Upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-bridge knots
Geometric Topology
2011-01-24 v2
Abstract
In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K we characterize all other 2-bridge knots J such that {K, J} has an upper bound. As an application we answer a question of Suzuki, showing that there is no upper bound for the set consisting of the trefoil and figure-eight knots.
Keywords
Cite
@article{arxiv.1007.3278,
title = {Upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-bridge knots},
author = {Scott M. Garrabrant and Jim Hoste and Patrick D. Shanahan},
journal= {arXiv preprint arXiv:1007.3278},
year = {2011}
}
Comments
21 pages, 2 figures. This is a significant revision of the paper "Two-bridge knots with common ORS covers," including a proof of Conjecture 1. We have retitled the paper and added an author