Braids in trivial braid diagrams

We show that every trivial $3$-strand braid diagramcontains a disk, defined as a ribbon ending inopposed crossings. Under a convenient algebraicform, the result extends to every Artin--Titsgroup of dihedral type, but it fails toextend to braids with $4$~strands and more. Theproof uses a partition of the Cayley graph and acontinuity argument.