## Algebraic & Geometric Topology

### Epimorphisms and boundary slopes of $2$–bridge knots

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 10, Number 2 (2010), 1221-1244.

Dates
Revised: 4 May 2010
Accepted: 10 May 2010
First available in Project Euclid: 19 December 2017

https://projecteuclid.org/euclid.agt/1513715131

Digital Object Identifier
doi:10.2140/agt.2010.10.1221

Mathematical Reviews number (MathSciNet)
MR2653061

Zentralblatt MATH identifier
1205.57011

#### Citation

Hoste, Jim; Shanahan, Patrick D. Epimorphisms and boundary slopes of $2$–bridge knots. Algebr. Geom. Topol. 10 (2010), no. 2, 1221--1244. doi:10.2140/agt.2010.10.1221. https://projecteuclid.org/euclid.agt/1513715131

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