Bridge distance and plat projections

Every knot or link K subset of S-3 can be put in a bridge position with respect to a 2-sphere for some bridge number m >= 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 >= 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 inverted right perpendicular n/(2(m-2))inverted left perpendicular, where inverted right perpendicular x inverted left perpendicular 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.