Connected sums and involutive knot Floer homology

Zemke, Ian

doi:10.1112/plms.12227

Cited by 28 publications

(74 citation statements)

References 30 publications

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“…Furthermore, using the grading change formula, the author computes the maps for closed surfaces with simple dividing sets when b 1 = 0 and b + 2 = 0 [31, Theorem 1.7]. The link cobordism maps also satisfy a version of conjugation invariance [30,Theorem 1.3] similar to the version satisfied by the standard 4-manifold invariants [23,Theorem 3.6]. Using this, and some further properties of the link cobordism maps, the author explores the relation between the cobordism maps and the Künneth theorem for connected sums, and in particular proves a connected sum formula for involutive knot Floer homology [30, Theorem 1.1].…”

Section: Further Developmentsmentioning

confidence: 99%

Link cobordisms and functoriality in link Floer homology

Zemke

2018

Journal of Topology

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Section: Further Developmentsmentioning

confidence: 99%

Link cobordisms and functoriality in link Floer homology

Zemke

2018

Journal of Topology

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“…In a recent paper , the second author showed that knot Floer homology gives an obstruction to ribbon concordance. In this short note, we prove an analogous result for Khovanov homology .…”

Section: Applicationsmentioning

confidence: 99%

“…By the functoriality of Khovanov homology,

Kh false(D false) = Kh false(\overset{C}{true} false) \circ Kh false(C false)

. The second author showed in that

D

has the following nice topological description: There exist unknotted, unlinked 2‐spheres

S_{1}, \dots S_{n} \subset false(R^{3} ∖ L_{0} false) \times false[0, 1 false]

, and disjointly embedded 3‐dimensional 1‐handles

h_{1}, \dots, h_{n}

{double-struckR}^{3} \times I

, where

h_{i}

joins

L_{0} \times false[0, 1 false]

S_{i}

and is disjoint from

S_{j}

for

j \neq i

, such that

D

is obtained from

false(L_{0} \times [0, 1] false) \cup S_{1} \cup \dots \cup S_{n}

by embedded surgery along the handles

h_{1}, \dots, h_{n}

. By Proposition , we have

Kh false(D false) = \pm Kh false(L_{0} \times [0, 1] false) = \pm {prefixid}_{Kh (L_{0})}

.…”

Section: Applicationsmentioning

confidence: 99%

“…Actually, Gordon defines

F

to be ribbon if it has only index 1 and 2 critical points; thus, a concordance

F

is ribbon in Gordon's sense if and only if

\overset{F}{true}

is ribbon in our sense. Our reversed convention was introduced by the second author in . One justification for the change is that we prefer to treat a ribbon disk for a knot

K

as a cobordism from the empty link to

K

, rather than vice versa.…”

mentioning

confidence: 99%

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Khovanov homology and ribbon concordances

Levine

Zemke

2019

Bull. London Math. Soc.

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“…They similarly considered the conjugation action on the knot Floer complex. Zemke [Zem17] showed that, under an appropriate equivalence relation, the set of knot Floer complexes together with the extra structure given by the conjugation action form a group, and that there is a homomorphism from the knot concordance group to this group. The aim of this note is to prove an involutive analog of [Hom17, Theorem 1].…”

Section: Introductionmentioning

confidence: 99%