Document Type
Article
Publication Date
2010
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 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.
Publisher Statement
Permission has been granted by Mathematical Sciences Publishers to supply this article for educational and research purposes. More info can be found about the Algebraic & Geometric Topology at http://msp.org/agt/about/journal/about.html. © Mathematical Sciences Publishers.
Recommended Citation
Hoste, Jim and Patrick Shanahan. "Epimorphisms and boundary slopes of 2-bridge knots." Algebraic and Geometric Topology, Vol. 10, 2010, 1221-1244.
