Upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-bridge knots

Scott M. Garrabrant, Jim Hoste, Patrick D. Shanahan

Published 2010-07-19, updated 2011-01-20Version 2

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.