $n\times n$ grid of squares together with sets

[TABLE]

of markings in the squares such that each row and column of $\mathbb{G}$ contains exactly one $O$ marking and one $X$ marking (we omit the subscripts indexing these markings when convienient); $n$ is called the grid number of $\mathbb{G}$. We will think of $\mathbb{G}$ as a torus by identifying its top and bottom sides and its left and right sides in the standard way, so that the horizontal grid lines become horizontal circles and the vertical grid lines become vertical circles, as indicated in Figure 4.

A grid diagram specifies an oriented link in $\mathbb{R}^{3}$, obtained as the union of vertical segments from the $X$s to the $O$s in each column with horizontal segments from the $O$s to the $X$s in each row, such that vertical segments pass over horizontal ones, as shown in Figure 4. Conversely, every oriented link in $\mathbb{R}^{3}$ can be represented by a grid diagram in this way.

Suppose $\mathbb{G}$ is a grid diagram as above, representing an oriented link $L$. (The use of $L$ for links will only occur in this subsection, and hence should not cause confusion with Lagrangian cobordisms.) Let

[TABLE]

denote the vertical and horizontal circles of $\mathbb{G}$, respectively. The minus flavor of the grid chain complex,

[TABLE]

is generated by one-to-one correspondences between the vertical and horizontal circles. Equivalently, a generator is a set of $n$ intersection points between these circles where each intersection point in the set belongs to exactly one $\alpha$ circle and one $\beta$ circle. Letting $\SS(\mathbb{G})$ denote the set of generators, $\mathrm{GC}^{-}(\mathbb{G})$ is defined to be the free $\mathbb{F}[U_{1},\dots,U_{n}]$-module generated by the elements of $\SS(\mathbb{G})$, where each $U_{i}$ is a formal variable corresponding to the marking $O_{i}$ and $\mathbb{F}$ is the 2-element field.

Given $\mathbf{x},\mathbf{y}\in\SS(\mathbb{G})$, let $\mathrm{Rect}_{\mathbb{G}}(\mathbf{x},\mathbf{y})$ be the set of rectangles in $\mathbb{G}$ with the following properties. $\mathrm{Rect}_{\mathbb{G}}(\mathbf{x},\mathbf{y})$ is empty unless $\mathbf{x}$ and $\mathbf{y}$ coincide in exactly $n-2$ intersection points. An element $r\in\mathrm{Rect}_{\mathbb{G}}(\mathbf{x},\mathbf{y})$ is an embedded rectangle in the toroidal grid whose edges are arcs contained in the vertical and horizontal circles, and whose four corners are points in $\mathbf{x}\cup\mathbf{y}$. Moreover, we require that, with respect to the induced orientation on $\partial r$, every vertical edge of $r$ is directed from a point in $\mathbf{y}$ to a point in $\mathbf{x}$, and vice versa for horizontal edges; that is,

[TABLE]

(The astute reader may have noticed that this does not seem to line up with the usual convention in Lagrangian Floer homology, but we are in fact computing Heegaard Floer homology for $(-\mathbb{T}^{2},\alpha,\beta)$.) If $\mathrm{Rect}_{\mathbb{G}}(\mathbf{x},\mathbf{y})$ is non-empty then it contains exactly two rectangles, as illustrated in Figure 5. Let $\mathrm{Rect^{o}}_{\mathbb{G}}(\mathbf{x},\mathbf{y})$ denote the subset consisting of $r\in\mathrm{Rect}_{\mathbb{G}}(\mathbf{x},\mathbf{y})$ with

[TABLE]

The differential

[TABLE]

is the $\mathbb{F}[U_{1},\dots,U_{n}]$-module endomorphism defined on $\SS(\mathbb{G})$ by

[TABLE]

where $O_{i}(r)$ denotes the number of times the marking $O_{i}$ appears in $r$.

This complex is equipped with two gradings, the Maslov grading and the Alexander, defined as follows. Consider the partial ordering on points in $\mathbb{R}^{2}$ given by

[TABLE]

if $p_{1}<q_{1}$ and $p_{2}<q_{2}$. Given two sets $P$ and $Q$ consisting of finitely many points in $\mathbb{R}^{2}$, let

[TABLE]

We symmetrize this quantity by defining

[TABLE]

A generator $\mathbf{x}\in\SS(\mathbb{G})$ can be viewed as a finite set of points in $\mathbb{R}^{2}$, as can the marking sets $\mathbb{X}$ and $\mathbb{O}$. It therefore makes sense to define

[TABLE]

The Maslov and Alexander gradings of a generator $\mathbf{x}$ are then given by

[TABLE]

where $|L|$ is the number of components of the link $L$. It follows that for $\mathbf{x},\mathbf{y}\in\SS(\mathbb{G})$ and $r\in\mathrm{Rect}_{\mathbb{G}}(\mathbf{x},\mathbf{y})$, the relative Maslov and Alexander gradings of these generators are given by

[TABLE]

These gradings are extended to gradings on the complex $\mathrm{GC}^{-}(\mathbb{G})$ by the rule that multiplication by any of the $U_{i}$ lowers Maslov grading by $2$ and Alexander grading by $1$. Note that the differential $\partial^{-}_{\mathbb{G}}$ lowers the Maslov grading by 1 and preserves the Alexander grading.

