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Enumerating neighborly polytopes and oriented matroids

Neighborly polytopes are those that maximize the number of faces in each dimension among all polytopes with the same number of vertices. Despite their extremal properties they form a surprisingly rich class of polytopes, which has been widely studied and is the subject of many open problems and conjectures. In this paper, we study the enumeration of neighborly polytopes beyond the cases that have been computed so far. To this end, we enumerate neighborly oriented matroids --- a combinatorial abstraction of neighborly polytopes --- of small rank and corank. In particular, if we denote by OM($r,n$) the set of all oriented matroids of rank $r$ and $n$ elements, we determine all uniform neighborly oriented matroids in OM($5,\leq 12$), OM($6,\leq 9$), OM($7,\leq 11$) and OM($9,\leq 12$) and all possible face lattices of neighborly oriented matroids in OM($6,10$) and OM($8,11$). Moreover, we classify all possible face lattices of uniform $2$-neighborly oriented matroids in OM($7,10$) and OM($8,11$). Based on the enumeration, we construct many interesting examples and test open conjectures.