Geodesic systems of tunnels in hyperbolic 3-manifolds

TL;DR

This paper investigates whether unknotting tunnels in hyperbolic 3-manifolds can always be isotoped to geodesics, providing examples where most tunnels cannot be isotoped to simple geodesic representatives.

Contribution

It constructs hyperbolic 3-manifolds with tunnel systems where all but one tunnel self-intersect, challenging assumptions about isotopy to geodesics.

Findings

Existence of hyperbolic 3-manifolds with non-isotopic tunnels to geodesics

Most tunnels in certain manifolds can be arbitrarily close to self-intersecting

A hyperbolic structure on a (1;n)-compression body with self-intersecting core tunnels

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 problem 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 at least two tunnels, such that all but one of the tunnels 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 core tunnels such that all but one of the core tunnels self-intersect.