Polytopes.Combinatorial.SmallSpheresDim4
Item from PolyDB: All combinatorial 3-spheres with up to 9 vertices together with information about their realizability.
All combinatorial 3-spheres with up to 9 vertices together with information about their realizability. For up to seven vertices this was obtained by Perles (see [Gr?nbaum 1967]), for eight vertices by Altshuler and Steinberg, and for nine vertices by Firsching. For all entries the database contains the incidence matrix (VERTICES_IN_FACETS) information about realizability (REALIZABLE), the f-vector (F_VECTOR), the number of vertices and facets (N_VERTICES, N_FACETS), and the information whether the sphere is simple of simplicial (SIMPLE, SIMPLICIAL). For realizable spheres with nine vertices (i.e. 4-polytopes with nine vertices) there is also a list of vertices (VERTICES), while for the non-realizable spheres with nine vertices there is also a proof of non-realizability that depends on a possibly incomplete computation of the chirotope (PARTIAL_CHIROTOPE). This proof can be of one of three types. The first two depend on the Grassmann-Pl?cker relations. In the first type the chirotope contains enough information to find a violated relation, and the three involved terms are given. In the second type two relations are given that would allow to deduce a not yet known entry of the chirotope, but do not infer the same value. In the last type there is a complete chirotope, but the linear program to the biquadratic final polynomial is infeasible, and the linear program is given. For a detailed explanation of the proofs see the paper of Firsching.
