Algebraic & Geometric Topology

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

Masaaki Suzuki

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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.

Article information

Source
Algebr. Geom. Topol., Volume 17, Number 4 (2017), 2413-2428.

Dates
Received: 15 June 2016
Revised: 1 October 2016
Accepted: 3 December 2016
First available in Project Euclid: 16 November 2017

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841446

Digital Object Identifier
doi:10.2140/agt.2017.17.2413

Mathematical Reviews number (MathSciNet)
MR3686401

Zentralblatt MATH identifier
1372.57021

Subjects
Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds

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

Citation

Suzuki, Masaaki. Epimorphisms between $2$–bridge knot groups and their crossing numbers. Algebr. Geom. Topol. 17 (2017), no. 4, 2413--2428. doi:10.2140/agt.2017.17.2413. https://projecteuclid.org/euclid.agt/1510841446


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