Metabelian SL(n,C) representations of knot groups, III: deformations

TL;DR

This paper investigates the deformation theory of metabelian SL(n,C) representations of knot groups, establishing conditions for smoothness and existence of large families of irreducible representations, linking twisted cohomology to branched cover cohomology.

Contribution

It provides new criteria for the smoothness of representation varieties and constructs large families of irreducible representations for knots in homology 3-spheres.

Findings

Dimension of first twisted cohomology equals a specific value under certain conditions.

Existence of smooth (n-1)-dimensional families of irreducible representations near specific points.

Connection between twisted cohomology conditions and untwisted cohomology of branched covers.

Abstract

Given a knot K and an irreducible metabelian SL(n,C) representation we establish an equality for the dimension of the first twisted cohomology. In the case of equality, we prove that the representation must have finite image and that it is conjugate to an SU(n) representation. In this case we show it determines a smooth point x in the SL(n,C) character variety, and we use a deformation argument to establish the existence of a smooth (n-1)-dimensional family of characters of irreducible SL(n,C) representations near x. Combining this with our previous existence results, we deduce the existence of large families of irreducible SU(n) and SL(n,C) non-metabelian representation for knots K in homology 3-spheres S with nontrivial Alexander polynomial. We then relate the condition on twisted cohomology to a more accessible condition on untwisted cohomology of a certain metabelian branched cover of S branched along K.