The grid homology of the grid diagram $\mathbb{G}$ is the Maslov–Alexander bigraded $\mathbb{F}[U_{1},\dots,U_{n}]$-module denoted by

[TABLE]

The grid diagram $\mathbb{G}$ is actually a multi-pointed Heegaard diagram for the link $L\subset S^{3}$, and the grid chain complex agrees with the minus version of the corresponding knot Floer chain complex. Therefore,

[TABLE]

Suppose $L$ has $\ell$ components. Label the markings so that $O_{1},\dots,O_{\ell}$ belong to the $\ell$ different components of $L$. Setting the corresponding $U_{i}$ equal to zero on the chain level results in a chain complex

[TABLE]

whose homology agrees with the hat flavor of knot Floer homology,

[TABLE]

Setting all of the $U_{i}$ to zero yields the tilde version of the grid complex,

[TABLE]

whose homology agrees with the tilde version of knot Floer homology, and is related to the hat flavor by

[TABLE]

where

[TABLE]

is the two-dimensional vector space supported in Maslov–Alexander bigradings $(0,0)$ and $(-1,-1)$. Finally, the quotient map

[TABLE]

induces an injection

[TABLE]

on homology [NOT08].

Suppose $\mathbb{G}$ is a grid diagram representing $L$. By changing all crossings in the associated link diagram, rotating $45$ degrees clockwise, smoothing the top and bottom pointing corners, and turning the left and right pointing corners into cusps, we obtain a front diagram for a Legendrian representative $\Lambda$ of $m(L)$, as indicated in Figure 4. We say that a grid diagram $\mathbb{G}$ represents a Legendrian link $\Lambda$ if the front diagram obtained from $\mathbb{G}$ in the manner above is isotopic to the front diagram for $\Lambda$. Every Legendrian link in $(\mathbb{R}^{3},\xi_{\mathrm{std}})$ can be represented by a grid diagram in this way.

Suppose $\mathbb{G}$ represents the smooth link $L$ and a Legendrian representative $\Lambda$ of $m(L)$ as above. As shown in [OSzT08], there are two canonical cycles

[TABLE]

consisting of the intersection points to the immediate upper right and lower left, respectively, of the markings in $\mathbb{X}$. These two generators give rise to cycles in the hat and tilde complexes as well, which we denote in the same way. The Maslov and Alexander gradings of these cycles are given by

[TABLE]

where $|\Lambda|$ is the number of components of $\Lambda$. In particular, the gradings of the two generators recover $\mathit{tb}(\Lambda)$ and $\mathit{r}(\Lambda)$. The hat and tilde versions of the GRID invariants of the Legendrian link $\Lambda$ are then defined [OSzT08] by

[TABLE]

and

[TABLE]

In particular,

[TABLE]

which implies that

[TABLE]

since $j_{*}$ is injective.555Unlike for the hat flavor of the GRID invariants, we do not denote $\widetilde{\lambda}^{\pm}(\mathbb{G})$ by $\widetilde{\lambda}^{\pm}(\Lambda)$ since these classes (and the group $\widetilde{\mathrm{GH}}(\mathbb{G})$) depend not only on the Legendrian link $\Lambda$ but also on the grid number $n$, as in (8).

Ozsváth, Szabó, and Thurston proved that $\widehat{\lambda}^{\pm}(\mathbb{G})$ are invariants of the Legendrian isotopy class of $\Lambda$. Specifically, if $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ are grid diagrams representing Legendrian isotopic links then there is an isomorphism [OSzT08, Theorem 1.1]

[TABLE]

of Maslov–Alexander bidegree $(0,0)$ that sends $\widehat{\lambda}^{\pm}(\mathbb{G}_{0})$ to $\widehat{\lambda}^{\pm}(\mathbb{G}_{1})$. This map is defined combinatorially, in terms of chain maps on the grid complex associated to grid diagram versions of the Legendrian Reidemeister moves. Their argument also gives rise to the following statement for the tilde flavor of the GRID invariants.

Proposition 2.1**.**

If $\mathbb{G}_{0}$ and $\mathbb{G}_{1}$ are grid diagrams representing Legendrian isotopic links then there is a homomorphism

[TABLE]

that sends $\widetilde{\lambda}^{\pm}(\mathbb{G}_{0})$ to $\widetilde{\lambda}^{\pm}(\mathbb{G}_{1})$. This map has Maslov–Alexander bidegree $(0,0)$.

Note that the homomorphism above may not be an isomorphism. The GRID invariants also behave as follows under stabilization [OSzT08, Theorem 1.3].

