SL(n,C)- Representation spaces of Knot Groups

TL;DR

This paper explores the structure of representation spaces of knot groups into SL(n,C), highlighting differences between various algebraic structures and presenting recent results on their properties for n ≥ 3.

Contribution

It provides a detailed analysis of the representation and character varieties of knot groups into SL(n,C), including examples of non-scheme reduced representations and recent advances for n ≥ 3.

Findings

Discussion of tangent space vs. representation scheme differences

Example of non scheme reduced representation by Lubotzky and Magid

Recent results on representation varieties for n ≥ 3

Abstract

The first part of this article is a general introduction to the the theory of representation spaces of discrete groups into SL(n,C). Special attention is paid to knot groups. In Section 2 we discuss the difference between the tangent space at the representation variety, and the representation scheme. We give an example of Lubotzky and Magid of a non scheme reduced representation (see Example 2.18). In the second part recent results about the representation and character varieties of knot groups into SL(n,C) with n $\ge$ 3 are presented. This second part concerns mostly joint work with L. Ben Abdelghani, O. Medjerab, V. Mu{\~n}os and J. Porti.