On the distance between the expressions of a permutation

We prove that the combinatorial distance between any two reduced expressions of a given permutation of { 1 , … , n } in terms of transpositions lies in O ( n 4 ) . We prove that this bound is sharp, and, using a connection with the intersection numbers of certain curves in van Kampen diagrams, we give a practical criterion for proving that the derivations provided by the reversing algorithm of Dehornoy [Groups with a complemented presentation, J. Pure Appl. Algebra, 116 (1997) 115–197] are optimal. We also show the existence of length ℓ expressions of different permutations whose reversing requires C ℓ 4 elementary steps.