Reduction of bridge positions along a bridge disk

TL;DR

This paper characterizes when reducing a knot in n-bridge position along a bridge disk decreases its bridge number, linking it to the existence of a cancelling pair of disks, and applies this to unknots in the 3-sphere.

Contribution

It provides a necessary and sufficient condition for bridge number reduction along a bridge disk and explores the implications for unknots and their bounding disks.

Findings

Reduction along a bridge disk decreases bridge number if and only if a cancelling pair exists.

If reduction yields an (n-1)-bridge position, the knot bounds a disk containing the bridge disk.

The disk intersects the bridge sphere in exactly n arcs.

Abstract

Suppose a knot in a $3$-manifold is in $n$-bridge position. We consider a reduction of the knot along a bridge disk $D$ and show that the result is an $(n-1)$-bridge position if and only if there is a bridge disk $E$ such that $(D, E)$ is a cancelling pair. We apply this to an unknot $K$, in $n$-bridge position with respect to a bridge sphere $S$ in the $3$-sphere, to consider the relationship between a bridge disk $D$ and a disk in the $3$-sphere that $K$ bounds. We show that if a reduction of $K$ along $D$ yields an $(n-1)$-bridge position, then $K$ bounds a disk that contains $D$ as a subdisk and intersects $S$ in $n$ arcs.