The concordance genus of a knot, II

Livingston, Charles

doi:10.2140/agt.2009.9.167

Cited by 5 publications

(5 citation statements)

References 24 publications

(60 reference statements)

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“…Therefore among the knots with up to twelve crossings only the sliceness status of the knot 12 a631 is unknown. (4) The concordance genus g c (K) of a knot K is defined to be the minimal genus among all knots concordant to K. Livingston [Liv09] uses twisted Alexander polynomials to show that the concordance genus of 10 82 equals three, which is its ordinary genus.…”

Section: Twistedmentioning

confidence: 99%

A Survey of Twisted Alexander Polynomials

Friedl

Vidussi

2011

The Mathematics of Knots

106

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Abstract. We give a short introduction to the theory of twisted Alexander polynomials of a 3-manifold associated to a representation of its fundamental group. We summarize their formal properties and we explain their relationship to twisted Reidemeister torsion. We then give a survey of the many applications of twisted invariants to the study of topological problems. We conclude with a short summary of the theory of higher order Alexander polynomials.

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Section: Twistedmentioning

confidence: 99%

A Survey of Twisted Alexander Polynomials

Friedl

Vidussi

2011

The Mathematics of Knots

106

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“…On the other hand for knots Kirk and Livingston [KL99a] used twisted torsion corresponding to 'Casson-Gordon'-type representations to give sliceness obstructions which go beyond Fox-Milnor. We also refer to [KL99b], [Ta02], [HKL08], [Liv09] and also [FV09] for more on twisted Alexander polynomials of knots and their relation to knot concordance.…”

Section: Introductionmentioning

confidence: 99%

Twisted torsion invariants and link concordance

Choon

Friedl

2011

Forum Mathematicum

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Abstract. The twisted torsion of a 3-manifold is well-known to be zero whenever the corresponding twisted Alexander module is non-torsion. Under mild extra assumptions we introduce a new twisted torsion invariant which is always non-zero. We show how this torsion invariant relates to the twisted intersection form of a bounding 4-manifold, generalizing a theorem of Milnor to the non-acyclic case. Using this result, we give new obstructions to 3-manifolds being homology cobordant and to links being concordant. These obstructions are sufficiently strong to detect that the Bing double of the figure eight knot is not slice.

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“…The twisted Alexander polynomials associated to cyclic covers of knots are powerful tools for distinguishing knots from their mutants, even up to topological concordance, as demonstrated by Livingston et al in [KL99b], [HKL10], and [Liv09]. For example, Herald, Kirk, and Livingston demonstrate in [HKL10] that the 24 distinct oriented mutants of P (3, 7, 9, 11, 15) are mutually distinct in the topological concordance group.…”

Section: Introductionmentioning

confidence: 97%

“…However, as observed by Long in [Lon14], the pretzel knots K ± m,n have only 2-torsion in their prime power cyclic branched covers. So we will need the following theorem, which follows immediately from the proof of Theorem 2.3, as observed by [Liv09].…”

Section: Introductionmentioning

confidence: 99%