Semisimple tunnels
Abstract
A knot in the 3-sphere in genus-1 1-bridge position (called a (1,1)-position) can be described by an element of the braid group of two points in the torus. Our main results tell how to translate between a braid group element and the sequence of slope invariants of the upper and lower tunnels of the (1,1)-position. After using them to verify previous calculations of the slope invariants for all tunnels of 2-bridge knots and (1,1)-tunnels of torus knots, we obtain characterizations of the slope sequences of tunnels of 2-bridge knots, and of a class of tunnels we call toroidal. The main results lead to a general algorithm to calculate the slope invariants of the upper and lower tunnels from a braid description. The algorithm has been implemented as software, and we give some sample computations.
Keywords
Cite
@article{arxiv.1006.5232,
title = {Semisimple tunnels},
author = {Sangbum Cho and Darryl McCullough},
journal= {arXiv preprint arXiv:1006.5232},
year = {2011}
}
Comments
Revision of first version. Minor rewriting throughout, and several sections of background material added. (-2,3,7)-pretzel knot example removed for writeup with related material, otherwise no significant change in content