Epimorphisms and Boundary Slopes of 2-Bridge Knots

TL;DR

This paper investigates the partial ordering of 2-bridge knots via epimorphisms of their groups, characterizes knots with few boundary slopes, and explores the structure of such orderings and their relation to boundary slopes.

Contribution

It characterizes 2-bridge knots with few boundary slopes and proves minimality of certain knots, supporting a conjecture about their ordering structure.

Findings

2-bridge knots with ≤4 boundary slopes are characterized

Knots with exactly 3 boundary slopes are minimal in the partial order

Knots with ≥5 boundary slopes are either torus knots or related to the Ohtsuki-Riley-Sakuma construction

Abstract

In this article we study a partial ordering on knots in the 3-sphere where K_1 is greater than or equal to K_2 if there is an epimorphism from the knot group of K_1 onto the knot group of K_2 which preserves peripheral structure. If K_1 is a 2-bridge knot and K_1 > K_2, then it is known that K_2 must also be 2-bridge. Furthermore, Ohtsuki, Riley, and Sakuma give a construction which, for a given 2-bridge knot K_{p/q}, produces infinitely 2-bridge knots K_{p'/q'} with K_{p'/q'}>K_{p/q}. After characterizing all 2-bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, K_{p'/q'} is either a torus knot or has 5 or more distinct boundary slopes. We also prove that 2-bridge knots with exactly 3 distinct boundary slopes are minimal with respect to the partial ordering. This result provides some evidence for the conjecture that all pairs of 2-bridge knots with K_{p'/q'}>K_{p/q} arise from the Ohtsuki-Riley-Sakuma construction.