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

TL;DR

This paper investigates the relationships between crossing numbers of 2-bridge knots connected by epimorphisms, establishing bounds and counting how many such mappings exist through generating functions.

Contribution

It provides a lower bound on crossing numbers under epimorphisms and introduces a generating function to count 2-bridge knot groups with epimorphisms onto a given knot group.

Findings

Crossing number of K is at least three times that of K' when there is an epimorphism.

Derived a generating function to count 2-bridge knot groups with epimorphisms.

Estimated the number of 2-bridge knot groups mapping onto a given knot group.

Abstract

Suppose that there exists an epimorphism from the knot group of a $2$-bridge knot $K$ onto that of another knot $K'$. In this paper, we study the relationship between their crossing numbers $c(K)$ and $c(K')$. Especially it is shown that $c(K)$ is greater than or equal to $3 c(K')$ 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.