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Geodesic Systems of Tunnels in Hyperbolic 3-Manifolds by Burton, Stephan D.; Purcell, Jessica S. is a scholarly article available to read on EtoBox.

What is Geodesic Systems of Tunnels in Hyperbolic 3-Manifolds about?

It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite volume hyperbolic 3-manifold. In this paper, we address the generalization of this problem to hyperbolic 3-manifolds admitting tunnel systems. We show that there exist finite volume hyperbolic 3-manifolds with a single cusp, with a system of at least two tunnels, such that all but one of the tunnels come arbitrarily close to self-intersecting. This gives evidence that systems of unknotting tunnels may not be isotopic to geodesics in tunnel number n manifolds. In order to show this result, we prove there is a geometrically finite hyperbolic structure on a (1;n)-compression body with a system of core tunnels such that all but one of the core tunnels self-intersect.

Author

Burton, Stephan D.; Purcell, Jessica S.

Published

2013

Language

EN