Character varieties of once-punctured torus bundles with tunnel number one (Q2842046)

In this paper, the authors determine the \(SL_2(\mathbb{C})\) and \(PSL_2(\mathbb{C})\)-character varieties of the once-punctured torus bundles with tunnel number one. The pleasant structure of the fundamental group of these manifolds enables the authors to compute explicitly ``natural'' models for the character varieties and identify them up to birational equivalence with smooth models and determine the genera of the canonical components. They also study the effect of the symmetries of these manifolds on their character varieties and the global structure of the latter by investigating the points of intersection of their various components. Additionally, the authors compute the twisted Alexander polynomial for the representations in \(SL_2(\mathbb{C})\) of these manifolds.