On the topology and geometric construction of oriented matroids and convex polytopes

Abstract

Abstract. This paper develops new combinatorial and geometric techniques for studying the topology of the real semialgebraic variety 31 (M) of all realizations of an oriented matroid M . We focus our attention on point configurations in general position, and as the main result we prove that the realization space of every uniform rank 3 oriented matroid with up to eight points is contractible. For these special classes our theorem implies the isotopy property which states the spaces 32(M) are path-connected.We further apply our methods to several related problems on convex polytopes and line arrangements. A geometric construction and the isotopy property are obtained for a large class of neighborly polytopes. We improve a result of M. Las Vergnas by constructing a smallest counterexample to a conjecture of G Ringel, and, finally, we discuss the solution to a problem of R. Cordovil and P. Duchet on the realizability of cyclic matroid polytopes.

Enumerating Neighborly Polytopes and Oriented Matroids

“…There are uniform neighborly oriented matroids without universal edges in OM (5,11), OM (5,12), OM (7,11) and OM(9, 12) (only one such example, in OM(5, 10), was known [19]). The latter (together with their realizability) gives a positive answer to a question by Richter and Sturmfels [50] concerning the existence of neighborly 2k-polytopes with 2k + 4 vertices without universal edges.…”

“…An edge {e, e } of a neighborly oriented matroid M of odd rank is called universal if the contraction M/{e, e } is also neighborly. Universal edges correspond to inseparable pairs of elements of the oriented matroid [50].…”

“…In [50] it is proved that every neighborly oriented matroid of rank 2k + 1 with 2k + 4 vertices that has at least 2k − 4 universal edges is realizable. This implies that the 1 968 neighborly oriented matroids in OM (9,12) with at least 4 universal edges are realizable.…”

“…Even more, the 2 589 neighborly oriented matroids in OM (9,12) with at least one universal edge are also easily classified into realizable and non-realizable. By the reduction sequence method in [50], realizability of these matroids is reduced to realizability of rank 3 oriented matroids with at most 11 elements, whose realizability is completely classified [3,4,49]. As remarked in Section 4.1.1, it is known that all non-realizable rank 3 oriented matroids with up to 11 elements can be recognized by the BFP method.…”

“…We can certify realizability for those polytopes obtained by sewing and Gale-sewing [46], and we have non-realizability certificates by biquadratic final polynomials [17]. Moreover, certain cases can be decided by studying their universal edges [50]. In particular, we are able to completely classify: 5. all possible combinatorial types of neighborly 8-polytopes with 12 vertices.…”