Abstract

Oriented matroids are a combinatorial model for configurations in real vector spaces. A central role in the theory is played by the realizability problem: Given an oriented matroid, find an associated vector configuration. In this paper we present two closely related oriented matroids\Omega + 14 and\Omega \Gamma 14 of rank 3 with 14 elements that have interesting properties with respect to realizability.\Omega + 14 and\Omega \Gamma 14 differ in exactly one basis orientation. The realizable oriented matroid\Omega + 14 has at least two interesting properties: First it has a combinatorial symmetry that has no metric realization, and second it has a disconnected realization space. In other words, there are different realizations of \Omega + 14 that cannot be continuously deformed into each other while staying in the same isotopy class. The oriented matroid\Omega \Gamma 14 is non-realizable but it has no bi-quadratic final polynomial. In other words, the only known effective al...

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