Celebratio Mathematica — Neumann — Complete Bibliography

Complete Bibliography

Works connected to Anastasiia Tsvietkova

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W. D. Neumann and A. Tsvietkova: “Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds,” Proc. Am. Math. Soc. 144 : 2 (2016), pp. 887–896. MR 3430862 Zbl 1360.57020 ArXiv 1402.5582 article

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.