Proposition 2.2**.**

Suppose $\mathbb{G}$ is a grid representative of a Legendrian link $\Lambda$, and that $\mathbb{G}_{\pm}$ are grid representatives of the positive and negative Legendrian stabilizations $S_{\pm}(\Lambda)$, respectively. Then

[TABLE]

and

[TABLE]

The analogous statement holds for the tilde invariants, by (11).

3. Proofs of main results

To define the map $\Phi_{L}$ in Theorem 1.5 associated to a decomposable Lagrangian cobordism $L$, we first define maps associated to Legendrian isotopies, pinches, and births, as discussed in the introduction. The maps associated to Legendrian isotopies were defined previously by Ozsváth, Szabó, and Thurston in [OSzT08], and are described in Proposition 2.1, so we will restrict our attention below to the maps associated to pinches and births.

3.1. Pinches

Proposition 3.1**.**

Suppose $\Lambda_{+}$ is obtained from $\Lambda_{-}$ via a pinch move. For any grid diagrams $\mathbb{G}_{+}$ and $\mathbb{G}_{-}$ representing $\Lambda_{+}$ and $\Lambda_{-}$, respectively, there is a homomorphism

[TABLE]

that sends $\widetilde{\lambda}^{\pm}(\mathbb{G}_{+})$ to $\widetilde{\lambda}^{\pm}(\mathbb{G}_{-})$. This map has Maslov–Alexander bidegree

[TABLE]

where $|\Lambda_{\pm}|$ is the number of components of $\Lambda_{\pm}$.

Proof.

By Proposition 2.1, it suffices to show that there exist some grid diagrams $\mathbb{G}_{+}$ and $\mathbb{G}_{-}$ representing links Legendrian isotopic to $\Lambda_{+}$ and $\Lambda_{-}$, respectively, for which the conclusions of Proposition 3.1 hold. For this, note that there are grid diagrams $\mathbb{G}_{\pm}$ representing $\Lambda_{\pm}$ that are identical except for the positions of two markings in adjacent rows, as shown in Figure 6. Since $L$ is an oriented cobordism, these two special markings must either both be $X$s, which we refer to as Case I, or both be $O$s, which we refer to as Case II.

We may combine the grid diagrams $\mathbb{G}_{-}$ and $\mathbb{G}_{+}$ into a single toroidal diagram, as shown in Figure 7, which we will refer to as the combined diagram. From this perspective, the markings in $\mathbb{X},\mathbb{O}$ are fixed and $\mathbb{G}_{-}$ and $\mathbb{G}_{+}$ differ in a single horizontal circle. We denote these differing horizontal circles by $\beta$ and $\gamma$, as shown in Figure 7. Let $a$ and $b$ be the intersection points of $\beta$ with $\gamma$ shown in the figure. Below, we define the map $\Phi$ for each of Cases I and II.

3.1.1. Case I

For $\mathbf{x}\in\SS(\mathbb{G}_{+})$ and $\mathbf{y}\in\SS(\mathbb{G}_{-})$, let $\mathrm{Pent}(\mathbf{x},\mathbf{y})$ be the space of pentagons in the combined diagram with the following properties. $\mathrm{Pent}(\mathbf{x},\mathbf{y})$ is empty unless $\mathbf{x}$ and $\mathbf{y}$ coincide in exactly $n-2$ intersection points, where $n$ is the grid number of $\mathbb{G}_{\pm}$. An element $p\in\mathrm{Pent}(\mathbf{x},\mathbf{y})$ is an embedded pentagon in the toroidal diagram whose edges are arcs contained in the vertical and horizontal circles, and whose five corners are points in $\mathbf{x}\cup\mathbf{y}\cup\{a\}$. We require that, with respect to the induced orientation on $\partial p$, the boundary of this pentagon may be traversed as follows: start at the point in $\mathbf{x}$ on $\beta$ and proceed along an arc of $\beta$ until arriving at $a$; next, proceed along an arc of $\gamma$ until arriving at a point in $\mathbf{y}$; next, follow an arc of a vertical circle until arriving at a point in $\mathbf{x}$; next, proceed along an arc of a horizontal circle until arriving at a point in $\mathbf{y}$; finally, follow an arc of a vertical circle back to the initial point in $\mathbf{x}$. See Figure 8 for such pentagons. Let $\mathrm{Pent^{o}}(\mathbf{x},\mathbf{y})$ be the subset consisting of $p\in\mathrm{Pent}(\mathbf{x},\mathbf{y})$ with

[TABLE]

Let

[TABLE]

be the linear map defined on generators by counting such pentagons,

[TABLE]

Lemma 3.2**.**

$\phi$* is a chain map.*

Proof.

