The tree of knot tunnels
Abstract
We present a new theory which describes the collection of all tunnels of tunnel number 1 knots in the 3-sphere (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.
Cite
@article{arxiv.math/0611921,
title = {The tree of knot tunnels},
author = {Sangbum Cho and Darryl McCullough},
journal= {arXiv preprint arXiv:math/0611921},
year = {2014}
}
Comments
This version has extensive minor rewriting for accuracy and clarity. The material on the depth invariant has been substantially expanded and moved into a new ArXiv preprint, The depth of a knot tunnel. Also moved there is the calculation of the slope sequences for the short tunnels of torus knots