On the Generation of Oriented Matroids

Abstract

We provide a multiple purpose algorithm for generating oriented matroids. An application disproves a conjecture of Grünbaum that every closed triangulated orientable 2-manifold can be embedded geometrically in R 3 , i.e., with flat triangles and without selfintersections. We can show in particular that there exists an infinite class of orientable triangulated closed 2-manifolds for each genus g ≥ 6 that cannot be embedded geometrically in Euclidean 3-space. Our algorithm is interesting in its own right as a tool for many investigations in which oriented matroids play a key role.

Degree-regular triangulations of the double-torus

“…If there exists an n-vertex {p, q}-equivelar polyhedral map of Euler characteristic −2 then −2 = n − nq 2 + nq p and 12,3,7) or (28,7,3). Since two polyhedral maps M and N are isomorphic if and only if their duals M and N are isomorphic (cf.…”

“…By using computer, Altshuler et al ([1]) have shown that there are exactly 59 orientable neighbourly combinatorial 2-manifolds on 12 vertices. In [3], Bokowski and Guedes de Oliveira have shown that one of these 59 combinatorial 2-manifolds (namely, N 12 54 ) is not geometrically realizable in R 3 . In [8], Datta and Nilakantan have shown that there are exactly 27 degree-regular combinatorial 2-manifolds on at most 11 vertices.…”

“…So, if K is weakly regular and χ(K) ≥ 0 then (n, d) = (4,3), (6,4), (6,5), (12,5) (6,4), (6,5), (12,5)}, there exists a unique combinatorial 2-manifold, namely, the 4-vertex 2-sphere, the boundary of the octahedron, the 6-vertex real projective plane and the boundary of the icosahedron (cf. [5,8]).…”

Hardness of embedding simplicial complexes in $\mathbb{R}^d$

“…Brehm [Bre83] constructed a triangulation of the Möbius strip that does not admit a linear embedding into R 3 . Using methods from the theory of oriented matroids, Bokowski and Guedes de Oliveira [BGdO00] showed that for any g ≥ 6, there is a triangulation of the orientable surface of genus g that does not admit a linear embedding into R 3 . In higher dimensions, Brehm and Sarkaria [BS92] showed that for every k ≥ 2, and every d with…”

Geometric Realization of Möbius Triangulations

“…Moreover, Archdeacon, Bonnington, and Ellis-Monanghan proved that every toroidal triangulation has a geometric realization [1]. In general, Grünbaum conjectured that every triangulation on any orientable closed surface has a geometric realization [7], but Bokowski and Guedes de Oliveira recently showed that a triangulation by K 12 on the orientable closed surface of genus 6 has no geometric realization [2]. (For related topics, see [5].…”