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Monotone paths on zonotopes and the natural generalization to maximal chains in the poset of topes of an oriented matroid or arrangement of pseudo-hyperplanes are studied with respect to a kind of local move, called polygon move or flip. It is proved that any monotone path on a 
$d$-dimensional zonotope with 
$n$ generators admits at least 
$\left\lceil 2n/\left( n-d+2 \right) \right\rceil -1$ flips for all 
$n\ge d+2\ge 4$ and that for any fixed value of 
$n-d$, this lower bound is sharp for infinitely many values of 
$n$. In particular, monotone paths on zonotopes which admit only three flips are constructed in each dimension 
$d\ge 3$. 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 matroid. An application in the context of Coxeter groups of a result known to be valid for monotone paths on simple zonotopes is included.
