Twisted Alexander polynomial and Reidemeister torsion

Abstract

In 1992, Wada [4] defined the twisted Alexander polynomial for finitely presentable groups. Let Γ be a finitely presentable group. We suppose that the abelianization Γ/[Γ, Γ] is a free abelian group Tr — ( t l 5 . . . , tr\tttj = tjti) of rank r. Then we will assign a Laurent polynomial Δr ) P (t i , . . . , tr) with a unique factorization domain i?-coefficients to each linear representation p : Γ -> GL(n R). We call it the twisted Alexander polynomial of Γ associated to p. For simplicity, we suppose that R is the real number field R and the image of p is included in SL(n] R). Because we are mainly interested in the case of the group of a knot, hereafter we suppose that Γ is a knot group. Let K C S be a knot and E its exterior of K. We denote the canonical abelianization of Γ by