Essential tangle decomposition from thin position of a link
Thin position for a link is a generalization of a bridge presentation, defined by minimizing a simply-defined intersection count for a height function on the link. It was introduced by \textit{D. Gabai} [J. Differ. Geom. 26, 479-536 (1987; Zbl 0639.57008)] and used effectively by him in studying decompositions of the link exterior and other manifolds constructed from the link. In this paper the authors consider the case where a thin position presentation is not a bridge presentation. They extend results of \textit{A. Thompson} [Topology 36, No. 2, 505-507 (1997; Zbl 0867.57009)] to find a maximal essential tangle decomposition for the link and also a canonical decomposition with incompressible 2-spheres which is unique, given the thin position. They give examples of links where there is more than one choice of thin position. Starting with a brief but helpful illustrated account of thin position the paper includes a number of other examples to help appreciating the general features of thin position presentations. Thompson showed that a thin position presentation which is not a bridge presentation gives rise to an incompressible punctured sphere in the link complement; one of the examples presented here exhibits an incompressible punctured sphere which does not arise in this way.
scientific article; zbMATH DE number 1066030