On high-dimensional representations of knot groups

Friedl, Stefan; Heusener, Michael

doi:10.2140/agt.2018.18.313

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On high-dimensional representations of knot groups

Stefan Friedl and Michael Heusener

Algebraic & Geometric Topology 18 (2018) 313–332

DOI: 10.2140/agt.2018.18.313

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 $SL (n, ℂ)$ –character variety. A component of the $SL (n, ℂ)$ –character variety of dimension $\geq n$ 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.

Keywords

knots, character, knot groups, representations

Mathematical Subject Classification 2010

Primary: 57M25, 57M27

Secondary: 57M50

References

Publication

Received: 14 October 2016

Revised: 4 April 2017

Accepted: 15 July 2017

Published: 10 January 2018

Authors

Stefan Friedl
Fakultät für Mathematik Universität Regensburg Regensburg Germany

Michael Heusener
Laboratoire de Mathématiques Blaise Pascal - UMR 6620 - CNRS Université Clermont Auvergne Campus des Cézeaux Aubière France
http://math.univ-bpclermont.fr/~heusener/

Volume 18, issue 1 (2018)