Unknotting tunnels in hyperbolic 3-manifolds

TL;DR

This paper establishes conditions under which certain vertical geodesics in hyperbolic 3-manifolds serve as unknotting tunnels, linking geometric properties like bracelet structures and horoball configurations to topological simplifications.

Contribution

It provides new criteria involving bracelet structures and the elder sibling property for vertical geodesics to be recognized as unknotting tunnels in hyperbolic 3-manifolds.

Findings

Vertical geodesics corresponding to 4-, 5-, or 6-bracelets with short length are unknotting tunnels.

Vertical geodesics with the elder sibling property and length less than ln(2) are unknotting tunnels.

Conditions connect geometric features to topological unknotting tunnels in hyperbolic 3-manifolds.

Abstract

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 in a hyperbolic 3-manifold M, we find sufficient conditions for it to be an unknotting tunnel. In particular, if the vertical geodesic 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 that satisfies the elder sibling property, which means that in the universal cover, every horoball except the one centered at infinity is connected to a larger horoball by a lift of the vertical geodesic. Such a vertical geodesic with length less than ln(2) is then shown to be an unknotting tunnel.