Character varieties for SL(3,C): the figure eight knot

TL;DR

This paper describes the structure of representation and character varieties for the figure eight knot in SL(3,C), providing explicit parametrizations and boundary projections, advancing understanding of knot group representations.

Contribution

It offers an explicit parametrization of SL(3,C) representations of the figure eight knot group and details the boundary character variety projection.

Findings

Explicit parametrization of representation matrices

Description of boundary torus character variety projection

Analysis of several representation varieties for the knot group

Abstract

We give a description of several representation varieties of the fundamental group of the complement of the figure eight knot in PGL(3,C) or SL(3,C). We moreover obtain an explicit parametrization of matrices generating the representation and a description of the projection of the representation variety into the character variety of the boundary torus into SL(3,C).