  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.
