Algebraic & Geometric Topology

Bridge distance and plat projections

Jesse Johnson and Yoav Moriah

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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.

Article information

Source
Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3361-3384.

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

Permanent link to this document
https://projecteuclid.org/euclid.agt/1510841263

Digital Object Identifier
doi:10.2140/agt.2016.16.3361

Mathematical Reviews number (MathSciNet)
MR3584261

Zentralblatt MATH identifier
1362.57021

Subjects
Primary: 57M27: Invariants of knots and 3-manifolds

Keywords
Heegaard splittings bridge sphere plats bridge distance train tracks

Citation

Johnson, Jesse; Moriah, Yoav. Bridge distance and plat projections. Algebr. Geom. Topol. 16 (2016), no. 6, 3361--3384. doi:10.2140/agt.2016.16.3361. https://projecteuclid.org/euclid.agt/1510841263


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