METABELIAN SL(N, ℂ) REPRESENTATIONS OF KNOT GROUPS, III: DEFORMATIONS

Given a knot K with complement NK and an irreducible metabelian SL(n, ℂ) representation α : π1(NK)→SL(n, ℂ), we establish the inequality dim H1(NK; sl(n, ℂ)Adα)≥n−1. In the case of equality, we prove that α must have finite image and is conjugate to an SU(n) representation. In this case, we show α determines a smooth point ξα in the SL(n, ℂ) character variety, and we use a deformation argument to establish the existence of a smooth …