## Algebraic & Geometric Topology

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

Masaaki Suzuki

#### 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
Revised: 1 October 2016
Accepted: 3 December 2016
First available in Project Euclid: 16 November 2017

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

#### 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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