BigObject GrassPluckerCertificate

from application topaz

This object encodes a decorated graph that certifies that a given simplicial complex is not realizable as the boundary of a convex polytope. The simplicial complex itself should either be a simplicial sphere, or an oriented simplicial complex with boundary. In the latter case, each component of the boundary will be coned over by a new point, and the resulting complex should then be a simplicial sphere. (This is important when handling Criado & Santos's topological prismatoids, for example.) The nodes of the graph are decorated with Grassmann-Plücker polynomials, and the edges with “undetermined solids”, ie, solids whose orientation can vary according to the realization. The point of the certificate is that no matter how these undetermined solids are oriented, there will always be some Grassmann-Plücker polynomial in the tree all of whose terms are positive. But this contradicts realizability, because the matrix of homogeneous coordinates of any putative convex realization of a d-sphere on n vertices determines a point in the Grassmann manifold G(d,n), which means that all GP-polynomials should vanish – but the special one can't, because all its terms are positive.

derived from:

Graph

These properties capture geometric information of the object. Geometric properties depend on geometric information of the object, like, e.g., vertices or facets.

EDGE_LABELS

A string representation of the UNDETERMINED_SOLIDS It indexes the participating solids using the ordering given in SOLIDS

Type:

EdgeMap<Undirected,String>

PLUCKER_RELATIONS

This property encodes the Grassmann-Plücker relations of a simplicial complex of dimension d on n vertices that participate in the nonrealizability certificate. Each relation is of the form Gamma(I|J) = sum_{j in J} sign(j,I,J) [I cup j] [J minus j], where I in ([n] choose d) and J in ([n] choose d+2), and where sign(j,I,J) in {-1,+1} is determined by j, I and J. Compare Thm 14.6 in Miller & Sturmfels, Combinatorial Commutative Algebra. The Array<Set<Int» has length 3, and consists of a singleton set with entry +-1 for the sign, followed by I and J.

Type:

NodeMap<Undirected,Array<Set<Int>>>

SOLIDS

The ordering of the solids used in NODE_LABELS and EDGE_LABELS

Type:

Array<Array<Int>>

UNDETERMINED_SOLIDS

For each edge, store the undetermined solids across that edge. There can be more than one in the case of dummy edges connecting a cube node to a cube vertex node. The sign of each undetermined solid is incorporated into the inner Array as a possible leading “-1”.

Type:

EdgeMap<Undirected,Array<Array<Int>>>