Epimorphisms between 2–bridge knot groups and their crossing numbers

Suzuki, Masaaki

doi:10.2140/agt.2017.17.2413

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Epimorphisms between $2$–bridge knot groups and their crossing numbers

Masaaki Suzuki

Algebraic & Geometric Topology 17 (2017) 2413–2428

DOI: 10.2140/agt.2017.17.2413

Abstract

Suppose that there exists an epimorphism from the knot group of a 2–bridge knot $K$ onto that of another knot $K^{'}$ . We study the relationship between their crossing numbers $c (K)$ and $c (K^{'})$ . More specifically, it is shown that $c (K)$ is greater than or equal to $3 c (K^{'})$ , and we estimate how many knot groups a 2–bridge knot group maps onto. Moreover, we formulate the generating function which determines the number of 2–bridge knot groups admitting epimorphisms onto the knot group of a given 2–bridge knot.

Keywords

epimorphism, $2$–bridge knot, knot group, crossing number

Mathematical Subject Classification 2010

Primary: 57M25

Secondary: 57M27

References

Publication

Received: 15 June 2016

Revised: 1 October 2016

Accepted: 3 December 2016

Published: 3 August 2017

Authors

Masaaki Suzuki
Department of Frontier Media Science School of Interdisciplinary Mathematical Sciences Meiji University 4-21-1 Nakano Tokyo 164-8525 Japan

Volume 17, issue 4 (2017)