Bridge distance and plat projections

Johnson, Jesse; Moriah, Yoav

doi:10.2140/agt.2016.16.3361

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Bridge distance and plat projections

Jesse Johnson and Yoav Moriah

Algebraic & Geometric Topology 16 (2016) 3361–3384

DOI: 10.2140/agt.2016.16.3361

Abstract

Every knot or link $K \subset S^{3}$ can be put in a bridge position with respect to a $2$ –sphere for some bridge number $m \geq m_{0}$ , where $m_{0}$ is the bridge number for $K$ . Such $m$ –bridge positions determine $2 m$ –plat projections for the knot. We show that if $m \geq 3$ 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 $n > 4 m (m - 2)$ 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

Volume 16, issue 6 (2016)