Nonconcordant links with homology cobordant zero-framed surgery manifolds

Choon, Jae; Powell, Mark

doi:10.2140/pjm.2014.272.1

Cited by 10 publications

(3 citation statements)

References 45 publications

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“…Since any element in can be realised by a suitable framing on , the framing of can be modified in a small neighbourhood until the element in vanishes; see e.g. [CP14, p. 13] for details.…”

Section: Setting Up the Surgery Problemmentioning

confidence: 99%

Embedding spheres in knot traces

Feller¹,

Miller²,

Nagel³

et al. 2021

Compositio Math.

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The trace of the $n$ -framed surgery on a knot in $S^{3}$ is a 4-manifold homotopy equivalent to the 2-sphere. We characterise when a generator of the second homotopy group of such a manifold can be realised by a locally flat embedded $2$ -sphere whose complement has abelian fundamental group. Our characterisation is in terms of classical and computable $3$ -dimensional knot invariants. For each $n$ , this provides conditions that imply a knot is topologically $n$ -shake slice, directly analogous to the result of Freedman and Quinn that a knot with trivial Alexander polynomial is topologically slice.

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“…Since any element in can be realised by a suitable framing on , the framing of can be modified in a small neighbourhood until the element in vanishes; see e.g. [CP14, p. 13] for details.…”

Section: Setting Up the Surgery Problemmentioning

confidence: 99%

Embedding spheres in knot traces

Feller¹,

Miller²,

Nagel³

et al. 2021

Compositio Math.

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“…We note that many concordance invariants are defined via the 0-surgery of a knot (often with the positive meridian), and two knots with the same 0-surgery have the same concordance invariants for relevant orientations (see Problem 1.19 in [24] and introduction in [13]). Furthermore, if every homotopy 4-sphere is diffeomorphic to the standard one, then this conjecture is true when one knot is slice (see [24]).…”

Section: Introductionmentioning

confidence: 99%

“…On the other hand, Cochran, Franklin, Hedden, and Horn [15] produced non-concordant pairs of knots under the weaker assumption that 0-framed surgeries on knots are homology cobordant. For related results in the link case, we refer to [13] and the references therein. Recently Abe and Tagami [1] proved that, if the slice-ribbon conjecture is true, then the Akbulut-Kirby conjecture is false.…”

Section: Introductionmentioning

confidence: 99%

Corks, exotic 4-manifolds and knot concordance

Yasui¹

2015

Preprint

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We show that, for each integer n, there exist infinitely many pairs of n-framed knots representing homeomorphic but non-diffeomorphic (Stein) 4-manifolds, which are the simplest possible exotic 4-manifolds regarding handlebody structures. To produce these examples, we introduce a new description of cork twists and utilize satellite maps. As an application, we produce knots with the same 0-surgery which are not concordant for any orientations, disproving the Akbulut-Kirby conjecture given in 1978.

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A Topological Approach to Cheeger‐Gromov Universal Bounds for von Neumann ρ‐Invariants

Choon

2015

Comm Pure Appl Math

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Using deep analytic methods, Cheeger and Gromov showed that for any smooth .4k 1/-manifold there is a universal bound for the von Neumann L 2 -invariants associated to arbitrary regular covers. We present a proof of the existence of a universal bound for topological .4k 1/-manifolds, using L 2 -signatures of bounding 4k-manifolds. We give explicit linear universal bounds for 3-manifolds in terms of triangulations, Heegaard splittings, and surgery descriptions. We show that our explicit bounds are asymptotically optimal. As an application, we give new lower bounds of the complexity of 3-manifolds that can be arbitrarily larger than previously known lower bounds. As ingredients of the proofs that seem interesting on their own, we develop a geometric construction of efficient 4-dimensional bordisms of 3-manifolds over a group and develop an algebraic topological notion of uniformly controlled chain homotopies.

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Nonconcordant links with homology cobordant zero-framed surgery manifolds

Cited by 10 publications

References 45 publications

Embedding spheres in knot traces

Embedding spheres in knot traces

Corks, exotic 4-manifolds and knot concordance

A Topological Approach to Cheeger‐Gromov Universal Bounds for von Neumann ρ‐Invariants

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