Volume 10, issue 2 (2010)

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Epimorphisms and boundary slopes of $2$–bridge knots

Jim Hoste and Patrick D Shanahan

Algebraic & Geometric Topology 10 (2010) 1221–1244

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 Kpq, produces infinitely many 2–bridge knots Kpq with Kpq Kpq. After characterizing all 2–bridge knots with 4 or less distinct boundary slopes, we use this to prove that in any such pair, Kpq 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 Kpq Kpq arise from the Ohtsuki–Riley–Sakuma construction.

knot, $2$–bridge, boundary slope, epimorphism
Mathematical Subject Classification 2000
Primary: 57M25
Received: 8 February 2010
Revised: 4 May 2010
Accepted: 10 May 2010
Published: 1 June 2010
Jim Hoste
Department of Mathematics
Pitzer College
1050 N Mills Avenue
Claremont, CA 91711
Patrick D Shanahan
Department of Mathematics
Loyola Marymount University
1 LMU Drive, Suite 2700
University Hall
Los Angeles, CA 90045