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

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- by Walter D. Neumann and Anastasiia Tsvietkova PDF
- Proc. Amer. Math. Soc.
**144**(2016), 887-896 Request permission

## 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.## References

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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
- Email: neumann@math.columbia.edu
**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
- Email: tsvietkova@math.ucdavis.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**144**(2016), 887-896 - MSC (2010): Primary 57M25, 57M50, 57M27
- DOI: https://doi.org/10.1090/proc/12704
- MathSciNet review: 3430862