topaz: Standard Properties

These properties and pseudo-targets are known to polymake when the default set of rules poly.rules is used. If the polymake script is called, it assumes by default that the file is an application polytope file. So, if you want your file be treated as a topaz file, call the polymake script followed by -A topaz. For any further use of that file, polymake knows that it is a topaz file and you do not need to specify it any longer. Have a look at the tutorial for a first few steps in topaz.

Basic sections The following sections describe the basic properties of a simplicial complex.
FACETS array of sets	Inclusion maximal faces of the (abstract) simplicial complex, encoded as their ordered set of vertices. The vertices must be numbered 0..n-1.
N_FACETS cardinal	Number of FACETS
DIM cardinal	Maximal dimension of the FACETS, where the dimension of a facet is defined as the number of its vertices less one.
PURE boolean	A simplicial complex is pure if all its facets have the same dimension.
N_VERTICES cardinal	Number of vertices
VERTEX_LABELS array of labels	Labels of the vertices.
GRAPH graph	The graph is defined by the subcomplex consisting of all 1-faces.
DUAL_GRAPH graph	The graph of facet neighborhood. Two facets are neighbors if they share a d-1-dimensional face.
HASSE_DIAGRAM face lattice	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.
BOUNDARY_OF_PSEUDO_MANIFOLD boolean	Co-dimension -1 faces of a PSEUDO_MANIFOLD which are contained in one facet only.
Combinatorics
MINIMAL_NON_FACES array of sets	Inclusion minimal non-faces (vertex subsets which are not faces of the simplicial complex).
N_MINIMAL_NON_FACES cardinal	Number of MINIMAL_NON_FACES
F_VECTOR vector of cardinals	f_k is the number of k-faces, for k = -1,..,d, where d is the dimension.
F2_VECTOR matrix of cardinals	f_ik is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.
ODD_SUBCOMPLEX array of sets	All faces of co-dimension -2 that are contained in an odd number of faces of co-dimension -1.
CONNECTIVITY cardinal	Node connectivity of the GRAPH, that is, the minimal number of nodes to be removed from the graph such that the result is disconnected.
DUAL_CONNECTIVITY cardinal	Node connectivity of the DUAL_GRAPH. Dual to CONNECTIVITY
PROJ_ORBITS powerset	Orbit decomposition of the group of projectivities acting on the set of vertices of facet 0.
PROJ_DICTIONARY array of cardinal	For each vertex the corresponding vertex of facet 0 with respect to the action of the group of projectivities.
Topology The following properties are topological invariant to different triangulations of the same topological space.
EULER_CHARACTERISTIC cardinal	Reduced Euler characteristic. Alternating sum of the F_VECTOR.
HOMOLOGY array of tuple(list of tuple(cardinal , cardinal) , cardinal)	Reduced simplicial homology groups H₀,...,H_d (integer coefficients), listed in increasing dimension order. Each group G is encoded as a sequence ({ (t₁ m₁) ... (t_n m_n) } f) of non-negative integers, with t₁ > t₂ > ... > t_n > 1, plus an extra non-negative integer f. The group G is isomorphic to (Z/t₁)^m_1 × ... × (Z/t_n)^m_n × Z^f, where Z⁰ is the trivial group.
COHOMOLOGY array of tuple(list of tuple(cardinal , cardinal) , cardinal)	Reduced cohomology groups, listed in increasing co-dimension order. Encoding similar to HOMOLOGY.
CYCLES array of tuple(sparse matrix of integer , array of sets)	Representatives of cycle groups, listed in increasing dimension order. The first component in each dimension is a matrix of integer coefficients, the second component is a vector of faces. To obtain the chains, one must multiply (symbolically) both components.
COCYCLES array of tuple(sparse matrix of integer , array of sets)	Representatives of co-cycle groups, listed in increasing co-dimension order. Encoding similar to CYCLES.
INTERSECTION_FORM tuple(cardinal , cardinal , cardinal)	Parity and signature of the intersection form of a closed oriented 4-manifold.
STIEFEL_WHITNEY array of powersets	Mod 2 cycle representation of Stiefel-Whitney classes. Each cycle is represented as a set of simplices.
PSEUDO_MANIFOLD boolean	A PURE simplicial complex with the property that each ridge is contained in either one or two facets.
MANIFOLD boolean	Compact simplicial manifold with boundary.
CLOSED_PSEUDO_MANIFOLD boolean	PURE simplicial complex with the property that each ridge is contained in exactly two facets.
ORIENTED_PSEUDO_MANIFOLD boolean	A PSEUDO_MANIFOLD with top level homology isomorphic to Z.
BALL_OR_SPHERE boolean	Topological space homeomorphic to either a ball or a sphere.
SPHERE boolean	Topological space homeomorphic to a sphere.
BALL boolean	Topological space homeomorphic to a ball.
CONNECTED boolean	A simplicial complex is connected if its GRAPH is a connected graph.
DUAL_CONNECTED boolean	A simplicial complex is dually connected if its DUAL_GRAPH is a connected graph.
CONNECTED_COMPONENTS array of sets	The connected components of the GRAPH, encoded by their node sets.
DUAL_CONNECTED_COMPONENTS array of sets	The connected components of the DUAL_GRAPH, encoded by their node sets.
SURFACE boolean	A CONNECTED MANIFOLD of dimension 2.
LOCALLY_STRONGLY_CONNECTED boolean	The vertex star of each vertex is DUAL_CONNECTED.
FUNDAMENTAL_GROUP tuple(list , list)	A finite representation of the fundamental group. You may use the fundamental2gap client to produce a `GAP` file.
heckenbach pseudo	Use the Heckenbach's homology program to compute the homology groups instead of the built-in client t_homology.
Visualization
GEOMETRIC_REALIZATION matrix	Coordinates for the vertices of the simplicial complex, such that the complex is embedded without crossings in some R^e. Vector "x₁ .. x_e" represents a point in Euclidean e-space .
G_DIM cardinal	Dimension e of the space the GEOMETRIC_REALIZATION of the complex is embedded in.
GraphVisualizer word	Name of a package implementing the graph visualization interface.
VISUAL pseudo	Uses the coordinates of GEOMETRIC_REALIZATION, if they exist and are at most three dimensional, to visualize the graph. Otherwise, the spring embedder is used.
VISUAL_FACETS pseudo	Uses the coordinates of GEOMETRIC_REALIZATION, if they are at most three dimensional, to visualize the graph and all faces with dimension smaller three. All faces of one facet build a geometry in the jvx-file, so you may use Method -> Effect -> Explode Group of Geometries in the javaview menu. If G_DIM > 3, the spring embedder is used to produce coordinates for the visualization.
VISUAL_GRAPH pseudo	Uses the spring embedder to visualize the graph.
VISUAL_DUAL_GRAPH pseudo	Uses the spring embedder to visualize the dual graph.
VISUAL_SUBCOMPLEX pseudo	Uses the coordinates of the GEOMETRIC_REALIZATION if possible or the spring embedder otherwise to visualize the facets and the subcomplex specified by the section SUBCOMPLEX.
SUBCOMPLEX array of sets	The subcomplex to be visualized by VISUAL_SUBCOMPLEX.
javaview pseudo	Runs javaview program - default tool for the visualization.

Basic sections

Combinatorics

Topology

Visualization