Homotopically equivalent simple loops on 2-bridge spheres in 2-bridge link complements (II)

TL;DR

This paper provides a complete criterion for when two essential simple loops on a 2-bridge sphere in certain 2-bridge link complements are homotopic, extending previous results to new classes of links.

Contribution

It establishes necessary and sufficient conditions for homotopy of loops on 2-bridge spheres in specific 2-bridge link complements, generalizing prior work.

Findings

Derived conditions for homotopy of loops in new link classes

Extended previous results from torus links to more complex links

Provided a comprehensive classification for the considered link types

Abstract

This is the second of a series of papers which give a necessary and sufficient condition for two essential simple loops on a 2-bridge sphere in a 2-bridge link complement to be homotopic in the link complement. The first paper of the series treated the case of the 2-bridge torus links. In this paper, we treat the case of 2-bridge links of slope $n/(2n+1)$ and $(n+1)/(3n+2)$, where $n \ge 2$ is an arbitrary integer.