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

Abstract

For each Montesinos knot $K$, we propose an efficient method to explicitly determine the irreducible $\mathrm{SL}(2,\mathbb{C})$-character variety, and show that it can be decomposed as ${\mathcal{X}}_0(K)\bigsqcup {\mathcal{X}}_1(K)\bigsqcup {\mathcal{X}}_2(K)\bigsqcup {\mathcal{X}}^{\prime }(K)$, 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 ${\mathcal{X}}^{\prime }(K)$ consists of finitely many points when $K$ satisfies a generic condition.

18 pages, 5 figures; final version

Cited by in corpus (2)

• The adjoint Reidemeister torsion for the connected sum of knots
• The $\mathrm{SL}(2,\mathbb{C})$-character variety of the Borromean link