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2010 Epimorphisms and boundary slopes of $2$–bridge knots
Jim Hoste, Patrick D Shanahan
Algebr. Geom. Topol. 10(2): 1221-1244 (2010). DOI: 10.2140/agt.2010.10.1221

Abstract

In this article we study a partial ordering on knots in S3 where K1K2 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 K1K2, 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 KpqKpq. 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 KpqKpq arise from the Ohtsuki–Riley–Sakuma construction.

Citation

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Jim Hoste. Patrick D Shanahan. "Epimorphisms and boundary slopes of $2$–bridge knots." Algebr. Geom. Topol. 10 (2) 1221 - 1244, 2010. https://doi.org/10.2140/agt.2010.10.1221

Information

Received: 8 February 2010; Revised: 4 May 2010; Accepted: 10 May 2010; Published: 2010
First available in Project Euclid: 19 December 2017

zbMATH: 1205.57011
MathSciNet: MR2653061
Digital Object Identifier: 10.2140/agt.2010.10.1221

Subjects:
Primary: 57M25

Rights: Copyright © 2010 Mathematical Sciences Publishers

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