Alexander–lin Twisted Polynomials

Silver, Daniel S.; Williams, Susan G.

doi:10.1142/s0218216511008838

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…also [Mo08]) that there exists an alternating knot such that a twisted Reidemeister torsion is monic, but which is not fibered. (3) Theorem 6.2 has been generalized by Silver and Williams [SW09d] to give an obstruction for a general group to admit an epimorphism onto Z such that the kernel is a finitely generated free group.…”

Section: Remarkmentioning

confidence: 99%

A Survey of Twisted Alexander Polynomials

Friedl

Vidussi

2011

The Mathematics of Knots

107

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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: Remarkmentioning

confidence: 99%

A Survey of Twisted Alexander Polynomials

Friedl

Vidussi

2011

The Mathematics of Knots

107

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“…We refer the reader to the survey papers [FV5,Mo] and the recent preprint [DFL] for details and references. As this article has been already referred in the papers [DFJ,DFV,FKK,FV2,FV3,FV4,FV5,FV6,FV7,KM,SW] and frequently suggested to be published, we think that it might be worthwhile to have it published.…”

Section: Introductionmentioning

confidence: 92%

Normalization of twisted Alexander invariants

Kitayama

2015

Int. J. Math.

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Twisted Alexander invariants of knots are well-defined up to multiplication of units. We get rid of this multiplicative ambiguity via a combinatorial method and define normalized twisted Alexander invariants. We then show that the invariants coincide with sign-determined Reidemeister torsion in a normalized setting, and refine the duality theorem. We further obtain necessary conditions on the invariants for a knot to be fibered, and study behavior of the highest degrees of the invariants.2010 Mathematics Subject Classification. Primary 57M25, Secondary 57M05, 57Q10. Key words and phrases. twisted Alexander invariant, Reidemeister torsion, fibered knot, free genus.(See Theorem 6.6.)This paper is organized as follows. In the next section, we first review the definition of twisted Alexander invariants for knots. We also describe how to compute them from a presentation of a knot group and the duality theorem for unitary representations. In Section 3, we review Turaev's sign-determined Reidemeister torsion and the relation with twisted Alexander invariants. In Section 4, we establish normalization of twisted Alexander invariants. In Section 5, we refine the correspondence with sign-determined Reidemeister torsion and the duality theorem for twisted Alexander invariants. Section 6 is devoted to applications. Here we extend the result of Cha [C], Goda-Kitano-Morifuji [GKM] and Friedl-Kim [FK] for fibered knots, and study behavior of the highest degrees of the normalized invariants to obtain (1.1).Note. This article appeared first in 2007 on the arXiv, and has remained long to be unpublished. Since then twisted Alexander invariants and Reidemeister torsion for knots and 3-manifolds have been further intensively studied by many researchers. We refer the reader to the survey papers [FV5, Mo] and the recent preprint [DFL] for details and references. As this article has been already referred in the papers [DFJ, DFV, FKK, FV2, FV3, FV4, FV5, FV6, FV7, KM, SW] and frequently suggested to be published, we think that it might be worthwhile to have it published.

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“…It is an invariant of the triple (G, ǫ, x 0 ) (equivalence defined as for pairs (G, ǫ) but respecting the distinguished group element x 0 ) and the conjugacy class of the representation ρ (see [16]).…”

Section: Twisted Alexander Polynomialsmentioning

confidence: 99%

Twisted Alexander Invariants of Twisted Links

Silver

Williams

2012

J. Knot Theory Ramifications

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Let L = ℓ 1 ∪ · · · ∪ ℓ d+1 be an oriented link in S 3 , and let L(q) be the d-component link ℓ 1 ∪ · · · ∪ ℓ d regarded in the homology 3-sphere that results from performing 1/q-surgery on ℓ d+1 . Results about the Alexander polynomial and twisted Alexander polynomials of L(q) corresponding to finite-image representations are obtained. The behavior of the invariants as q increases without bound is described.

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Alexander–lin Twisted Polynomials

Cited by 5 publications

References 20 publications

A Survey of Twisted Alexander Polynomials

A Survey of Twisted Alexander Polynomials

Normalization of twisted Alexander invariants

Twisted Alexander Invariants of Twisted Links

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