Character varieties of once-punctured torus bundles with tunnel number one

TL;DR

This paper thoroughly analyzes the character varieties of once-punctured torus bundles with tunnel number one, providing algebraic models, genus computations, and insights into their symmetries and trace fields.

Contribution

It introduces explicit models for the PSL_2(C) and SL_2(C) character varieties of these bundles, including genus calculations and symmetry actions, advancing understanding of their algebraic and geometric properties.

Findings

Determined algebraic models for character varieties

Computed genera of canonical components

Analyzed symmetry actions and trace fields

Abstract

We determine the PSL_2(C) and SL_2(C) character varieties of the once-punctured torus bundles with tunnel number one, i.e. the once-punctured torus bundles that arise from filling one boundary component of the Whitehead link exterior. In particular, we determine `natural' models for these algebraic sets, identify them up to birational equivalence with smooth models, and compute the genera of the canonical components. This enables us to compare dilatations of the monodromies of these bundles with these genera. We also determine the minimal polynomials for the trace fields of these manifolds. Additionally we study the action of the symmetries of these manifolds upon their character varieties, identify the characters of their lens space fillings, and compute the twisted Alexander polynomials for their representations to SL_2(C).