#### Volume 16, issue 6 (2016)

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Bridge distance and plat projections

### Jesse Johnson and Yoav Moriah

Algebraic & Geometric Topology 16 (2016) 3361–3384
##### 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 $m\ge {m}_{0}$, where ${m}_{0}$ is the bridge number for $K$. Such $m$–bridge positions determine $2m$–plat projections for the knot. We show that if $m\ge 3$ 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 $⌈n∕\left(2\left(m-2\right)\right)⌉$, where $⌈x⌉$ is the smallest integer greater than or equal to $x$. As a corollary, we conclude that if such a diagram has $n>4m\left(m-2\right)$ 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
Primary: 57M27