#### Volume 11, issue 4 (2011)

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Flipping bridge surfaces and bounds on the stable bridge number

### Jesse Johnson and Maggy Tomova

Algebraic & Geometric Topology 11 (2011) 1987–2005
##### 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 $3n-4$ bridges. We generalize both of these results to bridge surfaces for knots in any $3$–manifold.

##### Keywords
stable Euler characteristic, flipping genus, bridge surface, common stabilization, knot distance, bridge position, Heegaard splitting, strongly irreducible, weakly incompressible
##### Mathematical Subject Classification 2000
Primary: 57M25, 57M27, 57M50