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Classification of 3-bridge spheres of 3-bridge arborescent links. (English) Zbl 1275.57011

By the author’s paper “Classification of \(3\)-bridge arborescent links” [Hiroshima Math. J. 41, No. 1, 89-136 (2011; Zbl 1231.57006)], a link named in the title – unless a Montesinos link – belongs to one of three particular classes. A \(3\)-bridge sphere of such a link is a \(2\)-sphere that cuts the link into \(3\)-string trivial tangles. For any of the aforementioned links, along with a \(3\)-bridge sphere, a simple diagrammatic construction is given, and an isotopy classification including the \(3\)-bridge sphere is established for these links. For Montesinos links, only partial results are achieved. The methods applied rely heavily on the paper above and an extension of results of Birman and Hilden as well as Morimoto involving Heegaard splittings of \(3\)-manifolds. In this context, a question of Morimoto is answered.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M12 Low-dimensional topology of special (e.g., branched) coverings

Citations:

Zbl 1231.57006
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References:

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