A note on Riley polynomials of $2$-bridge knots

TL;DR

This paper demonstrates how Riley polynomials can be used to establish epimorphisms between 2-bridge knot groups and classifies these knots based on their Riley polynomials.

Contribution

It introduces an elementary method to relate Riley polynomials to group epimorphisms and provides a classification scheme for 2-bridge knots.

Findings

Existence of epimorphisms between 2-bridge knot groups shown using Riley polynomials

Classification of 2-bridge knots based on Riley polynomial properties

Elementary argument simplifies understanding of knot group relationships

Abstract

In this short note we show the existence of an epimorphism between groups of $2$-bridge knots by means of an elementary argument using the Riley polynomial. As a corollary, we give a classification of $2$-bridge knots by Riley polynomials.