Geometry of the SL(3,C)-character variety of torus knots

TL;DR

This paper provides a geometric description of the character variety of representations of torus knot groups into SL(3,C), GL(3,C), and PGL(3,C), enhancing understanding of their algebraic and geometric structures.

Contribution

It offers the first detailed geometric analysis of the SL(3,C)-character variety for torus knot groups, extending previous work on lower-dimensional cases.

Findings

Explicit geometric description of X(G) for SL(3,C)

Characterization of the structure of the variety in different groups

Insights into the algebraic and geometric properties of the 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(3,C), GL(3,C) and PGL(3,C).