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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 $$n/(2n+1)$$n/(2n+1) and $$(n+1)/(3n+2)$$(n+1)/(3n+2), where $$n \ge 2$$n≥2 is an arbitrary integer. In this paper, we first treat the case of 2-bridge links of slope $$n/(mn+1)$$n/(mn+1) and $$(n+1)/((m+1)n+m)$$(n+1)/((m+1)n+m), where $$m \ge 3$$m≥3 is an arbitrary integer, and then treat the remaining cases by induction.

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