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

TL;DR

This paper completes a series establishing a precise criterion for when simple loops on 2-bridge spheres are homotopic within 2-bridge link complements, covering new classes of slopes through inductive methods.

Contribution

It provides a complete characterization of homotopic simple loops on 2-bridge spheres for a broad class of link slopes, extending previous results with an inductive approach.

Findings

Necessary and sufficient conditions for homotopy of simple loops on 2-bridge spheres.

Extension of criteria to new classes of 2-bridge link slopes.

Inductive proof covering remaining cases.

Abstract

This is the last 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, and the second paper treated the case of 2-bridge links of slope $n/(2n+1)$ and $(n+1)/(3n+2)$, where $n \ge 2$ is an arbitrary integer. In this paper, we first treat the case of 2-bridge links of slope $n/(mn+1)$ and $(n+1)/((m+1)n+m)$, where $m \ge 3$ is an arbitrary integer, and then treat the remaining cases by induction.