On high-dimensional representations of knot groups

Abstract

Given a hyperbolic knot $K$ and any $n\ge 2$ the abelian representations and the holonomy representation each give rise to an $\left(n-1\right)$–dimensional component in the $SL\left(n,ℂ\right)$–character variety. A component of the $SL\left(n,ℂ\right)$–character variety of dimension $\ge n$ is called high-dimensional.

It was proved by D Cooper and D Long that there exist hyperbolic knots with high-dimensional components in the $SL\left(2,ℂ\right)$–character variety. We show that given any nontrivial knot $K$ and sufficiently large $n$ the $SL\left(n,ℂ\right)$–character variety of $K$ admits high-dimensional components.