The SL(2,C)-character varieties of torus knots

TL;DR

This paper provides a geometric description of the $SL(2,C)$-character variety for the fundamental group of torus knot complements, enhancing understanding of their representation spaces.

Contribution

It explicitly characterizes the structure of the $SL(2,C)$-character variety for torus knot groups, a novel geometric insight.

Findings

Explicit geometric description of $X(G)$

Characterization of representation spaces for torus knots

Advancement in understanding knot group representations

Abstract

Let $G$ be the fundamental group of the complement of the torus knot of type $(m,n)$. This has a presentation $G=<x,y|x^m=y^n>$. We find the geometric description of the character variety $X(G)$ of characters of representations of $G$ into $SL(2,C)$.