Abstract
We determine the PSL2(&complexes) and SL2(& complexes) 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 SL2(ℂ).
Original language | English (US) |
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Article number | 1350048 |
Journal | International Journal of Mathematics |
Volume | 24 |
Issue number | 6 |
DOIs | |
State | Published - Jun 2013 |
Externally published | Yes |
Bibliographical note
Funding Information:The authors would like to thank Eriko Hironaka and Ronald van Luijk for helpful conversations. The authors are also indebted to Farshid Hajir for the idea behind the proof of Lemma 6.7. This work was partially supported by Simons Foundation grant #209184 to Kenneth Baker and grant #209226 to Kathleen Petersen.
Keywords
- Character variety
- punctured torus bundle
- tunnel number one