## Algebraic & Geometric Topology

### On high-dimensional representations of knot groups

#### Abstract

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

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

#### Article information

Source
Algebr. Geom. Topol., Volume 18, Number 1 (2018), 313-332.

Dates
Revised: 4 April 2017
Accepted: 15 July 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.agt/1517454218

Digital Object Identifier
doi:10.2140/agt.2018.18.313

Mathematical Reviews number (MathSciNet)
MR3748245

Zentralblatt MATH identifier
1383.57004

#### Citation

Friedl, Stefan; Heusener, Michael. On high-dimensional representations of knot groups. Algebr. Geom. Topol. 18 (2018), no. 1, 313--332. doi:10.2140/agt.2018.18.313. https://projecteuclid.org/euclid.agt/1517454218

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