BigObject Cone

An abstract simplicial complex represented by its facets.

These properties are for input only. They allow redundant information.

Type: /common/property_types/Basic Types/Array
Any description of the faces of a simplicial complex with vertices v_0 < v_1 < v_2 < … arbitrary. Redundant faces allowed.

These properties are for visualization.

Type: /topaz/objects/SimplicialComplex/properties/Visualization/MIXED_GRAPH
The nodes of the mixed graph are the nodes of the primal GRAPH and the DUAL_GRAPH. Additional to the primal and dual edges, there is an edge between a primal and a dual node iff the primal node represents a vertex of the corresponding facet of the dual node.
Properties of MIXED_GRAPH:

• EDGE_WEIGHTS
  Type: /common/property_types/Graph Types/EdgeMap
  Associated edge weights.

Type: /common/property_types/Basic Types/Array
Labels of the vertices.

These properties capture information of the object that is concerned with the action of permutation groups.

Type: /group/objects/Group

These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

Type: /common/property_types/Basic Types/Bool
A simplicial complex is pure if all its facets have the same dimension.

Type: /common/property_types/Basic Types/Int
Number of MINIMAL_NON_FACES.

Type: /common/property_types/Basic Types/Array
f_k is the number of k-faces, for k = 0,… , d, where d is the dimension.

Type: /common/property_types/Basic Types/Array
Inclusion minimal non-faces (vertex subsets which are not faces of the simplicial complex).

Type: /common/property_types/Basic Types/Array
The h-vector of the simplicial complex.

Type: /common/property_types/Basic Types/Bool
True if GRAPH is (DIM + 1)-colorable.

Type: /graph/objects/Combinatorics/Graph
The subcomplex consisting of all 1-faces.

Type: /common/property_types/Basic Types/Array
Indices of the vertices from INPUT_FACES. That is, the map i \mapsto v_i.

Type: /topaz/objects/SimplicialComplex/properties/Combinatorics/BOUNDARY
Codimension-1-faces of a PSEUDO_MANIFOLD which are contained in one facet only.
Properties of BOUNDARY:
These properties capture combinatorial information of the object. Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice.

• VERTEX_MAP
  Type: /common/property_types/Basic Types/Array
  Maps vertices of the boundary complex to the corresponding ones in the supercomplex

Type: /topaz/objects/SimplicialComplex/properties/Combinatorics/HASSE_DIAGRAM
The face lattice of the simplical complex organized as a directed graph. Each node corresponds to some face of the simplical complex. It is represented as the list of vertices comprising the face. The outgoing arcs point to the containing faces of the next dimension. An artificial top node is added to represent the entire complex.

Type: /common/property_types/Basic Types/Array
A coloring of the vertices.

Type: /topaz/objects/SimplicialComplex/properties/Combinatorics/DUAL_GRAPH
The graph of facet neighborhood. Two FACETS are neighbors if they share a (d-1)-dimensional face.
Properties of DUAL_GRAPH:

• COLORING
  Type: /common/property_types/Graph Types/NodeMap
  UNDOCUMENTED

Type: /common/property_types/Basic Types/Array
For each vertex the corresponding vertex of facet 0 with respect to the action of the group of projectivities.

Type: /common/property_types/Set Types/PowerSet
Orbit decomposition of the group of projectivities acting on the set of vertices of facet 0.

Type: /common/property_types/Basic Types/Array
Faces which are maximal with respect to inclusion, encoded as their ordered set of vertices. The vertices must be numbered 0, …, n-1.

Type: /common/property_types/Basic Types/Int
Maximal dimension of the FACETS, where the dimension of a facet is defined as the number of its vertices minus one.

Type: /common/property_types/Basic Types/Bool
True if this is shellable.

Type: /common/property_types/Basic Types/Int
Number of FACETS.

Type: /common/property_types/Algebraic Types/Matrix
f_ik is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.

Type: /topaz/objects/SimplicialComplex
Subcomplex generated by faces of codimension 2 that are contained in an odd number of faces of codimension 1.

Type: /common/property_types/Basic Types/Int
Number of vertices.

Type: /common/property_types/Basic Types/Array
An ordered list of facets constituting a shelling.

The following properties are topological invariants.

