Monotone paths on zonotopes and oriented matroids

Monotone paths on zonotopes and the natural generalization to maximal chain s in the poset of topes of an oriented matroid or arrangement of pseudo-hyp erplanes are studied with respect to a kind of local move, called polygon m ove or flip. It is proved that any monotone path on a d-dimensional zonotop e with n generators admits at least inverted right perpendicular 2n/(n -d+2 ) inverted left perpendicular -1 flips for all n greater than or equal to d + 2 greater than or equal to 4 and that for any fixed value of n - d, this lower bound is sharp for infinitely many values of n. In particular, monot one paths on zonotopes which admit only three flips are constructed in each dimension d greater than or equal to 3. Furthermore, the previously known 2-connectivity of the graph of monotone paths on a polytope is extended to the 2-connectivity of the graph of maximal chains of topes of an oriented m atroid. An application in the context of Coxeter groups of a result known t o be valid for monotone paths on simple zonotopes is included.