On a Theorem of Burde and de Rham

TL;DR

This paper generalizes a theorem relating the zeros of the Alexander polynomial to group representations, introducing the Crowell group and linking it to twisted Alexander polynomials.

Contribution

It extends Burde and de Rham's theorem by defining the Crowell group and connecting the zeros of twisted Alexander polynomials to its representations.

Findings

Zeros of twisted Alexander polynomials correspond to Crowell group representations

Introduces the Crowell group as an extension of the knot group

Provides a new characterization of Alexander polynomial zeros

Abstract

We generalize a theorem of Burde and de Rham characterizing the zeros of the Alexander polynomial. Given a representation of a knot group $\pi$, we define an extension of $\pi$, the Crowell group. For any GL(n,C) representation of $\pi$, the zeros of the associated twisted Alexander polynomial correspond to representations of the Crowell group into the group of dilations of C^n.