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Monotone Paths on Zonotopes and Oriented Matroids

Published online by Cambridge University Press:  20 November 2018

Christos A. Athanasiadis  and
Francisco Santos
Christos A. Athanasiadis
Affiliation:
Department of Mathematics, Royal Institute of Technology, S-100 44 Stockholm, Sweden. email: athana@math.kth.se
Francisco Santos
Affiliation:
Departamento de Matemàticas, Estadística y Computación, Universidad de Cantabria, Santander, E-39005, Spain. email: santos@matesco.unican.es
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Abstract

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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.

Keywords

52C3552B1252C4020F55
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 53 , Issue 6 , 01 December 2001 , pp. 1121 - 1140
Copyright
Copyright © Canadian Mathematical Society 2001

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