Flipping bridge surfaces and bounds on the stable bridge number

We show that if $K$ is a knot in $S^{3}$ and $Σ$ is a bridge sphere for $K$ with high distance and $2 n$ punctures, the number of perturbations of $K$ required to interchange the two balls bounded by $Σ$ via an isotopy is $n$ . We also construct a knot with two different bridge spheres with $2 n$ and $2 n - 1$ bridges respectively for which any common perturbation has at least $3 n - 4$ bridges. We generalize both of these results to bridge surfaces for knots in any $3$ –manifold.

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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

References

Publication

Received: 17 April 2010

Revised: 29 March 2011

Accepted: 15 May 2011

Published: 16 July 2011

Authors

Jesse Johnson
Mathematics Department Oklahoma State University 401 Mathematical Sciences Stillwater OK 74078-1058 USA
http://www.math.okstate.edu/~jjohnson/
Maggy Tomova
Department of Mathematics The University of Iowa Iowa City IA 52242 USA
http://www.math.uiowa.edu/~mtomova

Volume 11, issue 4 (2011)

Jesse Johnson and Maggy Tomova

Abstract