Monotone Paths on Zonotopes and Oriented Matroids
نویسنده
چکیده
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 ip. It is proved that any monotone path on a d-dimensional zonotope with n generators admits at least d2n=(n ? d + 2)e ? 1 ips for all n d+2 4 and that for any xed value of n?d, this lower bound is sharp for innnitely many values of n. In particular, monotone paths on zonotopes which admit only three ips 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.
منابع مشابه
A Note on Space Tiling Zonotopes
In 1908 Voronoi conjectured that every convex polytope which tiles space face-to-face by translations is affinely equivalent to the Dirichlet-Voronoi polytope of some lattice. In 1999 Erdahl proved this conjecture for the special case of zonotopes. A zonotope is a projection of a regular cube under some affine transformation. In 1975 McMullen showed several equivalent conditions for a zonotope ...
متن کاملSymmetry, oriented matroids and two conjectures of Michel Las Vergnas
The paper has two parts. In the first part we survey the existing results on the cube conjecture of Las Vergnas. This conjecture claims that the orientation of the matroid of the cube is determined by the symmetries of the underlying matroid. The second part deals with euclidean representations of matroids as geometric simplicial complexes defined by symmetry properties abstracting those of zon...
متن کاملRandom Walk and Hyperplane Arrangements
Let C be the set of chambers of a real hyperplane arrangement. We study a random walk on C introduced by Bidigare, Hanlon, and Rockmore. This includes various shuuing schemes used in computer science, biology, and card games. It also includes random walks on zonotopes and zonotopal tilings. We nd the stationary distributions of these Markov chains, give good bounds on the rate of convergence to...
متن کاملMonotone clutters
Ding, G., Monotone clutters, Discrete Mathematics 119 (1993) 67-77. A clutter is k-monotone, completely monotone or threshold if the corresponding Boolean function is k-monotone, completely monotone or threshold, respectively. A characterization of k-monotone clutters in terms ofexcluded minors is presented here. This result is used to derive a characterization of 2-monotone matroids and of 3-m...
متن کامل