Volume 10, issue 2 (2010)

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Epimorphisms and boundary slopes of $2$–bridge knots

Jim Hoste and Patrick D Shanahan

Algebraic & Geometric Topology 10 (2010) 1221–1244
Abstract

In this article we study a partial ordering on knots in ${S}^{3}$ where ${K}_{1}\ge {K}_{2}$ 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 ${K}_{1}\ge {K}_{2}$, 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 many $2$–bridge knots ${K}_{{p}^{\prime }∕{q}^{\prime }}$ with ${K}_{{p}^{\prime }∕{q}^{\prime }}\ge {K}_{p∕q}$. After characterizing all $2$–bridge knots with $4$ or less distinct boundary slopes, we use this to prove that in any such pair, ${K}_{{p}^{\prime }∕{q}^{\prime }}$ 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 ${K}_{{p}^{\prime }∕{q}^{\prime }}\ge {K}_{p∕q}$ arise from the Ohtsuki–Riley–Sakuma construction.

Keywords
knot, $2$–bridge, boundary slope, epimorphism
Primary: 57M25