UNKNOTTING TUNNELS, BRACELETS AND THE ELDER SIBLING PROPERTY FOR HYPERBOLIC 3-MANIFOLDS

An unknotting tunnel in a 3-manifold with boundary is a properly embedded arc, the complement of an open neighborhood of which is a handlebody. A geodesic with endpoints on the cusp boundary of a hyperbolic 3-manifold and perpendicular to the cusp boundary is called a vertical geodesic. Given a vertical geodesic $\alpha $ in a hyperbolic 3-manifold $M$, we find sufficient conditions for it to be an unknotting tunnel. In particular, if $\alpha $ corresponds to a 4-bracelet, 5-bracelet or 6-bracelet in the universal cover and has short enough length, it must be an unknotting tunnel. Furthermore, we consider a vertical geodesic $\alpha $ that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at $\infty $ is connected to a larger horoball by a lift of $\alpha $. Such an $\alpha $ with length less than $\ln (2)$ is then shown to be an unknotting tunnel.

TWISTED TORUS KNOTS T(p, q; 3, s) ARE TUNNEL NUMBER ONE

We show that twisted torus knots T(p, q; 3, s) are tunnel number one. A short spanning arc connecting two adjacent twisted strands is an unknotting tunnel.

INVOLUTIONS OF KNOTS THAT FIX UNKNOTTING TUNNELS

Let K be a knot that has an unknotting tunnel τ. We prove that K admits a strong involution that fixes τ pointwise if and only if K is a two-bridge knot and τ its upper or lower tunnel.

TUNNELS OF 2-BRIDGE LINKS

In the class of links with one unknotting tunnel the 2-bridge links are characterized by the property of consisting of two trivial components. The tunnels of 2-bridge links are classified.

Examples of tunnel number one knots which have the property ‘1 + 1 = 3’

Let K be a knot in the 3-sphere S3, N(K) the regular neighbourhood of K and E(K) = cl(S3−N(K)) the exterior of K. The tunnel number t(K) is the minimum number of mutually disjoint arcs properly embedded in E(K) such that the complementary space of a regular neighbourhood of the arcs is a handlebody. We call the family of arcs satisfying this condition an unknotting tunnel system for K. In particular, we call it an unknotting tunnel if the system consists of a single arc.