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# Monotone Paths on Zonotopes and Oriented Matroids

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Published:2001-12-01
Printed: Dec 2001
• Christos A. Athanasiadis
• Francisco Santos
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## Abstract

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 $\lceil 2n/(n-d+2) \rceil-1$ 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.
 MSC Classifications: 52C35 - Arrangements of points, flats, hyperplanes [See also 32S22] 52B12 - Special polytopes (linear programming, centrally symmetric, etc.) 52C40 - Oriented matroids 20F55 - Reflection and Coxeter groups [See also 22E40, 51F15]

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