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2016 Bridge distance and plat projections
Jesse Johnson, Yoav Moriah
Algebr. Geom. Topol. 16(6): 3361-3384 (2016). DOI: 10.2140/agt.2016.16.3361


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.


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Jesse Johnson. Yoav Moriah. "Bridge distance and plat projections." Algebr. Geom. Topol. 16 (6) 3361 - 3384, 2016.


Received: 5 August 2015; Revised: 31 March 2016; Accepted: 8 May 2016; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 1362.57021
MathSciNet: MR3584261
Digital Object Identifier: 10.2140/agt.2016.16.3361

Primary: 57M27

Keywords: bridge distance , bridge sphere , Heegaard splittings , plats , train tracks

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.16 • No. 6 • 2016
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