The MathNet Korea
Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Canadian Journal of Mathematics
( Vol. 53 NO.6 / (2001))
Monotone Paths Zonotopes and Oriented Matroids
Christos A. Athanasiadis, Francisco Santos,
Pages. 1121-1140
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 $lceil 2n/(n-d+2)
ceil-1$ flips for all $n geq d+2 geq 4$ 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 $d geq 3$. 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.
Contents 1. Introduction
2. Preliminaries
3. Flips for monotone paths on zonotopes
4. The lower bound
5. Monotone paths with few flips
6. Oriented matroids and chains of topes
7. Simple zonotopes and coxeter arrangements
Key words
Mathmatical Subject Classification 52C35, 52B12, 52C40, 20F55