The tree of knot tunnels

Abstract

We present a new theory which describes the collection of all tunnels of tunnel number $1$ knots in $S^3$ (up to orientation-preserving equivalence in the sense of Heegaard splittings) using the disk complex of the genus–$2$ handlebody and associated structures. It shows that each knot tunnel is obtained from the tunnel of the trivial knot by a uniquely determined sequence of simple cabling constructions. A cabling construction is determined by a single rational parameter, so there is a corresponding numerical parameterization of all tunnels by sequences of such parameters and some additional data. Up to superficial differences in definition, the final parameter of this sequence is the Scharlemann–Thompson invariant of the tunnel, and the other parameters are the Scharlemann–Thompson invariants of the intermediate tunnels produced by the constructions. We calculate the parameter sequences for tunnels of $2$–bridge knots. The theory extends easily to links, and to allow equivalence of tunnels by homeomorphisms that may be orientation-reversing.