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

TL;DR

This paper establishes a precise criterion for when two simple loops on a 2-bridge sphere are homotopic within 2-bridge link complements, focusing on the case of (2,p)-torus links.

Contribution

It provides a necessary and sufficient condition for homotopy of simple loops on 2-bridge spheres in (2,p)-torus link complements, advancing understanding of their topological properties.

Findings

Characterizes homotopy conditions for simple loops on 2-bridge spheres

Focuses on (2,p)-torus links, expanding previous cases

Sets groundwork for analyzing remaining 2-bridge link cases

Abstract

In this paper and its two sequels, we 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. This paper treats the case when the 2-bridge link is a $(2,p)$-torus link, where more cases of homotopy arise, and its sequels will treat the remaining cases.