Research ArticleNo Access

Crosscap number and epimorphisms of two-bridge knot groups

Jim Hoste

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

E-mail Address: jim_hoste@pitzer.edu

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Patrick D. Shanahan

Department of Mathematics, Loyola Marymount University, 1 LMU Dr, Los Angeles, CA 90045, USA

E-mail Address: Patrick.Shanahan@lmu.edu

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, and

Cornelia A. Van Cott

https://orcid.org/0000-0001-8763-0192

Department of Mathematics and Statistics, University of San Francisco, 2130 Fulton Street, San Francisco, CA 94117, USA

E-mail Address: cvancott@usfca.edu

Corresponding author.

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https://doi.org/10.1142/S0218216521500243Cited by:0 (Source: Crossref)

Abstract

We consider the relationship between the crosscap number $γ$ $γ$ of knots and a partial order on the set of all prime knots, which is defined as follows. For two knots $K$ $K$ and $J$ $J$ , we say $K \geq J$ $K \geq J$ if there exists an epimorphism $f : π_{1} (S^{3} - K) \to π_{1} (S^{3} - J)$ $f : π_{1} (S^{3} - K) \to π_{1} (S^{3} - J)$ . We prove that if $K$ $K$ and $J$ $J$ are 2-bridge knots and $K > J$ $K > J$ , then $γ (K) \geq 3 γ (J) - 4$ $γ (K) \geq 3 γ (J) - 4$ . We also classify all pairs $(K, J)$ $(K, J)$ for which the inequality is sharp. A similar result relating the genera of two knots has been proven by Suzuki and Tran. Namely, if $K$ $K$ and $J$ $J$ are 2-bridge knots and $K > J$ $K > J$ , then $g (K) \geq 3 g (J) - 1$ $g (K) \geq 3 g (J) - 1$ , where $g (K)$ $g (K)$ denotes the genus of the knot $K$ $K$ .

Dedicated to our friend and colleague, Mark Kidwell, 1948–2019

Keywords:

AMSC: 57M25, 57M27

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