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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Epimorphisms between $2$–bridge knot groups and their crossing numbers

Masaaki Suzuki

Algebraic & Geometric Topology 17 (2017) 2413–2428
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 3c(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