Link cobordisms and functoriality in link Floer homology

Zemke, Ian

doi:10.1112/topo.12085

Cited by 48 publications

(135 citation statements)

References 31 publications

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“…In the above theorem, the vertical bar

false|

denotes the tensor product of maps. The maps

Φ_{i}

and

Ψ_{i}

appear frequently in the link Floer TQFT , however they appeared earlier in work of Sarkar on diffeomorphism maps on knot Floer homology . Using our formulation, they can algebraically be described as the formal derivatives of the differential with respect to

U

V

, respectively.…”

Section: Introductionmentioning

confidence: 99%

“…The chain homotopy equivalences appearing in Theorem are the link cobordism maps for certain decorated link cobordisms, using the link cobordism maps from . The first step towards proving Theorem is to analyze the interaction between the link cobordism maps from and the conjugation map

η

.…”

Section: Introductionmentioning

confidence: 99%

η

. The maps from use the following notion of decorated link cobordism (adapted from [, Definition 4.5]): Definition We say a pair

(W, F)

is a decorated link cobordism between two 3‐manifolds with multi‐based links, and write

false(W, scriptF false) : false(Y_{1}, L_{1}, p_{1}, q_{1} false) \to false(Y_{2}, L_{2}, p_{2}, q_{2} false)

, if the following are satisfied.…”

Section: Introductionmentioning

confidence: 99%

“…The first step towards proving Theorem is to analyze the interaction between the link cobordism maps from and the conjugation map

η

. The maps from use the following notion of decorated link cobordism (adapted from [, Definition 4.5]): Definition We say a pair

(W, F)

is a decorated link cobordism between two 3‐manifolds with multi‐based links, and write

false(W, scriptF false) : false(Y_{1}, L_{1}, p_{1}, q_{1} false) \to false(Y_{2}, L_{2}, p_{2}, q_{2} false)

, if the following are satisfied. (1)

W

is a four‐dimensional cobordism from

Y_{1}

Y_{2}

. (2)

F = (S, A)

, where

S

is an oriented surface, properly embedded in

W

, with

\partial S = - L_{1} ⊔ L_{2}

. (3)

A \subseteq S

is a properly embedded 1‐manifold dividing

S

into two disjoint subsurfaces

S_{w}

and

S_{z}

, which meet along

scriptA

. (4)Each component of

L_{i} ∖ A

contains exactly one basepoint. (5)

{boldp}_{1} \cup {boldp}_{2} \subseteq S_{boldw}

and

{boldq}_{1} \cup {boldq}_{2} \subseteq S_{boldz}

. …”

Section: Introductionmentioning

confidence: 99%

“…As a first step to proving Theorem , we prove that the link cobordism maps from satisfy a version of conjugation invariance similar to the one satisfied by the cobordism maps of Ozsváth and Szabó [, Theorem 3.6]. Theorem Suppose that

false(W, scriptF false) : false(Y_{1}, L_{1}, p_{1}, q_{1} false) \to false(Y_{2}, L_{2}, p_{2}, q_{2} false)

is a decorated link cobordism, and let

false(W, \bar{F} false) : false(Y_{1}, L_{1}, q_{1}, p_{1} false) \to false(Y_{2}, L_{2}, q_{2}, p_{2} false)

denote the cobordism obtained by switching the roles of

S_{w}

and

S_{z}

.…”

Section: Introductionmentioning

confidence: 99%

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Connected sums and involutive knot Floer homology

Zemke

2019

Proc. London Math. Soc.

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We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected sum formula, we perform some example computations of Hendricks and Manolescu's involutive invariants on large surgeries of connected sums of knots.

