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

Abstract

The purpose of this note is to announce complete answers tothe following questions.(1) For an essential simple loopon 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 loopson a 2-bridge sphere in a 2-bridge link complement,when are they homotopic in the link complement?We also announce applications of these resultsto character varieties andMcShane's identity.