Epimorphisms and boundary slopes of $2$–bridge knots

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.