Studying Links via Plats: The Unlink

TL;DR

This paper introduces new moves called pocket and flip that simplify plat presentations of unlinks without stabilization, extending Birman's theorem to unlink cases and providing a method to reduce complex plats to standard forms.

Contribution

It presents a stabilization-free version of Birman's theorem for unlinks and introduces pocket and flip moves to simplify plat presentations.

Findings

Any plat of the unknot can be simplified to the standard form.

The moves preserve link type and bridge index.

The method generalizes to unlinks.

Abstract

Our main result is a version of Birman's theorem about equivalence of plats, which does not involve stabilization, for the unlink. We introduce the pocket and flip moves, which modify a plat without changing its link type or bridge index. Theorem 1 shows that using the pocket and flip moves, one can simplify any closed $n$-plat presentation of the unknot to the standard 0-crossing diagram of the unknot, through a sequence of plats of non-increasing bridge index. The theorem readily generalises to the case of the unlink.