Unknotting tunnels in hyperbolic 3-manifolds
We examine the relationship between the unknotting tunnels of a hyperbolic 3-manifold and the hyperbolic structure on that manifold. In particular, we begin by proving that the maximal cusp volume of a 1- cusped hyperbolic 3-manifold with tunnel number greater than one is at least \(3\sqrt {3}/4\). This implies that the volume of the entire manifold is at least \(3v_0/ 2\), where \(v_0= 1.01494 \dots\;\). Several interesting corollaries of a purely topological nature follow. These results point toward a new invariant for any compact 3-manifold, which is the least volume of any hyperbolic knot complement within it. We investigate this ``parent volume'', determining the manifold of least parent volume and the manifold with boundary of least parent volume. We also give necessary conditions for a vertical geodesic to be an unknotting tunnel and use this to classify all of the isotopy classes of unknotting tunnels for the Whitehead link and a few other links. Moreover, in addition to other applications, we obtain a universal upper bound on the shortest representative of the homotopy class of an unknotting tunnel for a 2-cusped hyperbolic 3-manifold. Finally, we begin to delineate some restrictions on the hyperbolic structure of a tunnel number one hyperbolic 3-manifold.
unknotting tunnels of a hyperbolic 3-manifold
maximal cusp volume of a 1-cusped hyperbolic 3-manifold with tunnel number greater than one
isotopy classes of unknotting tunnels for the Whitehead link
scientific article; zbMATH DE number 769474
scientific article