Braid group and leveling of a knot

Abstract

Any knot [Formula: see text] in genus-[Formula: see text] [Formula: see text]-bridge position can be moved by isotopy to lie in a union of [Formula: see text] parallel tori tubed by [Formula: see text] tubes so that [Formula: see text] intersects each tube in two spanning arcs, which we call a leveling of the position. The minimal [Formula: see text] for which this is possible is an invariant of the position, called the level number. In this work, we describe the leveling by the braid group on two points in the torus, which yields a numerical invariant of the position, called the [Formula: see text]-length. We show that the [Formula: see text]-length equals the level number. We then find braid descriptions for [Formula: see text]-positions of all [Formula: see text]-bridge knots providing upper bounds for their level numbers and also show that the [Formula: see text]-pretzel knot has level number two.