Epimorphisms and boundary slopes of 2–bridge knots

In this article we study a partial ordering on knots in $S^{3}$ where $K_{1} \geq K_{2}$ if there is an epimorphism from the knot group of $K_{1}$ onto the knot group of $K_{2}$ which preserves peripheral structure. If $K_{1}$ is a $2$ –bridge knot and $K_{1} \geq K_{2}$ , then it is known that $K_{2}$ must also be $2$ –bridge. Furthermore, Ohtsuki, Riley and Sakuma give a construction which, for a given $2$ –bridge knot $K_{p ∕ q}$ , produces infinitely many $2$ –bridge knots $K_{p^{'} ∕ q^{'}}$ with $K_{p^{'} ∕ q^{'}} \geq K_{p ∕ q}$ . After characterizing all $2$ –bridge knots with $4$ or less distinct boundary slopes, we use this to prove that in any such pair, $K_{p^{'} ∕ q^{'}}$ 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 $K_{p^{'} ∕ q^{'}} \geq K_{p ∕ q}$ arise from the Ohtsuki–Riley–Sakuma construction.

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Keywords

knot, $2$–bridge, boundary slope, epimorphism

Mathematical Subject Classification 2000

Primary: 57M25

References

Publication

Received: 8 February 2010

Revised: 4 May 2010

Accepted: 10 May 2010

Published: 1 June 2010

Authors

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

Volume 10, issue 2 (2010)

Jim Hoste and Patrick D Shanahan

Abstract