Examples of character varieties in characteristic $p$ and ramification

Abstract

We study ${\mathrm{SL}}_2(\mathbb{F})$-character varieties of knots over algebraically closed fields $\mathbb{F}$. We give a sufficient condition in terms of the double branched cover of a $2$-bridge knot (or, equivalently, of its Alexander polynomial) on the characteristic of $\mathbb{F}$, an odd prime, for the ${\mathrm{SL}}_2(\mathbb{F})$-character variety to present ramification phenomena. Finally we provide several explicit computations of character varieties to illustrate the result, exhibiting also other types of ramification.