Khovanov homology and exotic surfaces in the 4-ball

Hayden, Kyle; Sundberg, Isaac

doi:10.48550/arxiv.2108.04810

Cited by 7 publications

(7 citation statements)

References 20 publications

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“…However, in [13, section 2.1], it is shown that 𝐷 and 𝐷 ′ are not smoothly isotopic (or even diffeomorphic) rel boundary. (See also [18,Theorem 3.2].) Note that 𝐽 admits a strong inversion 𝜏; a crucial part of the argument in [13] relies on the fact that 𝐷 and 𝐷 ′ are related by the obvious extension of 𝜏 over 𝐵 4 .…”

Section: Applicationsmentioning

confidence: 99%

Equivariant knots and knot Floer homology

Dai,

Mallick,

Stoffregen

2023

Journal of Topology

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We define several equivariant concordance invariants using knot Floer homology. We show that our invariants provide a lower bound for the equivariant slice genus and use this to give a family of strongly invertible slice knots whose equivariant slice genus grows arbitrarily large, answering a question of Boyle and Issa. We also apply our formalism to several seemingly nonequivariant questions. In particular, we show that knot Floer homology can be used to detect exotic pairs of slice disks, recovering an example due to Hayden, and extend a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route toward establishing the noncommutativity of the equivariant concordance group.

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Section: Applicationsmentioning

confidence: 99%

Equivariant knots and knot Floer homology

Dai,

Mallick,

Stoffregen

2023

Journal of Topology

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“…boundary, including exotic examples [HS21, LS21, SS21]. Moreover, the approach in [HS21] demonstrates the computability of such maps: by carefully choosing a cycle ψ ∈ CKh(L) from the Khovanov chain complex of L, we can control the complexity of calculating the induced chain maps CKh(S 0 )(ψ) and CKh(S 1 )(ψ). Moreover, the calculations in this paper only require a subset of the Morse and Reidemeister induced chain maps (c.f.…”

Section: Obstructions From Khovanov Homologymentioning

confidence: 99%

Seifert surfaces in the 4-ball

Hayden¹,

Kim²,

Miller³

et al. 2022

Preprint

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We answer a question of Livingston from 1982 by producing Seifert surfaces of the same genus for a knot in S 3 that do not become isotopic when their interiors are pushed into B 4 . We give examples where the surfaces are not topologically isotopic in B 4 , as well as examples that are topologically but not smoothly isotopic. These latter surfaces are distinguished by their associated cobordism maps on Khovanov homology, and our calculations demonstrate the stability and computability of these maps under certain satellite operations.

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“…For closed surfaces, this invariant turns out not to be interesting: it vanishes if some component of the surface is not a torus, and otherwise is 2 n if the surface consists of n tori [146,60]. On the other hand, for surfaces with boundary a nontrivial link in S 3 , Khovanov homology does give an interesting invariant [169], even distinguishing some surfaces that are topologically isotopic [64].…”

Section: Applicationsmentioning

confidence: 99%