An Elementary Fact About Unlinked Braid Closures

Abstract

Let \(n \in \mathbb {Z}^+\). We provide two short proofs of the following classical fact, one using Khovanov homology and one using Heegaard–Floer homology: if the closure of an n-strand braid \(\sigma \) is the n-component unlink, then \(\sigma \) is the trivial braid.