Bridge distance and plat projections

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 $\lceil n∕\left(2\left(m-2\right)\right)\rceil$, where $\lceil x\rceil$ 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.