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.