Metabelian SL(n, ℂ) representations of knot groups

Hans U. Boden and Stefan Friedl

Vol. 238 (2008), No. 1, 7–25

DOI: 10.2140/pjm.2008.238.7

Abstract

We give a classification of irreducible metabelian representations from a knot group into SL(n, ℂ) and GL(n, ℂ). If the homology of the n-fold branched cover of the knot is finite, we show that every irreducible metabelian SL(n, ℂ) representation is conjugate to a unitary representation and that the set of conjugacy classes of such representations is finite. In that case, we give a formula for this number in terms of the Alexander polynomial of the knot. These results are the higher rank generalizations of a result of Nagasato, who recently studied irreducible, metabelian SL(2, ℂ) representations of knot groups. Finally we deduce the existence of irreducible metabelian SL(n, ℂ) representations of the knot group for any knot with nontrivial Alexander polynomial.

Keywords

metabelian representation, knot group, Alexander polynomial, branched cover

Mathematical Subject Classification 2000

Primary: 57M25

Secondary: 20C15

Milestones

Received: 29 March 2008

Accepted: 24 July 2008

Published: 1 November 2008