To show that $\phi$ is a chain map, we must prove the equality of coefficients,

[TABLE]

for every pair of generators $\mathbf{x}\in\SS(\mathbb{G}_{+})$ and $\mathbf{y}\in\SS(\mathbb{G}_{-})$. The coefficients on the left and right count concatenations of rectangles and pentagons from $\mathbf{x}$ to $\mathbf{y}$ of the forms $p*r$ and $r*p$, respectively, where $p$ is a pentagon of the sort used to define $\phi$, and $r$ is a rectangle of the sort used to define the differentials. Every domain in the combined diagram that decomposes as the juxtaposition of a rectangle and pentagon in this way admits exactly one other such decomposition, exactly as in the proof of commutation invariance for grid homology [MOSzT07, Lemma 3.1]. In particular, the concatenations of pentagons and rectangles contributing to the coefficients above cancel in pairs, proving the lemma.∎

Lemma 3.3**.**

$\phi$* sends $\mathbf{x}^{\pm}(\mathbb{G}_{+})$ to $\mathbf{x}^{\pm}(\mathbb{G}_{-})$.*

Proof.

There is a unique pentagon contributing to each of $\phi(\mathbf{x}^{+}(\mathbb{G}_{+}))$ and $\phi(\mathbf{x}^{-}(\mathbb{G}_{+}))$, shown in Figure 8, that certifies that

[TABLE]

∎

Lemma 3.4**.**

$\phi$* is homogeneous of Maslov–Alexander bidegree*

[TABLE]

Proof.

This is a straightforward calculation from the definitions of the Maslov and Alexander gradings in (4) and (5), and the map $\phi$; see the proofs of [OSSz15, Lemma 5.3.1] and [Won17, Lemma 6.6], for example. ∎

Remark 3.5**.**

If one could show that $\phi$ is homogeneous (say, using the relative grading formulas in (6) and (7)), the bidegree of $\phi$ would be determined by the fact that this map sends $\mathbf{x}^{+}(\mathbb{G}_{+})$ to $\mathbf{x}^{+}(\mathbb{G}_{-})$, and the lemma would follow immediately from (9) and (10), together with the facts that

[TABLE]

3.1.2. Case II

For $\mathbf{x}\in\SS(\mathbb{G}_{+})$ and $\mathbf{y}\in\SS(\mathbb{G}_{-})$, let $\mathrm{Tri}(\mathbf{x},\mathbf{y})$ be the space of triangles in the combined diagram with the following properties. $\mathrm{Tri}(\mathbf{x},\mathbf{y})$ is empty unless $\mathbf{x}$ and $\mathbf{y}$ coincide in exactly $n-1$ intersection points. An element $p\in\mathrm{Tri}(\mathbf{x},\mathbf{y})$ is an embedded triangle in the torus whose edges are arcs contained in the vertical and horizontal circles, and whose three corners are points in $\mathbf{x}\cup\mathbf{y}\cup\{b\}$. We require that, with respect to the induced orientation on $\partial p$, the boundary of this triangle may be traversed as follows: start at the point in $\mathbf{x}$ on $\beta$ and proceed along an arc of $\beta$ until arriving at $b$; next, proceed along an arc of $\gamma$ until arriving at a point in $\mathbf{y}$; finally, follow an arc of a vertical circle back to the initial point in $\mathbf{x}$. See Figure 9 for such triangles. Note that all such triangles automatically satisfy

[TABLE]

Let $\mathrm{Tri^{o}}(\mathbf{x},\mathbf{y})$ be the subset consisting of $p\in\mathrm{Tri}(\mathbf{x},\mathbf{y})$ with $p\cap\mathbb{O}=\emptyset.$ Let

[TABLE]

be the linear map defined on generators by counting such triangles,

[TABLE]

Lemma 3.6**.**

$\phi$* is a chain map.*

Proof.

This follows from an argument identical to that in the proof of Lemma 3.2, except that here we consider canceling concatenations of rectangles with triangles rather than pentagons. See the proof of [Won17, Lemma 3.4] for details in this case. ∎

Lemma 3.7**.**

$\phi$* sends $\mathbf{x}^{\pm}(\mathbb{G}_{+})$ to $\mathbf{x}^{\pm}(\mathbb{G}_{-})$.*

Proof.

There is a unique triangle contributing to each of $\phi(\mathbf{x}^{+}(\mathbb{G}_{+}))$ and $\phi(\mathbf{x}^{-}(\mathbb{G}_{+}))$, shown in Figure 9, that certifies that

[TABLE]

∎

Lemma 3.8**.**

$\phi$* is homogeneous of Maslov–Alexander bidegree*

[TABLE]

Proof.

As with Lemma 3.4, this is a straightforward calculation from the definitions of these gradings in (4) and (5), and the map $\phi$; see the proof of [Won17, Lemma 6.6] for details. ∎

The map $\Phi$ induced by $\phi$ therefore satisfies the conclusions of Proposition 3.1. ∎

3.2. Births

Proposition 3.9**.**

Suppose $\Lambda_{+}$ is obtained from $\Lambda_{-}$ via a birth move. For any grid diagrams $\mathbb{G}_{+}$ and $\mathbb{G}_{-}$ representing $\Lambda_{+}$ and $\Lambda_{-}$, respectively, there is a homomorphism

[TABLE]

that sends $\widetilde{\lambda}^{\pm}(\mathbb{G}_{+})$ to $\widetilde{\lambda}^{\pm}(\mathbb{G}_{-})$. This map has Maslov–Alexander bidegree $(1,0)$.

