The Alexander module, Seifert forms, and categorification

Hom, Jennifer; Lidman, Tye; Watson, Liam

doi:10.1112/topo.12001

Cited by 10 publications

(10 citation statements)

References 38 publications

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“…We will not need the above gradings in this paper, but only a simpler grading by Z/2. Gradings by Z/2 on bordered Floer homology have appeared in [Pet18,Pet14,HLW17]. We lay out definitions here directly, specializing to the case of torus boundary, and encourage the reader to compare with earlier literature.…”

Section: Be the Left Typementioning

confidence: 99%

Bordered Floer homology with integral coefficients for manifolds with torus boundary

Knowles,

Petkova

2021

Preprint

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We provide a combinatorial definition of a bordered Floer theory with Z coefficients for manifolds with torus boundary. Our bordered Floer structures recover the combinatorial Heegaard Floer homology from [OSS14].2. BACKGROUND 2.1. A ∞ structures over Z. Since definitions for the relevant algebraic structures appear only over Z/2 in the bordered Floer literature, we begin our background discussion by reminding the reader the more general setup over Z. Our summary closely follows [OS17, Section 12]. Fix a ring R, not necessarily of characteristic two. In the later sections, R will be the ring Z 2 of idempotents of the torus algebra A P defined in Section 3. For the discussion below, we will assume the spaces are graded by Z/2. There are, of course, more general definitions in the literature. Definition 2.1. An A ∞ -algebra A is a Z/2-graded R-bimodule A, equipped with degree 0 R-linear multiplication maps for i ≥ 1 µ i : A ⊗i → A[2 − i] that satisfy the following conditions for n ≥ 1 r+s+t=n (−1) r+st µ r+1+t (Id ⊗r ⊗ µ s ⊗ Id ⊗t ) = 0.1 See Theorems 5.9 and 6.6 for a precise statement of this result.

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Section: Be the Left Typementioning

confidence: 99%

Bordered Floer homology with integral coefficients for manifolds with torus boundary

Knowles,

Petkova

2021

Preprint

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“…start by reviewing the relative Z 2 -grading gr on CF . The details are in [22,8]. Let H = (Σ g , α, β, z) be a Heegaard diagram for a closed 3-manifold Y .…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

“…gr induces a relative Z 2 -grading on HF (H), which we also call gr. With respect to both relative There's an analogous story for the bordered invariants CFA and CFD, due to Hom, Lidman, and Watson in [8]. To explain this, we'll need the notion of a bordered partial permutation.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

A Floer homology invariant for $3$-orbifolds via bordered Floer theory

Wong¹

2018

Preprint

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Using bordered Floer theory, we construct an invariant HFO(Y orb ) for 3orbifolds Y orb with singular set a knot that generalizes the hat flavor HF (Y ) of Heegaard Floer homology for closed 3-manifolds Y . We show that for a large class of 3-orbifolds HFO behaves like HF in that HFO, together with a relative Z 2 -grading, categorifies the order of H orb 1 . When Y orb arises as Dehn surgery on an integer-framed knot in S 3 , we use the {−1, 0, 1}-valued knot invariant ε to determine the relationship between HFO(Y orb ) and HF (Y ) of the 3-manifold Y underlying Y orb .

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“…Other related invariants, for example the determinant and Arf invariant, are consequently determined. Moreover, Hom et al have shown that both the Alexander module of a knot and the Seifert form are determined by Heegaard Floer theory [8].…”

Section: Introductionmentioning

confidence: 99%

Classical homological invariants are not determined by knot Floer homology and Khovanov homology

Choon

Tanaka

2016

J. Knot Theory Ramifications

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We illustrate that there are knots for which Heegaard knot Floer homology and Khovanov homology are identical but the Alexander module and torsion invariants differ. The examples are certain symmetric unions. We also give examples of similar flavor, concerning the Kauffman and Q-polynomials in place of the classical homological invariants. This shows there are nonmutant knots with the same knot Floer and Khovanov homology.

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The Alexander module, Seifert forms, and categorification

Cited by 10 publications

References 38 publications

Bordered Floer homology with integral coefficients for manifolds with torus boundary

Bordered Floer homology with integral coefficients for manifolds with torus boundary

A Floer homology invariant for $3$-orbifolds via bordered Floer theory

Classical homological invariants are not determined by knot Floer homology and Khovanov homology

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