We determine all 2- and 3-cocycles for Laver tables, an infinite sequence of
finite structures obeying the left-selfdistributivity law; in particular, we
describe simple explicit bases. This provides a number of new positive braid
invariants and paves the way for further potential topological applications.
Incidentally, we establish and study a partial ordering on Laver tables given
by the right-divisibility relation
