Characterisation of homotopy ribbon discs

Conway, Anthony; Powell, Mark

doi:10.1016/j.aim.2021.107960

Cited by 10 publications

(23 citation statements)

References 18 publications

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“…Most relevant to us are the exotic ribbon discs from [Hay20]. In order to prove that his discs in D 4 are topologically isotopic, Hayden showed that their exteriors have group Z and appealed to [CP21]. From the perspective of this paper and [CP20], any two Z-ribbon discs are isotopic rel.…”

Section: Statement Of Resultsmentioning

confidence: 99%

“…The classification of closed surfaces in 4-manifolds whose exterior is simply-connected was carried out by Boyer [Boy93]; see also [Sun15]. Literature on the classification of discs in D 4 where the complement has fixed fundamental group includes [FT05,CP21,Con22]. For surfaces in more general 4-manifolds, [CP20] gave necessary and sufficient conditions for a pair of Z-surfaces to be equivalent.…”

mentioning

confidence: 99%

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$4$-manifolds with boundary and fundamental group $\mathbb{Z}$

Conway¹,

Piccirillo²,

Powell³

2022

Preprint

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We classify topological 4-manifolds with boundary and fundamental group Z, under some assumptions on the boundary. We apply this to classify surfaces in simply-connected 4manifolds with S 3 boundary, where the fundamental group of the surface complement is Z. We then compare these homeomorphism classifications with the smooth setting. For manifolds, we show that every Hermitian form over Z[t ±1 ] arises as the equivariant intersection form of a pair of exotic smooth 4-manifolds with boundary and fundamental group Z. For surfaces we have a similar result, and in particular we show that every 2-handlebody with S 3 boundary contains a pair of exotic discs.

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Section: Statement Of Resultsmentioning

confidence: 99%

mentioning

confidence: 99%

$4$-manifolds with boundary and fundamental group $\mathbb{Z}$

Conway¹,

Piccirillo²,

Powell³

2022

Preprint

Self Cite

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“…Moreover, he showed that for 𝐾 the untwisted Whitehead double of the trefoil and figure eight knots, 𝜂(𝐾) ≠ 0, and therefore that ker 𝐹 ∩ ker Ψ is nontrivial. Note that these examples are isotopy-equivariantly (topologically) slice by [12], as they have Alexander polynomial one. (4) All of  𝑆𝐼 , , and  are abelian, whereas  𝑆𝐼 is not [16].…”

Section: Theorem 13mentioning

confidence: 99%

“…Moreover, he showed that for

K

the untwisted Whitehead double of the trefoil and figure eight knots,

\eta (K) \ne 0

, and therefore that

\ker F \cap \ker \Psi

is nontrivial. Note that these examples are isotopy‐equivariantly (topologically) slice by [12], as they have Alexander polynomial one. (4)All of

\mathcal {AC}^{SI}

\mathcal {AC}

, and

\mathcal {C}

are abelian, whereas

\mathcal {C}^{SI}

is not [16]. So, the nontrivial commutators found by Di Prisa also lie in

\ker F \cap \ker \Psi

.…”

Section: Introductionmentioning

confidence: 99%

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

Miller

Powell

2023

Journal of London Math Soc

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We use the Blanchfield form to obtain a lower bound on the equivariant slice genus of a strongly invertible knot. For our main application, let K$K$ be a strongly invertible genus one slice knot with nontrivial Alexander polynomial. We show that the equivariant slice genus of an equivariant connected sum #nK$\#^n K$ is at least n/4$n/4$. We also formulate an equivariant algebraic concordance group, and show that the kernel of the forgetful map to the classical algebraic concordance group is infinite rank.

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“…In [13], Hayden presents a certain pair of slice disks

D

and

D^{\prime }

for

J

, each with complement having fundamental group

\mathbb {Z}

. By a result of Conway–Powell [6, Theorem 1.2], this implies that

D

and

D^{\prime }

are topologically isotopic. However, in [13, section 2.1], it is shown that

D

and

D^{\prime }

are not smoothly isotopic (or even diffeomorphic) rel boundary.…”

Section: Introductionmentioning

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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Characterisation of homotopy ribbon discs

Cited by 10 publications

References 18 publications

$4$-manifolds with boundary and fundamental group $\mathbb{Z}$

$4$-manifolds with boundary and fundamental group $\mathbb{Z}$

Strongly invertible knots, equivariant slice genera, and an equivariant algebraic concordance group

Equivariant knots and knot Floer homology

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