#### Volume 18, issue 1 (2018)

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On high-dimensional representations of knot groups

### Stefan Friedl and Michael Heusener

Algebraic & Geometric Topology 18 (2018) 313–332
##### 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.

##### Keywords
knots, character, knot groups, representations
##### Mathematical Subject Classification 2010
Primary: 57M25, 57M27
Secondary: 57M50