The ${\rm SL}(2,\mathbb{C})$-character variety of $8_{18}$

TL;DR

This paper provides a detailed, software-free method to explicitly determine the ${ m SL}(2, ext{C})$-character variety of the hyperbolic knot $8_{18}$, enhancing understanding of its geometric structure.

Contribution

It introduces a new, accessible approach for computing character varieties without computer assistance, specifically applied to the hyperbolic knot $8_{18}$.

Findings

Explicit description of the ${ m SL}(2, ext{C})$-character variety for $8_{18}$

Development of an efficient method for simultaneous conjugacy classes in ${ m SL}(2, ext{C})$

Clarification of the structure of the character variety for a non-arborescent hyperbolic knot

Abstract

The knot $8_{18}$ is the first non-arborescent hyperbolic knot. In 2020, Paoluzzi and Porti found its ${\rm SL}(2,\mathbb{C})$-character variety with the aid of a computer, but many details were omitted. In this paper, we determine the character variety by a software-free procedure, which is easy to follow and enlightening. Along the way, we develop an efficient method for working with simultaneous conjugacy classes of four elements of ${\rm SL}(2,\mathbb{C})$.