Deformations of metabelian representations of knot groups into $SL(3,\mathbb{C})$

TL;DR

This paper investigates the deformation space of certain reducible metabelian representations of knot groups into SL(3,C), showing they are smooth points with potential for irreducible non-metabelian deformations.

Contribution

It demonstrates that reducible metabelian representations linked to double roots of the Alexander polynomial are smooth points and can deform into irreducible non-metabelian representations.

Findings

Reducible metabelian representations are smooth points in the representation variety.

Such representations can be deformed into irreducible non-metabelian representations.

Deformations are associated with double roots of the Alexander polynomial.

Abstract

Let K be a knot in $S^3$ and $X$ its complement. We study deformations of reducible metabelian representations of the knot group $\pi_1(X)$ into $SL(3,\mathbb{C})$ which are associated to a double root of the Alexander polynomial. We prove that these reducible metabelian representations are smooth points of the representation variety and that they have irreducible non metabelian deformations.