| Monotone Paths On Zonotopes And Oriented Matroids (2007) | |||||||||||||||
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 d2n=(n \Gamma d + 2)e \Gamma 1 flips for all n d+2 4 and that for any fixed value of n \Gamma 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 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. 1. Introduction Let P be a d-dimensional polytope in R ... | |||||||||||||||
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