Epimorphisms and boundary slopes of 2–bridge knots

Hoste, Jim; Shanahan, Patrick D

doi:10.2140/agt.2010.10.1221

¹Department of Mathematics, Pitzer College, 1050 N Mills Avenue, Claremont, CA 91711
²Department of Mathematics, Loyola Marymount University, 1 LMU Drive, Suite 2700, University Hall, Los Angeles, CA 90045

Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 1221-1244

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Résumé

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.

DOI : 10.2140/agt.2010.10.1221

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

Affiliations des auteurs :

Hoste, Jim ¹ ; Shanahan, Patrick D ²

¹ Department of Mathematics, Pitzer College, 1050 N Mills Avenue, Claremont, CA 91711
² Department of Mathematics, Loyola Marymount University, 1 LMU Drive, Suite 2700, University Hall, Los Angeles, CA 90045

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     title = {Epimorphisms and boundary slopes of 2{\textendash}bridge knots},
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     doi = {10.2140/agt.2010.10.1221},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2010.10.1221/}
}

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Hoste, Jim; Shanahan, Patrick D. Epimorphisms and boundary slopes of 2–bridge knots. Algebraic and Geometric Topology, Tome 10 (2010) no. 2, pp. 1221-1244. doi: 10.2140/agt.2010.10.1221

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