#### Volume 17, issue 4 (2017)

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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 ${K}^{\prime }$. We study the relationship between their crossing numbers $c\left(K\right)$ and $c\left({K}^{\prime }\right)$. More specifically, it is shown that $c\left(K\right)$ is greater than or equal to $3c\left({K}^{\prime }\right)$, 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
Primary: 57M25
Secondary: 57M27