Proof.

By Proposition 2.1, it suffices to show that there exist some grid diagrams $\mathbb{G}_{+}$ and $\mathbb{G}_{-}$ representing links Legendrian isotopic to $\Lambda_{+}$ and $\Lambda_{-}$, respectively, for which the conclusions of Proposition 3.9 hold. For this, let $\mathbb{G}_{-}$ be any grid diagram representing $\Lambda_{-}$, with marking sets $\mathbb{X},\mathbb{O}$. Fix a marking $X_{1}\in\mathbb{X}$. Let $\mathbb{G}_{+}$ be the grid diagram obtained from $\mathbb{G}_{-}$ by inserting two rows and two columns to the immediate bottom right of $X_{1}$, with four new markings $X_{2},X_{3},O_{2},O_{3}$, as shown in Figure 10. Let $a$ and $b$ be the intersection points between the new vertical and horizontal circles indicated in the figure. Note that $\mathbb{G}_{+}$ represents the disjoint union of $\Lambda_{-}$ with the $tb=-1$ Legendrian unknot, which is Legendrian isotopic to $\Lambda_{+}$.

The generating set $\SS(\mathbb{G}_{+})$ can be expressed as a disjoint union,

[TABLE]

where

• •

$\mathrm{AB}$ consists of $\mathbf{x}\in\SS(\mathbb{G}_{+})$ with $a,b\in\mathbf{x}$,

• •

$\mathrm{AN}$ consists of $\mathbf{x}\in\SS(\mathbb{G}_{+})$ with $a\in\mathbf{x}$ but $b\notin\mathbf{x}$,

• •

$\mathrm{NB}$ consists of $\mathbf{x}\in\SS(\mathbb{G}_{+})$ with $a\notin\mathbf{x}$ but $b\in\mathbf{x}$,

• •

$\mathrm{NN}$ consists of $\mathbf{x}\in\SS(\mathbb{G}_{+})$ with $a\notin\mathbf{x}$ and $b\notin\mathbf{x}$.

This induces a decomposition of the vector space $\widetilde{\mathrm{GC}}(\mathbb{G}_{+})$ as a direct sum,

[TABLE]

where these summands are the vector spaces generated by the corresponding subsets of $\SS(\mathbb{G}_{+})$. Note that we have a sequence of subcomplexes,

[TABLE]

This follows immediately from the observation that any rectangle either starting at $b$ or terminating at $a$ must pass through one of the new markings or $X_{1}$ (and therefore does not contribute to the differential). Let

[TABLE]

be the quotient complex of $\widetilde{\mathrm{AB}}\oplus\widetilde{\mathrm{NB}}$ by $\widetilde{\mathrm{NB}}$. In other words, for $\mathbf{x},\mathbf{y}\in\mathrm{AB}$, the coefficient

[TABLE]

counts the number of rectangles in $\mathrm{Rect^{o}}_{\mathbb{G}_{+}}(\mathbf{x},\mathbf{y})$, as usual.

Note that there is a bijection between generators in $\mathrm{AB}$ and generators in $\SS(\mathbb{G}_{-})$, given by

[TABLE]

This bijection extends linearly to an isomorphism of chain complexes,

[TABLE]

since for $\mathbf{x},\mathbf{y}\in\mathrm{AB}$ there is also a natural bijection

[TABLE]

which identifies rectangles avoiding the $O$ and $X$ markings in $\mathbb{G}_{+}$ with rectangles avoiding the $O$ and $X$ markings in $\mathbb{G}_{-}$. Moreover, it follows readily from (6) and (7), together with this bijection of rectangles, that $e$ is homogeneous with respect to the Maslov–Alexander bigrading.

For $\mathbf{x}\in\mathrm{NB}$ and $\mathbf{y}\in\mathrm{AB}$, let

[TABLE]

be the subset consisting of rectangles $p$ satisfying

• •

$p\cap(\mathbb{O}\cup\{O_{2},O_{3}\})=\{O_{2},O_{3}\}$,

• •

$p\cap(\mathbb{X}\cup\{X_{2},X_{3}\})=\{X_{2},X_{3}\}$,

• •

$\mathrm{Int}(p)\cap\mathbf{x}=\mathrm{Int}(p)\cap\mathbf{y}=\{b\}$.

Let

[TABLE]

be the linear map defined on generators by counting such rectangles,

[TABLE]

Let

[TABLE]

be projection onto the summand $\widetilde{\mathrm{NB}}$, and define

[TABLE]

to be the linear map given as the composition

[TABLE]

Lemma 3.10**.**

$\phi$* is a chain map.*

Proof.

