Simplicial cells in arrangements and mutations of oriented matroids

Abstract

We investigate the combinatorial and topological properties of simplicial cells in arrangements of (pseudo)hyperplanes, using their interpretations in terms of oriented matroids. Simplicial cells have various applications in computational geometry due to the fact that for an arrangement in general position they are in one-to-one correspondence to local changes ('mutations') of its combinatorial type. Several characterizations for mutations of oriented matroids, and their relation to geometric realizability questions are being discussed.We give two short proofs for a result of Shannon [30] that every arrangement of n hyperplanes in E d has at least n simplicial cells, this bound being sharp for every n and d. The concatenation operation, a construction introduced by Lawrence and Weinberg [21], will be used to generate a large class of representable oriented matroids with this minimal number of mutations.A homotopy theorem is proved, stating that any two arrangements in general position can be transformed into each other be a sequence of representability preserving mutations. Finally, we give an example of an oriented matroid on eight elements with only seven mutations. As a corollary we obtain a new proof for the non-polytopality of the smallest non-polytopal matroid sphere M9963.

Lawrence Oriented Matroids and a Problem of McMullen on Projective Equivalences of Polytopes

“…The idea to explore the class of Lawrence oriented matroids in relation to the McMullen problem was originally suggested by Las Vergnas (personal communication). Lawrence oriented matroids have nice properties; for instance, Roudneff and Sturmfels [9] gave an inductive proof that a Lawrence oriented matroid of rank r on n elements has exactly n r -simplices. † This work was partially supported by SNI and the Alexander von Humboldt Foundation.…”

On the Topology and Geometric Construction of Oriented Matroids and Convex Polytopes

“…Consider the graph J^"1,3 on the set of all uniform rank 3 oriented matroids with « points whose edges are precisely the pairs {M, -M} . For n < 8 all oriented matroids are realizable [16], and hence the graph S"1'3 is connected by [34,Theorem 3.7].…”

A two-dimensional topological representation theorem for matroid polytopes of rank 4

“…Pseudosphere arrangements on S r−1 (r ≥ 4) are usually very difficult to visualize, and there remain many open problems on rank r(≥ 4) oriented matroids. One of the outstanding problems is Cordovil-Las Vergnas conjecture (mentioned in [10]), which claims that every pair of uniform oriented matroids on the same ground set can be transformed into each other by a finite sequence of mutations. In terms of pseudosphere arrangements, this conjecture claims the existence of a higher-dimensional analogue of Ringel's homotopy theorem.…”

“…In terms of pseudosphere arrangements, this conjecture claims the existence of a higher-dimensional analogue of Ringel's homotopy theorem. The conjecture is known to be true for realizable oriented matroids [10], but in general case, the conjecture is wide open. To be able to apply mutation to an oriented matroid, there must exist a simplicial tope (in terms of pseudosphere arrangements, a simplicial cell) in it.…”