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

Haimiao Chen

Published 2021-02-03Version 1

For each Montesinos knot $K$, we find a simple method to determine the ${\rm SL}(2,\mathbb{C})$-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 "connected sums" of representations of the constituent rational links, $\mathcal{X}_2(K)$ is a high-genus algebraic curve, and $\mathcal{X}'(K)$ generically consists of finitely many points.