Character varieties of double twist links

TL;DR

This paper computes and analyzes the structure of $SL_2( ext{C})$ character varieties for double twist links, revealing their birational equivalence to surfaces and calculating their genus and irrationality degree.

Contribution

It provides explicit models and geometric invariants for the character varieties of an infinite family of double twist links, including genus and degree of irrationality.

Findings

Character varieties are birational to surfaces of the form C×C.

Genus of the associated curve is computed.

Degree of irrationality of the canonical component is determined.

Abstract

We compute both natural and smooth models for the $SL_2(\mathbb C)$ character varieties of the two component double twist links, an infinite family of two-bridge links indexed as $J(k,l)$. For each $J(k,l)$, the component(s) of the character variety containing characters of irreducible representations are birational to a surface of the form $C\times \mathbb C$ where $C$ is a curve. The same is true of the canonical component. We compute the genus of this curve, and the degree of irrationality of the canonical component. We realize the natural model of the canonical component of the $SL_2(\mathbb C)$ character variety of the $J(3,2m+1)$ links as the surface obtained from $\mathbb{P}^1\times \mathbb{P}^1$ as a series of blow-ups.