Since $e$ is a chain map, it suffices to prove that $\psi\circ\Pi$ is a chain map. Note that both

[TABLE]

vanish for generators $\mathbf{x}\notin\mathrm{AB}\cup\mathrm{NB}$. The first vanishes on such generators $\mathbf{x}$ because $\Pi(\mathbf{x})=0.$ The second vanishes on such generators $\mathbf{x}$ because

[TABLE]

Indeed, for every $\mathbf{y}\in\mathrm{NB}$, the coefficient

[TABLE]

since every rectangle from a generator not containing $b$ to a generator containing $b$ must pass through a marking. To prove that $\phi$ is a chain map, it therefore suffices to prove the equality of coefficients

[TABLE]

for every pair of generators $\mathbf{x}\in\mathrm{AB}\cup\mathrm{NB}$ and $\mathbf{y}\in\mathrm{AB}$. These coefficients on the left and right count concatenations of rectangles from $\mathbf{x}$ to $\mathbf{y}$ of the forms $p*r$ and $r*p$, respectively, where $p$ is a rectangle of the sort used to define $\psi$, and $r$ is a rectangle of the sort used to define the differentials. For $\mathbf{x}\in\mathrm{NB}$, every domain in $\mathbb{G}_{+}$ that decomposes as the juxtaposition of the form $p*r$ admits exactly one other decomposition into rectangles $p$ and $r$, of the form $r*p$, exactly as in the proof that the grid differential squares to zero [MOSzT07, Proposition 2.10]. There are additional domains that decompose as the juxtaposition of the form $r*p$, but these cancel in pairs as well. In particular, the concatenations of rectangles contributing to the coefficients above cancel in pairs, proving the lemma in this case. For $\mathbf{x}\in\mathrm{AB}$, the first coefficient is zero since $\Pi(\mathbf{x})=0,$ and there are exactly two concatenations of the form $r*p$ contributing to the second coefficient. These two cancelling concatenations correspond to vertical and horizontal annular domains of width 2, as shown in Figure 11.

∎

Lemma 3.11**.**

$\phi$* sends $\mathbf{x}^{\pm}(\mathbb{G}_{+})$ to $\mathbf{x}^{\pm}(\mathbb{G}_{-})$.*

Proof.

Note that $\mathbf{x}^{\pm}(\mathbb{G}_{+})\in\mathrm{NB}$, so

[TABLE]

There is a unique rectangle contributing to each of $\psi(\mathbf{x}^{+}(\mathbb{G}_{+}))$ and $\psi(\mathbf{x}^{-}(\mathbb{G}_{+}))$, as shown in Figure 12. It is then clear from the figure that

[TABLE]

∎

Lemma 3.12**.**

$\phi$* is homogeneous of Maslov–Alexander bidegree $(1,0)$.*

Proof.

It is clear from the definition of $\psi$, together with (6) and (7), that $\psi$ is homogeneous. Since $e$ and $\Pi$ are also homogeneous, the same is true of $\phi$. The bidegree of $\phi$ is then determined by the fact that this map sends $\mathbf{x}^{+}(\mathbb{G}_{+})$ to $\mathbf{x}^{+}(\mathbb{G}_{-})$, and the lemma follows immediately from (9) and (10), together with the facts that

[TABLE]

∎

The map $\Phi$ induced by $\phi$ therefore satisfies the conclusions of Proposition 3.9. ∎

3.3. Putting it together

Proof of Theorem 1.5.

Suppose $L$ is a decomposable Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$. Then there is a sequence

[TABLE]

of Legendrian links such that for each $i$, $\Lambda_{i+1}$ is obtained from $\Lambda_{i}$ by either Legendrian isotopy, a pinch, or a birth. Let $\mathbb{G}_{i}$ be a grid diagram representing $\Lambda_{i}$. Then Proposition 2.1, 3.1, or 3.9 provides a map

[TABLE]

that sends $\widetilde{\lambda}^{\pm}(\mathbb{G}_{i+1})$ to $\widetilde{\lambda}^{\pm}(\mathbb{G}_{i})$, for each $i=0,\dots,m-1$. Note that the bidegree of each $\Phi_{i}$ may be expressed simply as

[TABLE]

where $L_{i}$ is the corresponding elementary cobordism from $\Lambda_{i}$ to $\Lambda_{i+1}$. We define the map

[TABLE]

This map then sends $\widetilde{\lambda}^{\pm}(\mathbb{G}_{+})$ to $\widetilde{\lambda}^{\pm}(\mathbb{G}_{-})$ and has bidegree

[TABLE]

as desired. ∎

Proof of Theorem 1.2.

As explained in the introduction, this theorem follows from Theorem 1.5 and the fact that

[TABLE]

for any grid diagram $\mathbb{G}$, as in (11). ∎

Proof of Corollary 1.3.

A decomposable Lagrangian filling of $\Lambda$ is a decomposable Lagrangian cobordism from the empty link to $\Lambda$, and can thus be described as a composition of elementary cobordisms starting with a birth. The rest of the filling is therefore a decomposable Lagrangian cobordism from the $tb=-1$ Legendrian unknot $\Lambda_{U}$ to $\Lambda$. The corollary then follows from Theorem 1.2 combined with the fact that $\widehat{\lambda}^{\pm}(\Lambda_{U})\neq 0.$ ∎

Proof of Corollary 1.4.

