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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 Kp′/q′ with Kp′∕q′≥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, Kp′∕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 Kp′/q′≥Kp∕q arise from the Ohtsuki–Riley–Sakuma construction.

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Recommended Citation

Hoste, Jim and Patrick Shanahan. "Epimorphisms and boundary slopes of 2-bridge knots." Algebraic and Geometric Topology, Vol. 10, 2010, 1221-1244.