Lagrangian Cobordisms and Legendrian Invariants in Knot Floer Homology

Baldwin, John A.; Lidman, Tye; Wong, C.-M. Michael

doi:10.1307/mmj/20195786

Cited by 5 publications

(10 citation statements)

References 29 publications

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“…Our results fit into an existing body of work showing that knot Floer homology effectively obstructs Lagrangian link cobordisms. For the hat theory, in the case that the Weinstein cobordism is the symplectization of the tight 3-sphere, the above results have already been established by Baldwin, Lidman and Wong [BLMW19]. They work with the so-called GRID invariants defined in [OST08]; their arguments utilize grid diagrams, and are specific to the setting they work in.…”

Section: Introductionmentioning

confidence: 76%

Quasipositive surfaces and decomposable Lagrangians

Tovstopyat-Nelip¹

2020

Preprint

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We show that a quasipositive surface with disconnected boundary induces a map between the knot Floer homology groups of its boundary components preserving the transverse invariant. As an application, we show that this invariant can be used to obstruct decomposable Lagrangian cobordisms of arbitrary genus within Weinstein cobordisms. The construction of our maps rely on the comultiplicativity of the transverse invariant. Along the way, we also recover various naturality statements for the invariant under contact +1 surgery.1 decomposable Lagrangians in the symplectization of (S 3 , ξ std ) are built up from elementary cobordisms associated to births and pinches, we extend this notion to the setting of Weinstein cobordisms in Section 7.

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Section: Introductionmentioning

confidence: 76%

Quasipositive surfaces and decomposable Lagrangians

Tovstopyat-Nelip¹

2020

Preprint

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“…The goal of the present article is to extend the result in [BLW22] to the context of filtered knot Floer chain complexes, which will then imply that certain invariants in the associated spectral sequences also provide effective obstructions to exact Lagrangian cobordisms. The existence of these invariants is known to experts in knot Floer homology, but their definitions have not appeared in the literature thus far; in Section 1.1 below, we first provide the definitions.…”

Section: Introductionmentioning

confidence: 85%

“…Functoriality. Analogous to [BLW22], Theorem 1.7 follows from the following theorem, which states that the spectral GRID invariants satisfy a weak functoriality under decomposable Lagrangian cobordisms, in the style of Theorem 1.2. Theorem 1.10.…”

Section: Introductionmentioning

confidence: 99%

“…Another source of effective invariants is knot (Heegaard) Floer homology. In earlier work, Baldwin, Lidman, and the fifth author [BLW22] prove that the so-called GRID invariants in knot Floer homology can effectively obstruct decomposable Lagrangian cobordisms (of any genus), which are cobordisms that can be obtained by concatenating elementary cobordisms associated to Legendrian isotopies, pinches, and births, as depicted in Figure 2. Decomposable cobordisms are exact, and most connected exact Lagrangian cobordisms between non-empty Legendrian links are known to be decomposable; whether all connected exact Lagrangian cobordisms between non-empty Legendrian links are decomposable remains a major open problem.…”

Section: Introductionmentioning

confidence: 99%

“…Obstructions. Baldwin, Lidman, and the fifth author [BLW22] prove that, if there exists a decomposable Lagrangian cobordism L : Λ − → Λ + , then there exists a homomorphism Φ L : HFL(−S 3 , Λ + ) → HFL(−S 3 , Λ − ) that sends λ ± (Λ + ) to λ ± (Λ − ). In other words:…”

Section: Introductionmentioning

confidence: 99%

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Spectral GRID invariants and Lagrangian cobordisms

Jubeir¹,

Petkova²,

Schwartz³

et al. 2023

Preprint

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We prove that the filtered GRID invariants of Legendrian links in link Floer homology, and consequently their associated invariants in the spectral sequence, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on R 3 , strengthening a result by Baldwin, Lidman, and the fifth author.1 This contrasts with the situation in smooth topology, where a cobordism exists between any pair of links, but the minimum genus of such a cobordism is difficult to determine.2 In particular, this also illustrates the important fact that exact Lagrangian cobordisms are directed.

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