Almost all strongly quasipositive braid closures are fibered

TL;DR

This paper proves that most strongly quasipositive braid closures are fibered, classifies such links geometrically, and relates Rudolph's fibered links condition to their criteria, highlighting the braid index's role in fiberability.

Contribution

It introduces a new classification of fibered strongly quasipositive links using the Birman-Ko-Lee presentation and relates Rudolph's condition to this framework.

Findings

All closures of certain strongly quasipositive braids are fibered.

Classified links as boundaries of plumbings of positive Hopf bands.

Braid index bounds the number of crossing changes needed for fiberization.

Abstract

We use the Birman-Ko-Lee presentation of the braid group to show that all closures of strongly quasipositive braids whose normal form contains a positive power of the dual Garside element $\delta$ are fibered. We classify links which admit such a braid representative in geometric terms as boundaries of plumbings of positive Hopf bands to a disk. Rudolph constructed fibered strongly quasipositive links as closures of positive words on certain generating sets of $B_n$ and we prove that Rudolph's condition is equivalent to ours. Finally, we show that the braid index is a strict upper bound for the number of crossing changes required to fiber a strongly quasipositive braid.