Two Interesting Oriented Matroids

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 $\Omega_{14}^-$ of rank~3 with 14 elements that have interesting properties with respect to realizability. $\Omega_{14}^+$ and $\Omega_{14}^-$ 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 $\Omega_{14}^-$ is non-realizable but it has no bi-quadratic final polynomial. In other words, the only known effective algorithmic method fails to prove the non-realizability of $\Omega_{14}^-$.

1991 Mathematics Subject Classification: Primary 52B40; Secondary 14P10, 51A25, 52B30.

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