Geodesic systems of tunnels in hyperbolic 3-manifolds

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.

19 pages, 4 figures. V2 contains minor updates to references and exposition. To appear in Algebr. Geom. Topol