Generating function on epimorphisms between $2$-bridge knot groups

TL;DR

This paper refines a generating function to count epimorphisms between 2-bridge knot groups by incorporating genus, and explores epimorphisms among fibered 2-bridge knots, degree one maps, and unknotting number one knots.

Contribution

It advances the understanding of epimorphisms between 2-bridge knot groups by including genus in the counting formula and analyzing specific classes of knots.

Findings

Refined generating function including genus

Count of epimorphisms between fibered 2-bridge knots

Discussion on degree one maps and unknotting number one knots

Abstract

We have the generating function which determines the number of $2$-bridge knot groups admitting epimorphisms onto the knot group of a given $2$-bridge knot, in terms of crossing number. In this paper, we will refine this formula by taking account into genus as well as crossing number. Next, we determine the number of epimorphisms between fibered $2$-bridge knot groups. Moreover, we discuss degree one maps and $2$-bridge knots uknotting number one.