  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.

Group Theory;Geometric Topology