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 $(n-1)$-dimensional component in the $\mathrm{SL}(n,\mathbb{C})$-character variety. A component of the $\mathrm{SL}(n,\mathbb{C})$-character variety of dimension $\ge n$ is called high-dimensional. It was proved by Cooper and Long that there exist hyperbolic knots with high-dimensional components in the $\mathrm{SL}(2,\mathbb{C})$-character variety. We show that given any non-trivial knot $K$ and sufficiently large $n$ the $\mathrm{SL}(n,\mathbb{C})$-character variety of $K$ admits high-dimensional components.

15 pages

References in corpus (1)

• Geometry of the SL(3,C)-character variety of torus knots