Bordered Heegaard Floer homology

Lipshitz, Robert; Thurston, Dylan P.; Ozsváth, Peter

doi:10.1090/memo/1216

Cited by 149 publications

(476 citation statements)

References 33 publications

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“…Given a surface F , together with a handle decomposition of F with a single 0-handle and a single 2-handle, and 2g 1-handles, bordered Floer homology [27,29] associates a differential algebra A(F ) = g i=−g A(F, i). Given a three-dimensional cobordism Y from F 1 to F 2 , together with a framed arc γ in Y connecting the 0-handle in F 1 and the 0-handle in F 2 , bordered Floer homology associates a dg (A(F 1 ), A(F 2 ))-bimodule CFDA(Y ), well-defined up to quasiisomorphism.…”

Section: Equivalence Of the Hendricks And Lipshitz-treumann Spectral mentioning

confidence: 99%

A flexible construction of equivariant Floer homology and applications

Hendricks

Lipshitz²,

Sarkar³

2016

J Topology

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Seidel–Smith and Hendricks used equivariant Floer cohomology to define some spectral sequences from symplectic Khovanov homology and Heegaard Floer homology. These spectral sequences give rise to Smith‐type inequalities. Similar‐looking spectral sequences have been defined by Lee, Bar–Natan, Ozsváth–Szabó, Lipshitz–Treumann, Szabó, Sarkar–Seed–Szabó, and others. In this paper, we give another construction of equivariant Floer cohomology with respect to a finite group action and use it to prove some invariance properties of these spectral sequences; prove that some of these spectral sequences agree; improve Hendricks's Smith‐type inequalities; give some theoretical and practical computability results for these spectral sequences; define some new spectral sequences conjecturally related to Sarkar–Seed–Szabó's; and introduce a new concordance homomorphism and concordance invariants. We also digress to prove invariance of Manolescu's reduced symplectic Khovanov homology.

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Section: Equivalence Of the Hendricks And Lipshitz-treumann Spectral mentioning

confidence: 99%

A flexible construction of equivariant Floer homology and applications

Hendricks

Lipshitz²,

Sarkar³

2016

J Topology

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“…We define the category of type D structures over

scriptA

{prefixCx}^{0} false(prefixMat (C) false)

. (Again, note that we need to pass to the underlying ordinary category to obtain the definitions in . )…”

Section: Preliminaries: Algebraic Structures From Dg Categoriesmentioning

confidence: 99%

“…Definition Let

M

be a (right) type D structure and

N

a (left) type A structure over the same dg algebra

scriptA

over a ring

scriptI

of idempotents, together with a fixed basis of

scriptI

and

scriptA

. We now reformulate the definition of the chain complex

(M ⊠ N, \partial^{⊠})

from [, Definition 7.4] and [, Section 2.4] in terms of the graphs associated with

M

and

N

. The generators of

M ⊠ N

are defined by pairs of vertices in

M

and

N

labelled by the same idempotents.…”

Section: Preliminaries: Algebraic Structures From Dg Categoriesmentioning

confidence: 99%

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Peculiar modules for 4‐ended tangles

Zibrowius

2020

Journal of Topology

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With a 4‐ended tangle T, we associate a Heegaard Floer invariant prefixCFT∂false(Tfalse), the peculiar module of T. Based on Zarev's bordered sutured Heegaard Floer theory (Zarev, PhD Thesis, Columbia University, 2011), we prove a glueing formula for this invariant which recovers link Floer homology HFL̂. Moreover, we classify peculiar modules in terms of immersed curves on the 4‐punctured sphere. In fact, based on an algorithm of Hanselman, Rasmussen and Watson (Preprint, 2016, arXiv:1604.03466v2), we prove general classification results for the category of curved complexes over a marked surface with arc system. This allows us to reinterpret the glueing formula for peculiar modules in terms of Lagrangian intersection Floer theory on the 4‐punctured sphere. We then study some applications: Firstly, we show that peculiar modules detect rational tangles. Secondly, we give short proofs of various skein exact triangles. Thirdly, we compute the peculiar modules of the 2‐stranded pretzel tangles T2n,−false(2m+1false) for n,m>0 using nice diagrams. We then observe that these peculiar modules enjoy certain symmetries which imply that mutation of the tangles T2n,−false(2m+1false) preserves δ‐graded, and for some orientations even bigraded link Floer homology.

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“…Proof of Proposition 2.5. We will prove Proposition 2.5 using Lipshitz-Ozsváth-Thurston's bordered Floer homology [LOT08], specifically Hanselman-Watson's [HW15] loop calculus. That is, we will decompose the result of surgery on K r,rn into two pieces, one that is surgery on a torus link in the solid torus and the other the knot complement, and then apply a gluing result of Hanselman-Watson to conclude that the result of this surgery along K r,rn is not an L-space.…”

Section: Dehn Surgery and Cable Linksmentioning

confidence: 99%

Cable links and L-space surgeries

Gorsky

Hom²

2017

Quantum Topol.

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Abstract. An L-space link is a link in S 3 on which all sufficiently large integral surgeries are L-spaces. We prove that for m, n relatively prime, the r-component cable link Krm,rn is an L-space link if and only if K is an L-space knot and n/m ≥ 2g(K) − 1. We also compute HFL -and HFL of an L-space cable link in terms of its Alexander polynomial. As an application, we confirm a conjecture of Licata [Lic12] regarding the structure of HFL for (n, n) torus links.

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Bordered Heegaard Floer homology

Cited by 149 publications

References 33 publications

A flexible construction of equivariant Floer homology and applications

A flexible construction of equivariant Floer homology and applications

Peculiar modules for 4‐ended tangles

Cable links and L-space surgeries

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