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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).

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).

Reviewer: Kazuhiro Ichihara (Nara)

### MSC:

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 |

### Keywords:

bridge position; Heegaard splitting; strongly irreducible; weakly incompressible; Conway spheres### Citations:

Zbl 0858.57020
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XMLCite

\textit{M. Scharlemann} and \textit{M. Tomova}, Mich. Math. J. 56, No. 1, 113--144 (2008; Zbl 1158.57011)

### References:

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