Metabelian SL(n, ℂ) representations of knot groups, II: Fixed points

Hans U. Boden and Stefan Friedl

Vol. 249 (2011), No. 1, 1–10

DOI: 10.2140/pjm.2011.249.1

Abstract

Given a knot K in an integral homology sphere Σ with exterior N_K, there is a natural action of the cyclic group ℤ∕n on the space of SL(n, ℂ) representations of the knot group π₁(N_K), which induces an action on the SL(n, ℂ) character variety. We identify the fixed points of this action in terms of characters of metabelian representations, and we apply this in order to show that the twisted Alexander polynomial Δ_K,1^α(t) associated to an irreducible metabelian SL(n, ℂ) representation α is actually a polynomial in tⁿ.

Keywords

metabelian representation, knot group, character variety, group action, fixed point

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 20C15

Milestones

Received: 20 September 2009

Revised: 9 February 2010

Accepted: 16 February 2010

Published: 3 January 2011