Volume 16, issue 6 (2016)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 6, 2971–3570
Issue 5, 2389–2970
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
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