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 question 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 n tunnels, n − 1 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 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 n core tunnels, n − 1 of which self-intersect.

Cusp shape and tunnel number

“…where d(·, ·) denotes minimal distance in C(S). The distance in the handlebody graph is bounded by one more than the distance in C(S); see [8,Lemma 4.11]. Hence X 0 has bounded diameter in H(S).…”

“…For example, take any homeomorphism of C to the compression body in a Heegaard splitting of any onecusped 3-manifold of genus n + 1, and mark the curves P on S. Let H be the handlebody of the splitting, also with curves P marked on S. Then the fact that there is a maximally pinched structure on H with curves of P pinched to parabolics follows from Thurston's Uniformization theorem (see Morgan [28]). The second part follows as in the proof of Theorem 4.12 of [8]. Let H(S) denote the handlebody graph, consisting of a vertex for each handlebody with boundary S and an edge between handlebodies whose disk sets intersect in the curve complex C(S).…”

“…To prove Theorem 1.1, we consider geometric structures on compression bodies, similar to work in [11], [8], and [24]. Given any framed Euclidean similarity structure on a torus, we first find hyperbolic structures on compression bodies with that cusp shape [T ], then show these lead to maximally pinched structures with cusp shape approaching [T ].…”

Determining isotopy classes of crossing arcs in alternating links

“…Sakuma [14]). Work of Adams, Burton, Cooper, Futer, Purcell provided information about isotopy classes of certain tunnel arcs under additional restrictions ( [1,2,4,5]).…”

“…Hence, the condition (3) from Definition 4.1 is satisfied. Now let us check the conditions (1), (2) and (4).…”