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

Neumann, Walter; Tsvietkova, Anastasiia

doi:10.1090/proc/12704

Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds
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by Walter D. Neumann and Anastasiia Tsvietkova

Proc. Amer. Math. Soc. 144 (2016), 887-896

DOI: https://doi.org/10.1090/proc/12704

Published electronically: October 7, 2015

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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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