On keen weakly reducible bridge spheres

TL;DR

This paper introduces infinitely many examples of keen weakly reducible bridge spheres, which are unique in having disjoint compressing disks and relate to the link’s position in the width complex.

Contribution

It provides the first infinite family of keen weakly reducible bridge spheres for links in bridge position with at least four bridges.

Findings

Keen weakly reducible bridge spheres are not perturbed or topologically minimal.

Such spheres are distance one from a local minimum in the width complex.

Infinitely many examples are constructed for links with $b \,\geq\, 4$ bridges.

Abstract

A bridge sphere is said to be keen weakly reducible if it admits a unique pair of disjoint compressing disks on opposite sides. In particular, such a bridge sphere is weakly reducible, not perturbed, and not topologically minimal in the sense of David Bachman. In terms of Jennifer Schultens' width complex, a link in bridge position with respect to a keen weakly reducible bridge sphere is distance one away from a local minimum. In this paper, we give infinitely many examples of keen weakly reducible bridge spheres for links in $b$ bridge position for $b \geq 4.$