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

摘要

This is the last of a series of papers which 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. The first paper of the series treated the case of the 2- bridge torus links, and the second paper treated the case of 2-bridge links of slope n/(2n+1) and (n+1)/(3n+2), where n≥2 is an arbitrary integer. In this paper, we first treat the case of 2-bridge links of slope n/(mn +1) and (n +1)/((m +1)n +m), where m≥3 is an arbitrary integer, and then treat the remaining cases by induction.