The SL(3,C)-character variety of the figure eight knot

TL;DR

This paper explicitly describes the SL(3,C)-character variety of the figure eight knot, detailing its components, reducibility types, and symmetry actions, providing a comprehensive algebraic and geometric analysis.

Contribution

It provides explicit equations for the character variety of the figure eight knot in SL(3,C), including its decomposition into components and symmetry group actions.

Findings

Five components of the character variety identified

One component contains irreducible representations from Sym^2:SL(2,C)

Two components are induced by exceptional Dehn fillings

Abstract

We give explicit equations that describe the character variety of the figure eight knot for the groups SL(3,C), GL(3,C) and PGL(3,C). This has five components of dimension 2, one consisting of totally reducible representations, another one consisting of partially reducible representations, and three components of irreducible representations. Of these, one is distinguished as it contains the curve of irreducible representations coming from $Sym^2:SL(2,C) \to SL(3,C)$. The other two components are induced by exceptional Dehn fillings of the figure eight knot. We also describe the action of the symmetry group of the figure eight knot on the character varieties.