The SU(2)-character varieties of torus knots

TL;DR

This paper explores the relationship between SU(2) and SL(2,C) representations of torus knot groups, providing a geometric description of the SU(2)-character variety within the SL(2,C)-character variety.

Contribution

It offers a detailed geometric characterization of SU(2)-representations of torus knot groups based on their SL(2,C)-character varieties, advancing understanding of their structure.

Findings

Description of the SU(2)-character variety as a subset of the SL(2,C)-character variety

Geometric insights into the structure of representations of torus knot groups

Connections between representation types and character varieties

Abstract

Let G be the fundamental group of the complement of the torus knot of type (m,n). We study the relationship between SU(2) and SL(2,C)-representations of this group, looking at their characters. Using the description of the SL(2,C)-character variety of G, X(G), we give a geometric description of Y(G)\subset X(G), the set of characters arising from SU(2)-representations.