Corks, covers, and complex curves

Hayden, Kyle

doi:10.48550/arxiv.2107.06856

Cited by 5 publications

(13 citation statements)

References 20 publications

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“…Surprisingly, our invariants also have applications to several (seemingly) non-equivariant questions. We first show that our formalism can be used to detect exotic pairs of slice disks, recovering an example originally due to Hayden [Hay21]. Note that while knot Floer homology has previously been used to detect exotic higher-genus surfaces (see the work of Juhász-Miller-Zemke [JMZ20]), the current work represents the first such application of knot Floer homology in the genus-zero case.…”

Section: Introductionmentioning

confidence: 80%

“…The topological intuition behind these examples is quite straightforward: the involutions τ # and τ sw on T 2n,2n`1 #T 2n,2n`1 are very different, so one should expect the equivariant slice genus of pK n , τ n q to be large. We also consider a particular knot J due to Hayden [Hay21], displayed in Figure 2. In [Hay21], Hayden presents a certain pair of slice disks D and D 1 for J, each with complement having fundamental group Z.…”

Section: Introductionmentioning

confidence: 99%

“…We also consider a particular knot J due to Hayden [Hay21], displayed in Figure 2. In [Hay21], Hayden presents a certain pair of slice disks D and D 1 for J, each with complement having fundamental group Z. By a result of Conway-Powell [CP19, Theorem 1.2], this implies that D and D 1 are topologically isotopic.…”

Section: Introductionmentioning

confidence: 99%

“…(See also [HS21,Theorem 3.2]. ) Note that J admits a strong inversion τ ; a crucial part of the argument in [Hay21] relies on the fact that D and D 1 are related by the obvious extension of τ over B 4 . Figure 2.…”

Section: Introductionmentioning

confidence: 99%

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Equivariant knots and knot Floer homology

Dai¹,

Mallick²,

Stoffregen³

2022

Preprint

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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 provide a re-proof of a result due to Miller and Powell regarding stabilization distance. Our formalism suggests a possible route towards establishing the non-commutativity of the equivariant concordance group.

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Section: Introductionmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Equivariant knots and knot Floer homology

Dai¹,

Mallick²,

Stoffregen³

2022

Preprint

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“…If pS 3 `1pKq, τ q is a cork, then the cork-twist action on HF ˝is identified with the action of τ K on a sub(quotient)-complex of CF K 8 pKq via the Theorem 1.1. We can apply this identification to many well-known corks in the literature, they include p`1q-surgery on the Stevedore knot, p`1q-surgery on the P p´3, 3, ´3q pretzel knot (also known as the Akbulut cork), and the positron cork [AM97,Hay21]. In fact, the identification of the two involution for the positron cork was useful in [DMS22] to re-prove a result due to [Hay21].…”

Section: Cf ˝Psmentioning

confidence: 99%

Knot Floer homology and surgery on equivariant knots

Mallick¹

2022

Preprint

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Given an equivariant knot K of order 2, we study the induced action of the symmetry on the knot Floer homology. We relate this action with the induced action of the symmetry on the Heegaard Floer homology of large surgeries on K. This surgery formula can be thought of as an equivariant analog of the involutive large surgery formula proved by Hendricks and Manolescu. As a consequence, we obtain that for certain double branched covers of S 3 and corks, the induced action of the involution on Heegaard Floer homology can be identified with an action on the knot Floer homology. As an application, we calculate equivariant correction terms which are invariants of a generalized version of the spin rational homology cobordism group, and define two knot concordance invariants. We also compute the action of the symmetry on the knot Floer complex of K for several equivariant knots.

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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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Corks, covers, and complex curves

Cited by 5 publications

References 20 publications

Equivariant knots and knot Floer homology

Equivariant knots and knot Floer homology

Knot Floer homology and surgery on equivariant knots

Equivariant knots and knot Floer homology

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