Parabolic generating pairs of genus-one 2-bridge knot groups

TL;DR

This paper classifies parabolic generating pairs in genus-one hyperbolic 2-bridge knot groups, showing they are equivalent to standard meridian pairs, and fully classifies epimorphisms between these groups.

Contribution

It proves all parabolic generating pairs are equivalent to meridian pairs and provides a complete classification of epimorphisms between these knot groups.

Findings

Parabolic generating pairs are equivalent to meridian pairs.

Complete classification of epimorphisms between 2-bridge knot groups.

Simplifies understanding of the structure of genus-one hyperbolic 2-bridge knot groups.

Abstract

We show that any parabolic generating pair of a genus-one hyperbolic 2-bridge knot group is equivalent to the upper or lower meridian pair. As an application, we obtain a complete classification of the epimorphisms from 2-bridge knot groups to genus-one hyperbolic 2-bridge knot groups.