Unknotting tunnels, bracelets and the elder sibling property for hyperbolic 3-manifolds (Q2866890)

A properly embedded arc in a \(3\)-manifold with boundary is called an \textsl{unknotting tunnel} if the complement of a regular open neighbourhood of the arc is a handlebody. When the boundary of the \(3\)-manifold consists of two tori and its interior admits a complete hyperbolic structure, it was proved by \textit{C. Adams} in [Math. Ann. 302, No. 1, 177--195 (1995; Zbl 0830.57009)] that each unknotting tunnel is isotopic to a \textsl{vertical geodesic}, that is a geodesic that goes from one cusp to the other and meets each cusp section orthogonally.NEWLINENEWLINEFor one-cusped hyperbolic \(3\)-manifolds admitting an unknotting tunnel, it is not known whether each unknotting tunnel is isotopic to a vertical geodesic, although this is indeed the case for \(2\)-bridge knot complements by work of Adam and Reid.NEWLINENEWLINEIn this paper, the authors give several criteria for a vertical geodesic to be an unknotting tunnel. All criteria, which hold for vertical geodesic of short enough ``length'', are based on combinatorial properties of the \textsl{ball-and-beam pattern}, that is the preimage in the hyperbolic space of the cusp and the geodesic: this consists in a maximal family of horospheres and a family of geodesic segments joining them.