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Suppose that there exists an epimorphism from the knot group of a $2$-bridgeknot $K$ onto that of another knot $K'$. In this paper, we study therelationship between their crossing numbers $c(K)$ and $c(K')$. Especially itis shown that $c(K)$ is greater than or equal to $3 c(K')$ and we estimate howmany knot groups a $2$-bridge knot group maps onto. Moreover, we formulate thegenerating function which determines the number of $2$-bridge knot groupsadmitting epimorphisms onto the knot group of a given $2$-bridge knot.

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