Classification of 3-bridge arborescent links
The class of arborescent links is a natural extension of rational and Montesinos links. It can be defined in terms of a recursive definition starting at rational links, or using sequences of plumbings given by weighted trees. Note that Figure 1 in the paper, which is used to give the recursive definition, is incorrect. In the lower right-hand corner, the characters `\(T_1\)' should appear reflected in the diagonal line from the top left to the bottom right. Building on a(n unpublished) classification of arborescent links by \textit{F. Bonahon} and \textit{L. Siebenmann} [New geometric splittings of classical knots, and the classification and symmetries of arborescent links (available at \url{http://www-bcf.usc.edu/fbonahon})], the paper under review gives a complete classification of those arborescent links that are 3--bridge links (that is, that have a decomposition along a sphere into two 3--string trivial tangles). In addition, the task of identifying all 3--bridge spheres for these links, up to isotopy, is begun in this paper, and is to be completed in a later one. The proofs rely on a number of earlier works by various authors, of which the author identifies \textit{T. Kobayashi} [``Structures of the Haken manifolds with Heegaard splittings of genus two.'' Osaka J.{} Math.{} 21, 437--455 (1984; Zbl 0568.57007)] and \textit{K. Morimoto} [``On minimum genus Heegaard splittings of some orientable closed 3-manifolds.'' Tokyo J.{} Math.{} 12, 321--355 (1989; Zbl 0714.57007)] as significant. The main element of the proof is to study, for a link \(L\), the double cover of \(\mathbb{S}^3\) branched over \(L\). For arborescent links, the resulting manifold is either Seifert fibred or a graph manifold (a union of Seifert fibred spaces glued along their boundaries). Known classifications of these spaces are sufficient to distinguish most of the links being studied. In some of the cases, however, it is necessary to consider also a certain subgroup of the mapping class group of the manifold.
scientific article; zbMATH DE number 5953599