The ${\rm SL}(2,\mathbb{C})$-character variety of a Montesinos knot

TL;DR

This paper introduces an efficient method to explicitly compute the irreducible ${ m SL}(2,bC)$-character variety of Montesinos knots, revealing its decomposition into distinct algebraic subsets with concrete descriptions.

Contribution

It provides a novel, explicit approach to determine the character variety of Montesinos knots and describes its natural decomposition into several algebraic components.

Findings

Decomposition of the character variety into four algebraic parts.

Explicit description of each component, including trace-free characters and algebraic curves.

Identification of finitely many points in the generic case.

Abstract

For each Montesinos knot $K$, we propose an efficient method to explicitly determine the irreducible ${\rm SL}(2,\mathbb{C})$-character variety, and show that it can be decomposed as $\mathcal{X}_0(K)\sqcup\mathcal{X}_1(K)\sqcup\mathcal{X}_2(K)\sqcup\mathcal{X}'(K)$, where $\mathcal{X}_0(K)$ consists of trace-free characters, $\mathcal{X}_1(K)$ consists of characters of "unions" of representations of rational knots (or rational link, which appears at most once), $\mathcal{X}_2(K)$ is an algebraic curve, and $\mathcal{X}'(K)$ consists of finitely many points when $K$ satisfies a generic condition.