Concordance groups of links

Donald, Andrew; Owens, Brendan

doi:10.2140/agt.2012.12.2069

Cited by 16 publications

(16 citation statements)

References 41 publications

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“…Proof of Proposition By restriction

Σ (L)

inherits a spin structure

frakturt

from

W

. Turaev [, Section 2.2] has shown that to each orientation

o

L

one can associate a spin structure

t_{o}

Σ (L)

, and Donald and Owens [, Proposition 3.3] gave the following interpretation of

t_{o}

. Fix a Seifert surface for the oriented link

(L, o)

, and push it into the 4‐ball, obtaining a surface

F_{o}

; the branched double cover

Σ (B^{4}, F_{o})

admits a spin structure

s_{F_{o}}

(the pull‐back of the spin structure on

B^{4}

), and

t_{o}

is the restriction of

s_{F_{o}}

Σ (L)

.…”

Section: Lens Spacesmentioning

confidence: 99%

Embedding 3‐manifolds in spin 4‐manifolds

Aceto

Golla

Larson

2017

Journal of Topology

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An invariant of orientable 3-manifolds is defined by taking the minimum n such that a given 3-manifold embeds in the connected sum of n copies of S 2 × S 2 , and we call this n the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of lens spaces and Brieskorn spheres. We show how to construct rational and integral homology spheres whose embedding numbers grow arbitrarily large, and which can be calculated exactly if we assume the 11/8-Conjecture. In a different direction we show that any simply connected 4-manifold can be split along a rational homology sphere into a positive definite piece and a negative definite piece.

show abstract

“…Proof of Proposition By restriction

Σ (L)

inherits a spin structure

frakturt

from

W

. Turaev [, Section 2.2] has shown that to each orientation

o

L

one can associate a spin structure

t_{o}

Σ (L)

, and Donald and Owens [, Proposition 3.3] gave the following interpretation of

t_{o}

. Fix a Seifert surface for the oriented link

(L, o)

, and push it into the 4‐ball, obtaining a surface

F_{o}

; the branched double cover

Σ (B^{4}, F_{o})

admits a spin structure

s_{F_{o}}

(the pull‐back of the spin structure on

B^{4}

), and

t_{o}

is the restriction of

s_{F_{o}}

Σ (L)

.…”

Section: Lens Spacesmentioning

confidence: 99%

Embedding 3‐manifolds in spin 4‐manifolds

Aceto

Golla

Larson

2017

Journal of Topology

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“…For nonsplit alternating links with more than one component, algorithmically ribbon does not imply slice; rather it implies that the link bounds a surface of Euler characteristic one, not necessarily orientable, with no closed components. Such links were called χ-slice in [8].…”

Section: Algorithmic Band Moves and Isotopiesmentioning

confidence: 99%

An algorithm to find ribbon disks for alternating knots

Owens¹,

Swenton²

2021

Preprint

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We describe an algorithm to find ribbon disks for alternating knots, and the results of a computer implementation of this algorithm. The algorithm is underlain by a slice link obstruction coming from Donaldson's diagonalisation theorem. It successfully finds ribbon disks for slice two-bridge knots and for the connected sum of any alternating knot with its reverse mirror, as well as for 81,560 prime alternating knots of 19 or fewer crossings. We also identify some examples of ribbon alternating knots for which the algorithm fails to find ribbon disks.

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“…This does not give rise to a group in the same way for links of more than one component. For various approaches to defining concordance groups of links, see [8,24,28].…”

Section: The Concordance Group Of Knotsmentioning

confidence: 99%

Knots and 4-manifolds

Owens¹

2021

Winter Braids Lecture Notes

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These notes are based on the lectures given by the author during Winter Braids IX in Reims in March 2019. We discuss slice knots and why they are interesting, as well as some ways to decide if a given knot is or is not slice. We describe various methods for drawing diagrams of double branched covers of knots in the 3-sphere and surfaces in the 4-ball, and how these can be useful to decide if an alternating knot is slice. We include a description of the computer search for slice alternating knots due to the author and Frank Swenton.

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Concordance groups of links

Cited by 16 publications

References 41 publications

Embedding 3‐manifolds in spin 4‐manifolds

Embedding 3‐manifolds in spin 4‐manifolds

An algorithm to find ribbon disks for alternating knots

Knots and 4-manifolds

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