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Remarks on Suzuki’s knot epimorphism number

Jim Hoste

http://orcid.org/0000-0002-9354-7229

Pitzer College, 1050 N. Mills Ave., Claremont, CA 91711, USA

E-mail Address: Jhoste@pitzer.edu

Corresponding author.

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Joshua Ocana Mercado

Loyola Marymount University, 1 LMU Drive, Los Angeles, CA 90045, USA

E-mail Address: jocana@lion.lmu.edu

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, and

Patrick D. Shanahan

Loyola Marymount University, 1 LMU Drive, Los Angeles, CA 90045, USA

E-mail Address: patrick.shanahan@lmu.edu

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https://doi.org/10.1142/S0218216519500603Cited by:1 (Source: Crossref)

Abstract

A partial order on prime knots can be defined by declaring $J \geq K$ $J \geq K$ , if there exists an epimorphism from the knot group of $J$ $J$ onto the knot group of $K$ $K$ . Suppose that $J$ $J$ is a 2-bridge knot that is strictly greater than $m$ $m$ distinct, nontrivial knots. In this paper, we determine a lower bound on the crossing number of $J$ $J$ in terms of $m$ $m$ . Using this bound, we answer a question of Suzuki regarding the 2-bridge epimorphism number $EK (n)$ $EK (n)$ which is the maximum number of nontrivial knots which are strictly smaller than some 2-bridge knot with crossing number $n$ $n$ . We establish our results using techniques associated with parsings of a continued fraction expansion of the defining fraction of a 2-bridge knot.

Keywords:

AMSC: 57M25

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