On the distance between the expressions of a permutation

TL;DR

This paper establishes a sharp polynomial bound on the combinatorial distance between reduced expressions of permutations, connecting it with geometric intersection numbers and analyzing the reversing algorithm's optimality.

Contribution

It proves a tight O(n^4) bound on the distance between permutation expressions and links it to geometric intersection theory, also analyzing the reversing algorithm's efficiency.

Findings

Bound on permutation expression distance is sharp and polynomial.

Connection established between combinatorial distance and geometric intersection numbers.

Reversing algorithm's optimality criterion is provided.

Abstract

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), a sharp bound. Using a connection with the intersection numbers of certain curves in van Kampen diagrams, we prove that this bound is sharp, and give a practical criterion for proving that the derivations provided by the reversing algorithm of [Dehornoy, JPAA 116 (1997) 115-197] are optimal. We also show the existence of length l expressions whose reversing requires C l^4 elementary steps.