Metabelian SL(n,C) representations of knot groups

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 irreducible metabelian SL(n,C) representations of the knot group for any knot with nontrivial Alexander polynomial.

18 pages

References in corpus (2)

• Non-triviality of the A-polynomial for knots in S^3
• Eta invariants as sliceness obstructions and their relation to Casson-Gordon invariants

Cited by in corpus (9)

• Left-orderability and cyclic branched coverings
• Symmetric chain complexes, twisted Blanchfield pairings, and knot concordance
• Representations of knot groups into ${\mathrm{SL}}_n(\mathbf{C})$ and twisted Alexander polynomials
• Metabelian SL(n,C) representations of knot groups, III: deformations
• Metabelian SL(n,C) representations of knot groups IV: twisted Alexander polynomials
• Metabelian SL(n,C) representations of knot groups II: fixed points
• Small Dehn surgery and SU(2)
• Quantum statistical mechanics in arithmetic topology
• Sutured Manifolds and Polynomial Invariants from Higher Rank Bundles