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“…In the above theorem, the vertical bar

false|

denotes the tensor product of maps. The maps

Φ_{i}

and

Ψ_{i}

U

V

, respectively.…”

Section: Introductionmentioning

confidence: 99%

η

.…”

Section: Introductionmentioning

confidence: 99%

η

. The maps from use the following notion of decorated link cobordism (adapted from [, Definition 4.5]): Definition We say a pair

(W, F)

is a decorated link cobordism between two 3‐manifolds with multi‐based links, and write

false(W, scriptF false) : false(Y_{1}, L_{1}, p_{1}, q_{1} false) \to false(Y_{2}, L_{2}, p_{2}, q_{2} false)

, if the following are satisfied.…”

Section: Introductionmentioning

confidence: 99%

“…The first step towards proving Theorem is to analyze the interaction between the link cobordism maps from and the conjugation map

η

. The maps from use the following notion of decorated link cobordism (adapted from [, Definition 4.5]): Definition We say a pair

(W, F)

is a decorated link cobordism between two 3‐manifolds with multi‐based links, and write

false(W, scriptF false) : false(Y_{1}, L_{1}, p_{1}, q_{1} false) \to false(Y_{2}, L_{2}, p_{2}, q_{2} false)

, if the following are satisfied. (1)

W

is a four‐dimensional cobordism from

Y_{1}

Y_{2}

. (2)

F = (S, A)

, where

S

is an oriented surface, properly embedded in

W

, with

\partial S = - L_{1} ⊔ L_{2}

. (3)

A \subseteq S

is a properly embedded 1‐manifold dividing

S

into two disjoint subsurfaces

S_{w}

and

S_{z}

, which meet along

scriptA

. (4)Each component of

L_{i} ∖ A

contains exactly one basepoint. (5)

{boldp}_{1} \cup {boldp}_{2} \subseteq S_{boldw}

and

{boldq}_{1} \cup {boldq}_{2} \subseteq S_{boldz}

. …”

Section: Introductionmentioning

confidence: 99%

false(W, scriptF false) : false(Y_{1}, L_{1}, p_{1}, q_{1} false) \to false(Y_{2}, L_{2}, p_{2}, q_{2} false)

is a decorated link cobordism, and let

false(W, \bar{F} false) : false(Y_{1}, L_{1}, q_{1}, p_{1} false) \to false(Y_{2}, L_{2}, q_{2}, p_{2} false)

denote the cobordism obtained by switching the roles of

S_{w}

and

S_{z}

.…”

Section: Introductionmentioning

confidence: 99%

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Connected sums and involutive knot Floer homology

Zemke

2019

Proc. London Math. Soc.

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Heegaard Floer Homology

Wong,

Zampa

2024

Bolyai Society Mathematical Studies

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Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

Juhász

Marengon

2017

Sel. Math. New Ser.

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We study the maps induced on link Floer homology by elementary decorated link cobordisms. We compute these for births, deaths, stabilizations, and destabilizations, and show that saddle cobordisms can be computed in terms of maps in a decorated skein exact triangle that extends the oriented skein exact triangle in knot Floer homology. In particular, we completely determine the Alexander and Maslov grading shifts.As a corollary, we compute the maps induced by elementary cobordisms between unlinks. We show that these give rise to a (1 + 1)-dimensional TQFT that coincides with the reduced Khovanov TQFT. Hence, when applied to the cube of resolutions of a marked link diagram, it gives the complex defining the reduced Khovanov homology of the knot. Finally, we define a spectral sequence from (reduced) Khovanov homology using these cobordism maps, and we prove that it is an invariant of the (marked) link.

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Link cobordisms and functoriality in link Floer homology

Abstract: We construct cobordism maps on link Floer homology associated to decorated link cobordisms. The maps are defined on a curved chain homotopy type invariant. We describe the construction and prove invariance. We also make a comparison with the graph TQFT for Heegaard Floer homology.

Cited by 48 publications

References 31 publications

Connected sums and involutive knot Floer homology

Connected sums and involutive knot Floer homology

Heegaard Floer Homology

Computing cobordism maps in link Floer homology and the reduced Khovanov TQFT

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