This follows immediately from Theorem 1.2 and the fact that $\widehat{\lambda}^{\pm}$ vanishes for positive and negative Legendrian stabilizations, respectively, as in Proposition 2.2. ∎

4. Examples

In this section, we illustrate the effectiveness of Theorem 1.2 via examples, proving Theorem 1.6 along the way.

4.1. Examples of genus zero

Let $K$ denote either one of the (oriented) smooth knot types given by $m(10_{132})$ or $m(12n_{200})$. In [NOT08, Section 3], Ng, Ozsváth, and Thurston describe two Legendrian representatives $\Lambda_{0}$ and $\Lambda_{1}$ of $K$ with

[TABLE]

but

[TABLE]

It follows that $\Lambda_{0}$ and $\Lambda_{1}$ are not Legendrian isotopic despite having the same classical invariants, by [OSzT08]. These are among the smallest crossing examples known that demonstrate the effectiveness of the GRID invariants in obstructing Legendrian isotopy. Ng, Ozsváth, and Thurston further observed, using an argument by Ng and Traynor from the proof of [NT04, Proposition 5.9], that these Legendrians are orientation reversals of one another,

[TABLE]

(Note that $m(10_{132})$ and $m(12n_{200})$ are reversible.) Thus, [OSzT08, Proposition 1.2] implies that

[TABLE]

as well. Combining (13) and (14) with Theorem 1.2, we obtain the following.

Proposition 4.1**.**

There is no decomposable Lagrangian concordance from $\Lambda_{0}$ to $\Lambda_{1}$ or from $\Lambda_{1}$ to $\Lambda_{0}$.

Proof.

The first is obstructed by $\widehat{\lambda}^{-}$, the second by $\widehat{\lambda}^{+}$. ∎

Note that the Thurston–Bennequin and rotation numbers do not obstruct the existence of decomposable Lagrangian concordances between $\Lambda_{0}$ and $\Lambda_{1}$ via (3).

Remark 4.2**.**

One of the two directions in Proposition 4.1 is proven by Baldwin and Sivek [BS18b] and independently Golla and Juhász in [GJ19, Proposition 1.7]. In particular, they use the equivalence between $\widehat{\lambda}^{+}$ and $\widehat{\mathfrak{L}}$ to obstruct a decomposable Lagrangian concordance from $\Lambda_{1}$ to $\Lambda_{0}$.

4.2. Examples of genus one

Let $K$ denote one of $m(10_{132})$ or $m(12n_{200})$, and let $\Lambda_{0}$ and $\Lambda_{1}$ be the Legendrian representatives of $K$ discussed above. Front diagrams for these Legendrians are given in [NOT08, Figures 2 and 3]. By modifying these front diagrams for $\Lambda_{0}$ and $\Lambda_{1}$ first by a Legendrian Reidemeister I move, and then by adding a positive clasp, as in Figure 13, we obtain new Legendrian knots $\Lambda_{0}^{\prime}$ and $\Lambda_{1}^{\prime}$, respectively, whose front diagrams are shown in Figure 14. These two Legendrian knots belong to the smooth knot type

[TABLE]

(The knot types were found using the program Knotscape by Hoste and Thistlethwaite [HT99].)

Observe that there is a smooth cobordism of genus one between $K$ and $K^{\prime}$ since the latter is obtained from the former via the addition of a positive clasp. Moreover, it is easy to see that the local modification in Figure 13 increases the Thurston–Bennequin number by $2$ and preserves the rotation number. Combined with (12), this implies that for any $i,j\in\{0,1\}$,

[TABLE]

In particular, the classical invariants and smooth topology do not obstruct the existence of a decomposable genus one Lagrangian cobordism from $\Lambda_{0}$ to $\Lambda_{1}^{\prime}$ or from $\Lambda_{1}$ to $\Lambda_{0}^{\prime}$.

However, a direct computer calculation using the program [MQR*+*19],666This program is a newer version of the program written by Ng, Ozsváth, and Thurston [NOT07], with minor bug fixes and improvements in computational efficiency. applied to the diagrams in Figure 14, shows that

[TABLE]

The argument used by Ng, Ozsváth, and Thurston to show that $\Lambda_{0}=-\Lambda_{1}$ shows that $\Lambda_{0}^{\prime}$ and $\Lambda_{1}^{\prime}$ are also orientation reversals of one another, which then implies that

[TABLE]

These calculations, combined with Theorem 1.2, lead immediately to the following.

Proposition 4.3**.**

There is no decomposable Lagrangian cobordism from $\Lambda_{0}$ to $\Lambda_{1}^{\prime}$ or from $\Lambda_{1}$ to $\Lambda_{0}^{\prime}$.

Proof.

