Volume 11, issue 4 (2011)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 S3 and Σ 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 Σ 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
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