Twisted Alexander polynomial for finitely presentable groups
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].
scientific article; zbMATH DE number 590276