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

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

2011-11-06

arxiv.org/abs/1111.3562v1

Mathematics

Group Theory

60 pages, 28 figures

Scientific paper

This is the last of the series of the papers with the aim to 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 [7] treated the case of the 2-bridge torus links, and the second paper [8] treated the case of 2-bridge links of slope $n/(2n+1)$ and $(n+1)/(3n+2)$, where $n \ge 2$ is an arbitrary integer. This paper first treats the case of 2-bridge links of slope $n/(mn+1)$ and $(n+1)/((m+1)n+m)$, where $m \ge 3$ is an arbitrary integer, and then the remaining cases by induction.

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