Simple loops on 2-bridge spheres in 2-bridge link complements

TL;DR

This paper provides complete characterizations of when simple loops on 2-bridge spheres in 2-bridge link complements are null-homotopic or homotopic, with applications to character varieties and McShane's identity.

Contribution

It offers the first comprehensive answers to homotopy questions for simple loops on 2-bridge spheres in 2-bridge link complements, linking topology with character varieties.

Findings

Complete criteria for null-homotopic simple loops.

Conditions for homotopy between two simple loops.

Applications to character varieties and McShane's identity.

Abstract

The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when are they homotopic in the link complement? We also announce applications of these results to character varieties and McShane's identity.