Type: /common/property_types/Basic Types/Array
Reduced simplicial homology groups H₀, …, H_d (integer coefficients), listed in increasing dimension order. See HomologyGroup for explanation of encoding of each group.

Type: /common/property_types/Basic Types/Array
Edge-subset of a 3-sphere which is a knot or link, that is, a collection of pairwise disjoint cycles.

Type: /common/property_types/Basic Types/Array
Representatives of cocycle groups, listed in increasing codimension order. See CycleGroup for explanation of encoding of each group.

Type: /common/property_types/Basic Types/Array
Representatives of cycle groups, listed in increasing dimension order. See CycleGroup for explanation of encoding of each group.

Type: /common/property_types/Basic Types/Int
Reduced Euler characteristic. Alternating sum of the F_VECTOR minus 1.

Type: /common/property_types/Basic Types/Bool
Determines if this is homeomorphic to a sphere. In general, this is undecidable; therefore, the implementation depends on heuristics. May be true or false or undef (if heuristic does not succeed).

Type: /topaz/property_types/Topology/IntersectionForm
The integral quadratic form obtained from restricting the multiplication of the cohomology of a closed 4k-manifold to H^{2k} x H^{2k} → H^{4k} = Z. As a quadratic form over the reals it is characterized by its dimension and its index of inertia (or, equivalenty, by the number of positive and negative ones in its canonical form). An integral quadratic form is even if it takes values in 2Z.

Type: /common/property_types/Basic Types/Pair
A finite representation of the fundamental group. The fundamental group is represented as a pair of an integer, the number of generators, and a list of relations. The generators are numbered consecutively starting with zero. A relation is encoded as a list of pairs, each pair consisting of a generator and its exponent.

You may use the fundamental2gap method to produce a GAP file.

Type: /common/property_types/Basic Types/Bool
True if this is a PURE simplicial complex with the property that each ridge is contained in either one or two facets.

Type: /common/property_types/Basic Types/Bool
True if the vertex star of each vertex is DUAL_CONNECTED.

Type: /common/property_types/Basic Types/Array
Mod 2 cycle representation of Stiefel-Whitney classes. Each cycle is represented as a set of simplices.

Type: /common/property_types/Basic Types/Array
An orientation of the facets of an ORIENTED_PSEUDO_MANIFOLD, such that the induced orientations of a common ridge of two neighboring facets cancel each other out. Each facet is marked with true if the orientation agrees with the (chosen) orientation of the first facet, and is marked with false otherwise.

Type: /common/property_types/Basic Types/Array
Reduced cohomology groups, listed in increasing codimension order. See HomologyGroup for explanation of encoding of each group.

Type: /common/property_types/Basic Types/Bool
Determines if this is homeomorphic to a ball. In general, this is undecidable; therefore, the implementation depends on heuristics. May be true or false or undef (if heuristic does not succeed).

Type: /common/property_types/Basic Types/Bool
Determines if this is a compact simplicial manifold with boundary. Depends on heuristic SPHERE recognition. May be true or false or undef (if heuristic does not succeed).

Type: /common/property_types/Basic Types/Int
The genus of a surface.

Type: /common/property_types/Basic Types/Array
Labels of the generators of the FUNDAMENTAL_GROUP. The labels can be chosen freely. If the FUNDAMENTAL_GROUP is computed by polymake, the generators correspond to the edges of the complex. Hence they are labeled g followed by the vertices of the edge, e.g. g3_6 corresponds to the edge {3 6}.

Type: /common/property_types/Basic Types/Bool
True if this is a PSEUDO_MANIFOLD with top level homology isomorphic to Z.

Type: /topaz/objects/SimplicialComplex/properties/Topology/MORSE_MATCHING
Morse matching in the Hasse diagram of the simplicial complex
Properties of MORSE_MATCHING:

• MATCHING
  Type: /common/property_types/Graph Types/EdgeMap
  The matching in the HasseDiagram of the SimplicialComplex

Type: /common/property_types/Basic Types/Bool
True if this is a PURE simplicial complex with the property that each ridge is contained in exactly two facets.

Type: /common/property_types/Basic Types/Bool
True if this is a CONNECTED MANIFOLD of dimension 2.