an:05496993
Zbl 1158.57011
Scharlemann, Martin; Tomova, Maggy
Conway products and links with multiple bridge surfaces
EN
Mich. Math. J. 56, No. 1, 113-144 (2008).
00243356
2008
j
57M25 57M27 57M50
bridge position; Heegaard splitting; strongly irreducible; weakly incompressible; Conway spheres
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 \textit{H. Rubinstein} and \textit{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).
Kazuhiro Ichihara (Nara)
Zbl 0858.57020