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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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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


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
  • Email:
  • 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:
  • 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:
  • MathSciNet review: 3430862