Disks in trivial braid diagrams. (Q1879799)

A `disk' in a braid diagram is a ribbon bounded by two strands which is disjoint from all the remaining strands of the diagram, up to isotopy, and which begins and ends in two crossings of the diagram with opposite orientations. Algebraically a disk is just a word of the form \(\sigma_i^ew\sigma_j^{-e}\), where \(e=\pm 1\), such that \(\sigma_i^ew\sigma_j^{-e}=w\) in the braid group. Using this algebraic definition of a disk, the author shows that in an Artin group whose associated Coxeter group is finite and dihedral, each word representing the trivial element must contain a subword which is a disk. This implies that each braid on three strands representing the trivial braid contains a disk. The proof is based on algebraic methods (Garside structure, partition of the Cayley graph). The author observes that his result cannot extend to braids with four or more strands, however computational experiments seem to suggest that his result is ``generically'' true even in this case.