Twisted Alexander invariants of complex hypersurface complements

Laurenţiu Maxim; Kaiho Tommy Wong

doi:10.1017/s0308210518000094

Twisted Alexander invariants of complex hypersurface complements

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210518000094 ◽

2018 ◽

Vol 148 (5) ◽

pp. 1049-1073 ◽

Cited By ~ 2

Author(s):

Laurenţiu Maxim ◽

Kaiho Tommy Wong

Keyword(s):

Alexander Polynomials ◽

Alexander Invariants

We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.

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Twisted Alexander Invariants Detect Trivial Links

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-097-x ◽

2017 ◽

Vol 60 (2) ◽

pp. 283-299

Author(s):

Stefan Friedl ◽

Stefano Vidussi

Keyword(s):

Alexander Polynomials ◽

Alexander Invariants

AbstractIt follows fromearlier work of Silver andWilliams and the authors that twisted Alexander polynomials detect the unknot and theHopf link. We now show that twisted Alexander polynomials also detect the trefoil and the ûgure-Ç knot, that twisted Alexander polynomials detect whether a link is split and that twisted Alexander modules detect trivial links. We use this result to provide algorithms for detecting whether a link is the unlink, whether it is split, and whether it is totally split.

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Knot adjacency from a surgical view of Alexander invariants

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517500985 ◽

2017 ◽

Vol 26 (14) ◽

pp. 1750098

Author(s):

Yasutaka Nakanishi ◽

Masahiro Shimoda

Keyword(s):

Alexander Polynomials ◽

Alexander Invariants ◽

Empty Subset

For two knots [Formula: see text] and [Formula: see text], [Formula: see text] is said to be [Formula: see text]-adjacent to [Formula: see text], if [Formula: see text] admits a knot diagram containing [Formula: see text] crossings such that crossing changes at any non-empty subset of them yield a knot diagram of [Formula: see text]. Using a surgical view of Alexander invariants, we will characterize the Alexander polynomials of knots which are [Formula: see text]-adjacent to the trivial knot.

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Distinguishing mutant pretzel knots in concordance

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216517500419 ◽

2017 ◽

Vol 26 (07) ◽

pp. 1750041 ◽

Cited By ~ 1

Author(s):

Allison N. Miller

Keyword(s):

Prime Power ◽

Reidemeister Torsion ◽

Alexander Polynomials ◽

Cyclic Covers ◽

Pretzel Knots ◽

Alexander Invariants

We prove that many four-strand pretzel knots of the form [Formula: see text] are not topologically slice, even though their positive mutants [Formula: see text] are ribbon. We use the sliceness obstruction of Kirk and Livingston [Twisted Alexander invariants, Reidemeister torsion, and Casson–Gordon invariants, Topology 38 (1999) 635–661], related to the twisted Alexander polynomials associated to prime power cyclic covers of knots.

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TWISTED ALEXANDER INVARIANTS OF TWISTED LINKS

Journal of Knot Theory and Its Ramifications ◽

10.1142/s0218216512501180 ◽

2012 ◽

Vol 21 (11) ◽

pp. 1250118

Author(s):

DANIEL S. SILVER ◽

SUSAN G. WILLIAMS

Keyword(s):

Alexander Polynomial ◽

Oriented Link ◽

Alexander Polynomials ◽

Finite Image ◽

Alexander Invariants ◽

Image Representations

Let L = ℓ1 ∪⋯∪ℓd+1 be an oriented link in 𝕊3, and let L(q) be the d-component link ℓ1 ∪⋯∪ℓd regarded in the homology 3-sphere that results from performing 1/q-surgery on ℓd+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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Profinite rigidity for twisted Alexander polynomials

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2020-0014 ◽

2020 ◽

Vol 0 (0) ◽

Author(s):

Jun Ueki

Keyword(s):

Rigidity Theorem ◽

Homology Groups ◽

Alexander Polynomials ◽

Twisted Homology ◽

Hyperbolic Volumes

AbstractWe formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a {{\mathbb{Z}}}-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley’s parabolic representations of two-bridge knot groups and give a remark on hyperbolic volumes.

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