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

Abstract

AbstractAn 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.