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Twisted Alexander polynomial for finitely presentable groups. (English) Zbl 0822.57006
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].

MSC:
 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57M05 Fundamental group, presentations, free differential calculus 20F05 Generators, relations, and presentations of groups
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