Epimorphisms and boundary slopes of 2-bridge knots

In this article we study a partial ordering on knots in S(3) where K(1) >= 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 many 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.