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Proceedings of the American Mathematical Society

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Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds

Authors: Walter D. Neumann and Anastasiia Tsvietkova
Journal: Proc. Amer. Math. Soc. 144 (2016), 887-896
MSC (2010): Primary 57M25, 57M50, 57M27
Published electronically: October 7, 2015
MathSciNet review: 3430862
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Abstract: Given a cusped hyperbolic 3-manifold with finite volume, we define two types of complex parameters which capture geometric information about the preimages of geodesic arcs traveling between cusp cross-sections. We prove that these parameters are elements of the invariant trace field of the manifold, providing a connection between the intrinsic geometry of a 3-manifold and its number-theoretic invariants. Further, we explore the question of choosing a minimal collection of arcs and associated parameters to generate the field. We prove that for a tunnel number $k$ manifold it is enough to choose $3k$ specific parameters. For many hyperbolic link complements, this approach allows one to compute the field from a link diagram. We also give examples of infinite families of links where a single parameter can be chosen to generate the field, and the polynomial for it can be constructed from the link diagram as well.

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Additional Information

Walter D. Neumann
Affiliation: Department of Mathematics, Barnard College, Columbia University, 2990 Broadway MC4429, New York, New York 10027
MR Author ID: 130560
ORCID: 0000-0001-6916-1935

Anastasiia Tsvietkova
Affiliation: Department of Mathematics, University of California - Davis, One Shields Ave, Davis, California 95616
MR Author ID: 885824
ORCID: 0000-0002-4623-2785

Keywords: Link complement, hyperbolic 3-manifold, invariant trace field, cusp, arithmetic invariants
Received by editor(s): October 10, 2014
Received by editor(s) in revised form: December 25, 2014
Published electronically: October 7, 2015
Communicated by: Martin Scharlemann
Article copyright: © Copyright 2015 American Mathematical Society