Algebraic & Geometric Topology
- Algebr. Geom. Topol.
- Volume 16, Number 6 (2016), 3361-3384.
Bridge distance and plat projections
Every knot or link can be put in a bridge position with respect to a –sphere for some bridge number , where is the bridge number for . Such –bridge positions determine –plat projections for the knot. We show that if and the underlying braid of the plat has 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 , where is the smallest integer greater than or equal to . As a corollary, we conclude that if such a diagram has 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.
Algebr. Geom. Topol., Volume 16, Number 6 (2016), 3361-3384.
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
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 57M27: Invariants of knots and 3-manifolds
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