Flipping bridge surfaces and bounds on the stable bridge number

TL;DR

This paper investigates the relationship between bridge surfaces of knots, their perturbations, and the minimal number of moves needed to interchange the bounded regions, providing bounds and constructions for knots in various 3-manifolds.

Contribution

It establishes bounds on the number of perturbations needed to interchange regions for high-distance bridge spheres and constructs knots with specific bridge sphere properties, generalizing results to all 3-manifolds.

Findings

Number of perturbations equals n for high-distance bridge spheres with 2n punctures.

Existence of knots with two bridge spheres differing by one in bridge number, requiring many perturbations to relate.

Generalization of bounds and constructions to knots in arbitrary 3-manifolds.

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.