ALGORITHMS FOR POSITIVE BRAIDS 论文

摘要

We give an easily handled algorithm for the word problem in each of Artin’s braid groups, Bn, based on Garside’s methods, but framed more directly in terms of the set of positive braids in which each pair of strings crosses at most once. We develop a natural partial order on each braid group defined in terms of positive braids, and apply this to compare braids with different powers ∆r of the fundamental half-twist braid ∆. This leads to an improvement of Garside’s conjugacy algorithm, using a much smaller finite subset of each conjugacy class, which we term the super summit set, to represent the class, in place of Garside’s summit set.