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Bridge distance and plat projections

Jesse Johnson and Yoav Moriah

Algebraic & Geometric Topology 16 (2016) 3361–3384
Abstract

Every knot or link K S3 can be put in a bridge position with respect to a 2–sphere for some bridge number m m0, where m0 is the bridge number for K. Such m–bridge positions determine 2m–plat projections for the knot. We show that if m 3 and the underlying braid of the plat has n 1 rows of twists and all the twisting coefficients have absolute values greater than or equal to three then the distance of the bridge sphere is exactly n(2(m 2)), where x is the smallest integer greater than or equal to x. As a corollary, we conclude that if such a diagram has n > 4m(m 2) rows then the bridge sphere defining the plat projection is the unique, up to isotopy, minimal bridge sphere for the knot or link. This is a crucial step towards proving a canonical (thus a classifying) form for knots that are “highly twisted” in the sense we define.

Keywords
Heegaard splittings, bridge sphere, plats, bridge distance, train tracks
Mathematical Subject Classification 2010
Primary: 57M27
References
Publication
Received: 5 August 2015
Revised: 31 March 2016
Accepted: 8 May 2016
Published: 15 December 2016
Authors
Jesse Johnson
Department of Mathematics
Oklahoma State University
Stillwater, OK 74078
United States
39 Chilton St #2
Cambridge, MA 02138
United States
Yoav Moriah
Department of Mathematics
Technion
32000 Haifa
Israel