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

Kenneth L. Baker; Kathleen L. Petersen

doi:10.1142/S0129167X13500481

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

Kenneth L. Baker, Kathleen L. Petersen

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

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(&complexes;).

Original language	English (US)
Article number	1350048
Journal	International Journal of Mathematics
Volume	24
Issue number	6
DOIs	https://doi.org/10.1142/S0129167X13500481
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

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title = "Character varieties of once-punctured torus bundles with tunnel number one",

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(&complexes;).",

keywords = "Character variety, punctured torus bundle, tunnel number one",

author = "Baker, {Kenneth L.} and Petersen, {Kathleen L.}",

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.",

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AU - Petersen, Kathleen L.

N1 - 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.

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N2 - 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(&complexes;).

AB - 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(&complexes;).

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KW - punctured torus bundle

KW - tunnel number one

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

Abstract

Bibliographical note

Keywords

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