Flipping bridge surfaces and bounds on the stable bridge number

Abstract

We show that if $K$ is a knot in $S^3$ and $\Sigma$ is a bridge sphere for $K$ with high distance and $2n$ punctures, the number of perturbations of $K$ required to interchange the two balls bounded by $\Sigma$ via an isotopy is $n$. We also construct a knot with two different bridge spheres with $2n$ and $2n-1$ bridges respectively for which any common perturbation has at least $3n-1$ bridges. We generalize both of these results to bridge surfaces for knots in any 3-manifold.

20 pages, 7 figures

References in corpus (2)

• Flipping and stabilizing Heegaard splittings
• Heegaard splittings of sufficiently complicated 3-manifolds I: Stabilization

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• High Distance Bridge Surfaces
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• Stabilizing Heegaard Splittings of High-Distance Knots
• Diagram uniqueness of even plats
• Diagram uniqueness for highly twisted knots