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arXiv · 1302.5469

Geodesic systems of tunnels in hyperbolic 3-manifolds

Stephan D. Burton·Jessica S. Purcell
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Abstract

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.

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Stephan D. Burton, Jessica S. Purcell. 2013-09-11. Geodesic systems of tunnels in hyperbolic 3-manifolds. https://doi.org/10.2140/agt.2014.14.925

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