Epimorphisms and boundary slopes of 2–bridge knots

Abstract

In this article we study a partial ordering on knots in S3 where K1 K2 if there is an epimorphism from the knot group of K1 onto the knot group of K2 which preserves peripheral structure. If K1 is a 2–bridge knot and K1 K2, then it is known that K2 must also be 2–bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given 2–bridge knot Kp=q, produces infinitely many 2–bridge knots Kp0=q0 with Kp0=q0 Kp=q. After characterizing all 2–bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, Kp0=q0 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 Kp0=q0 Kp=q arise from the Ohtsuki–Riley–Sakuma construction. 57M25 1