We show that if is a knot in and is a bridge sphere for with high distance and punctures, the number of perturbations of required to interchange the two balls bounded by via an isotopy is . We also construct a knot with two different bridge spheres with and bridges respectively for which any common perturbation has at least bridges. We generalize both of these results to bridge surfaces for knots in any –manifold.
"Flipping bridge surfaces and bounds on the stable bridge number." Algebr. Geom. Topol. 11 (4) 1987 - 2005, 2011. https://doi.org/10.2140/agt.2011.11.1987