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On the local structure of the $\mathrm{SL}(n,\mathbb{C})$ representation variety of knot groups

Leila Ben Abdelghani and Michael Heusener

Vol. 8 (2026), No. 1, 1–40
Abstract

We study the local structure of the representation variety of a knot group into the special linear group of degree n over the complex numbers at certain diagonal representations. In particular we determine the tangent cone of the representation variety at these diagonal representations, and show that the latter can be deformed into irreducible representations. Furthermore, we use Luna’s slice theorem to analyze the local structure of the character variety.

Keywords
knot group, variety of representations, deformations of reducible representations
Mathematical Subject Classification
Primary: 57K31
Secondary: 20C99, 57M05
Milestones
Received: 20 July 2024
Revised: 25 January 2025
Accepted: 9 February 2025
Published: 13 January 2026
Authors
Leila Ben Abdelghani
Département de Mathématiques
Faculté des Sciences
Université de Monastir
Monastir
Tunisia
Michael Heusener
Laboratoire de Mathématiques Blaise Pascal - UMR 6620 - CNRS
Université Clermont Auvergne
Aubière
France