Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 14, Number 2 (2014), 925-952.
Geodesic systems of tunnels in hyperbolic $3$–manifolds
It is unknown whether an unknotting tunnel is always isotopic to a geodesic in a finite-volume hyperbolic –manifold. In this paper, we address the generalization of this question to hyperbolic –manifolds admitting tunnel systems. We show that there exist finite-volume hyperbolic –manifolds with a single cusp, with a system of tunnels, of which come arbitrarily close to self-intersecting. This gives evidence that systems of unknotting tunnels may not be isotopic to geodesics in tunnel number manifolds. In order to show this result, we prove there is a geometrically finite hyperbolic structure on a –compression body with a system of core tunnels, of which self-intersect.
Algebr. Geom. Topol., Volume 14, Number 2 (2014), 925-952.
Received: 12 March 2013
Revised: 2 August 2013
Accepted: 6 September 2013
First available in Project Euclid: 19 December 2017
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Burton, Stephan D; Purcell, Jessica S. Geodesic systems of tunnels in hyperbolic $3$–manifolds. Algebr. Geom. Topol. 14 (2014), no. 2, 925--952. doi:10.2140/agt.2014.14.925. https://projecteuclid.org/euclid.agt/1513715853