On high-dimensional representations of knot groups

TL;DR

This paper demonstrates that for any non-trivial knot and sufficiently large n, the SL(n,C)-character variety contains high-dimensional components, extending previous results known for SL(2,C).

Contribution

It proves the existence of high-dimensional components in the SL(n,C)-character variety for any non-trivial knot when n is large, generalizing earlier work on SL(2,C).

Findings

High-dimensional components exist for large n and any non-trivial knot.

The result generalizes previous findings from SL(2,C) to SL(n,C).

Provides new insights into the structure of knot group representations.

Abstract

Given a hyperbolic knot $K$ and any $n\geq 2$ the abelian representations and the holonomy representation each give rise to an $(n-1)$-dimensional component in the $\operatorname{SL}(n,\Bbb{C})$-character variety. A component of the $\operatorname{SL}(n,\Bbb{C})$-character variety of dimension $\geq n$ is called high-dimensional. It was proved by Cooper and Long that there exist hyperbolic knots with high-dimensional components in the $\operatorname{SL}(2,\Bbb{C})$-character variety. We show that given any non-trivial knot $K$ and sufficiently large $n$ the $\operatorname{SL}(n,\Bbb{C})$-character variety of $K$ admits high-dimensional components.