Gonality and genus of canonical components of character varieties

Kathleen L. Petersen, Alan W. Reid

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

Let M be a finite volume hyperbolic 3-manifold with two cusps, and X0 a canonical component of the SL2(C) character variety of any hyperbolic filling of a single cusp of M. We show that the gonality of X0 is bounded above independent of the filling, and obtain bounds for the genus of X0 and the degree of a canonical component of the associated A-polynomial curve. We explicitly compute the gonality for several examples, including the double twist knots.

Original languageEnglish (US)
Title of host publicationContemporary Mathematics
PublisherAmerican Mathematical Society
Pages263-292
Number of pages30
DOIs
StatePublished - 2020
Externally publishedYes

Publication series

NameContemporary Mathematics
Volume760
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627

Bibliographical note

Funding Information:
2020 Mathematics Subject Classification. Primary 57M27; Secondary 14H51, 57M50. Key words and phrases. character variety, cusped hyperbolic 3-manifold, gonality. This work was partially supported by grants from the Simons Foundation (#209226 and #430077 to the first author), and by the National Science Foundation (to the second author).

Funding Information:
The first author would like to thank the University of Texas at Austin and the Max Planck Institut f?r Mathematik for their hospitality while working on this manuscript, and Martin Hils for helpful discussions. We would also like to thank two referees for helpful comments on a previous version of the paper that helped clarify some of the arguments.

Publisher Copyright:
© 2020 American Mathematical Society.

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