The first is obstructed by $\widehat{\lambda}^{-}$, the second by $\widehat{\lambda}^{+}$. ∎

4.3. An infinite family

Let $K$ be one of the knot types $m(10_{145})$, $m(10_{161})$, or $12n_{591}$. In [CN13, Proposition 6], Chongchitmate and Ng provide two Legendrian representatives $\Lambda_{0}$ and $\Lambda_{1}$ of $K$ (denoted by $L_{1}$ and $L_{3}$ there) satisfying

[TABLE]

and

[TABLE]

Fix any positive crossing in the front diagram for $\Lambda_{0}$. Let $\Lambda_{0}^{\prime}$ be the Legendrian knot representing the smooth knot type $K^{\prime}$ obtained from $\Lambda_{0}$ by first performing a Legendrian Reidemeister I move near this crossing, and then adding $m$ positive clasps, as in Figure 15. It is easy to see that this local modification increases the Thurston–Bennequin number by $2m$ and preserves rotation number,

[TABLE]

This implies, by (2) combined with (15) and (17), that

[TABLE]

With this, we may now prove Theorem 1.6.

Proof of Theorem 1.6.

Let us adopt the notation from above. Take $g=m$ and let

[TABLE]

The first is a Legendrian representative of $K$, the second of $K^{\prime}$, and there is a smooth genus $g$ cobordism from $K$ to $K^{\prime}$ since $K^{\prime}$ is obtained from $K$ by adding $g$ clasps, fulfilling the first bullet point of the theorem. The second bullet point is fulfilled by (18). Finally,

[TABLE]

by Proposition 2.2 since $\Lambda_{+}$ is a positive Legendrian stabilization, and

[TABLE]

by (16), fulfilling the third bullet point of the theorem. ∎

4.4. DGA versus GRID

We assume below that the reader is familiar with the Chekanov–Eliashberg DGA; for a survey, see [EN18].

As mentioned in the introduction, an exact Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$ induces a DGA morphism [EHK16]

[TABLE]

so if the first DGA is trivial while the second is not then there cannot be such a cobordism. The DGA is trivial for stabilized Legendrians [Che02], so, as noted in Section 1.3, this functoriality cannot obstruct decomposable Lagrangian cobordisms between stabilizations of the examples in Sections 4.1-4.3, while the GRID invariants can.

We mentioned in Section 1.3 that there are also cases in which the DGA obstruction applies where the GRID obstruction does not. For example, there is a Legendrian representative $\Lambda_{-}$ of the figure eight with

[TABLE]

such that $(\mathcal{A}_{\Lambda_{-}},\partial_{\Lambda_{-}})$ admits an augmentation, and is therefore nontrivial (see e.g. [CN13]). This example was pointed out to the authors by Steven Sivek. Now, let $\Lambda_{+}$ be the Legendrian representative of the right-handed trefoil with

[TABLE]

obtained by stabilizing the $\mathit{tb}=1$ representative twice, once with each sign, so that $(\mathcal{A}_{\Lambda_{+}},\partial_{\Lambda_{+}})$ is trivial. These DGAs therefore obstruct an exact Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$. There is a smooth genus one cobordism from the figure eight to the right-handed trefoil, as indicated in Figure 16, so the classical invariants and smooth topology do not obstruct such a cobordism. On the other hand, the figure eight has trivial Heegaard Floer tau-invariant, so that

[TABLE]

Since the figure eight is also thin, this implies that the GRID invariants of $\Lambda_{-}$ vanish [NOT08, Proposition 3.4], and therefore do not obstruct a decomposable Lagrangian cobordism from $\Lambda_{-}$ to $\Lambda_{+}$ via Theorem 1.2.

Finally, in the interest of completeness, we remark that the DGAs of the Legendrian representatives of $m(10_{132})$, $m(12n_{200})$, $m(10_{145})$, $m(10_{161})$, and $12n_{591}$, which served as examples of $\Lambda_{-}$ in Sections 4.1-4.3 admit no augmentations and therefore no linearized Legendrian contact homologies. Indeed, Rutherford showed in [Rut06] that the Kauffman polynomial bound on $\mathit{tb}(\Lambda)$ is sharp if and only if $(\mathcal{A}_{\Lambda},\partial_{\Lambda})$ admits an augmentation, and one can check on KnotInfo [CL] that the Kauffman bounds are not sharp for the Legendrians above. This does not completely rule out the possibility that their DGAs are nontrivial (and could therefore perhaps obstruct the Lagrangian cobordisms we are considering), but it eliminates the most tractable approach to proving nontriviality. Pan proved in [Pan17, Theorem 1.6] that if there exists an exact Lagrangian cobordism (with Maslov number 0) from $\Lambda_{-}$ to $\Lambda_{+}$, then the number of graded augmentations of $\Lambda_{-}$ up to a certain equivalence is less than or equal to the number of graded augmentations of $\Lambda_{+}$ up to equivalence. The fact that the Legendrians above admit no augmentations at all also rules out the possibility of applying Pan’s more refined obstruction in these examples.