... In order to obtain shorter notation, we shall in the sequel use a, b, c... for σ 1 , σ 2 , σ 3 ..., and, symmetrically, A, B... for σ −1 1 , σ −1 2 ... (as in the caption of Figure 1). Then, a typical greedy normal form for a 4-braid is the sequence (−2; ac, abcb, bcba, a), i.e., equivalently, using (f (1), ..., f (n)) to specify a permutation f of {1, ..., n}, (−2; (2, 1, 4, 3), (2, 4, 3, 1), (4, 1, 3, 2), (2, 1, 3, 4)), consisting of an integer and four simple 4-braids, or, equivalently, four permutations of {1, ..., 4}: for instance, (2,1,4,3) is the permutation associated with ac, i.e., with σ 1 σ 3 . To check that we have a greedy normal form, we observe for instance that the descents of (2, 1, 4, 3) are 1 and 3, while the recoils of (2, 4, 3, 1), i.e., the descents of (4, 1, 2, 3), are 1 and 3 as well, so the normality condition is satisfied between (2,1,4,3) and (2,4,3,1). ...