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

TL;DR

This paper investigates the partial order on 2-bridge knots using continued fractions, establishing conditions for upper bounds and characterizing related knots, with applications to specific knot sets like the trefoil and figure-eight.

Contribution

It provides necessary and sufficient conditions for upper bounds in the Ohtsuki-Riley-Sakuma partial order on 2-bridge knots and characterizes knots related to a given knot.

Findings

No upper bound exists for the set containing the trefoil and figure-eight knots.

Conditions for the existence of upper bounds are fully characterized.

The work answers a previously open question by Suzuki.

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.