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

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.