Twisted Alexander polynomial for finitely presentable groups
type: Item
instance of: scholarly article
title: Twisted Alexander polynomial for finitely presentable groups
zbMATH Open document ID: 0822.57006
DOI: 10.1016/0040-9383(94)90013-2
author: Masaaki Wada
published in: Topology
publication date: 1995-10-17T00:00:00Z
review text: The usual definition of the Alexander polynomial of a link extends readily to a Laurent polynomial invariant for any epimorphism \(\alpha : \Gamma \to Z^ r\), where \(\Gamma\) is a finitely presentable group. This paper presents a further extension to an invariant depending also on a linear representation \(\rho : \Gamma \to \text{GL} (n,R)\), where \(R\) is a unique factorization domain. The resulting invariant \(\Delta_{\Gamma,\rho} (t_ 1, \dots, t_ r)\) is a rational function in the quotient field of \(R [t_ 1, \dots, t_ r]\), well defined up to multiplication by units of this ring. For example, if \(\Gamma = Z\) (generated by \(t\)) and \(\alpha = \text{id}_ Z\) then \(\Delta_{\Gamma, \rho} (t) = \text{det} (I - t\rho(t))^{-1}\). If \(r > 1\) then \(\Delta_{\Gamma, \rho}\) is a Laurent polynomial with coefficients in the field of fractions of \(R\). In the final section twisted polynomials associated with representations into \(\text{GL} (2,Z/ 7Z)\) are used to distinguish the two 11 crossing knots with ordinary Alexander polynomial 1. [Reviewer's remark. The general case can be subsumed into the special case \(r = 0\) (\(\alpha\) trivial), provided we assume that there is a generator whose image under \(\rho\) does not have 1 as an eigenvalue].
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57M05
Mathematics Subject Classification ID: 20F05
zbMATH DE Number: 590276
zbMATH Keywords: Alexander polynomial of a link
zbMATH Keywords: finitely presentable group
zbMATH Keywords: linear representation
zbMATH Keywords: twisted polynomials
reviewed by: Jonathan A. Hillman
MaRDI profile type: Publication
full work available at URL: 0040-9383(94)90013-2
OpenAlex ID: W1983365813
Wikidata QID: Q127491863
description: scientific article; zbMATH DE number 590276