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Conway products and links with multiple bridge surfaces. (English) Zbl 1158.57011
For a given link in a 3-manifold, it is an interesting problem to compare two bridge surfaces. One can follow the program developed by H. Rubinstein and M. Scharlemann [Topology 35, No. 4, 1005–1026 (1996; Zbl 0858.57020)] to compare distinct Heegaard splittings of a given non-Haken 3-manifold. There the restriction to non-Haken manifolds was introduced to ensure that the relevant Heegaard splittings were strongly irreducible.
In the paper under review, under the analogous condition that the considered bridge surfaces are \(c\)-weakly incompressible, it is shown that, given two different bridge surfaces for a knot, either they can be properly isotoped to intersect in a nonempty collection of curves that are essential (including non-meridional) on both surfaces, or the knot is a Conway product with respect to an incompressible Conway sphere that naturally decomposes both surfaces into bridge surfaces for the respective factor link(s).

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M50 General geometric structures on low-dimensional manifolds
Full Text: DOI arXiv
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