# Examples of character varieties in characteristic $p$ and ramification

## TL;DR

This paper investigates how the characteristic of an algebraically closed field influences the ramification phenomena in SL_2-character varieties of knots, providing theoretical criteria and explicit examples.

## Contribution

It establishes a criterion based on the Alexander polynomial for ramification in character varieties over fields of prime characteristic, with explicit computations illustrating the phenomena.

## Key findings

- Ramification occurs under specific conditions related to the Alexander polynomial.
- Explicit examples demonstrate different types of ramification phenomena.
- The criterion links the field's characteristic to the topology of the knot via the double branched cover.

## 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.