Alternating normal forms for braids and locally Garside monoids

We describe new types of normal forms for braid monoids, Artin–Tits monoids, and, more generally, for all monoids in which divisibility has some convenient lattice properties (“locally Garside monoids”). We show that, in the case of braids, one of these normal forms coincides with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach leads to a new, simple description for the standard order (“Dehornoy order”) of BnBn in terms of that of Bn−1Bn−1, and to a quadratic upper bound for the complexity of this order.