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| playground:playground [2019/03/03 19:05] – oroehrig | playground:playground [2020/05/21 12:24] (current) – removed benmuell |
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| ====== BigObject Cone ====== | |
| An abstract simplicial complex represented by its facets. | |
| ===== Properties ===== | |
| |
| ==== Input Properties ==== | |
| These properties are for input only. They allow redundant information. | |
| |
| == INPUT_FACES == | |
| ''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. | |
| |
| ---- | |
| ==== Visualization ==== | |
| These properties are for visualization. | |
| |
| == MIXED_GRAPH == | |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Visualization/MIXED_GRAPH\\ The nodes of the mixed graph are the nodes of the primal [[/topaz/objects/SimplicialComplex/properties/Combinatorics/GRAPH]] and | |
| the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/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. | |
| |
| == VERTEX_LABELS == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Labels of the vertices. | |
| |
| ---- | |
| ==== Symmetry ==== | |
| These properties capture information of the object that is concerned with the action of permutation groups. | |
| |
| == GROUP == | |
| ''Type:'' /group/objects/Group | |
| ---- | |
| ==== Combinatorics ==== | |
| These properties capture combinatorial information of the object. | |
| Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice. | |
| |
| == PURE == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ A simplicial complex is __pure__ if all its facets have the same dimension. | |
| |
| == N_MINIMAL_NON_FACES == | |
| ''Type:'' /common/property_types/Basic Types/Int\\ Number of [[/topaz/objects/SimplicialComplex/properties/Combinatorics/MINIMAL_NON_FACES]]. | |
| |
| == F_VECTOR == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ f<sub>k</sub> is the number of k-faces, for k = 0,... , d, where d is the dimension. | |
| |
| == MINIMAL_NON_FACES == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Inclusion minimal non-faces (vertex subsets which are not faces of the simplicial complex). | |
| |
| == H_VECTOR == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ The h-vector of the simplicial complex. | |
| |
| == FOLDABLE == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if [[/topaz/objects/SimplicialComplex/properties/Combinatorics/GRAPH]] is ([[/topaz/objects/SimplicialComplex/properties/Combinatorics/DIM]] + 1)-colorable. | |
| |
| == GRAPH == | |
| ''Type:'' /graph/objects/Combinatorics/Graph\\ The subcomplex consisting of all 1-faces. | |
| |
| == VERTEX_INDICES == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Indices of the vertices from [[/topaz/objects/SimplicialComplex/properties/Input property/INPUT_FACES]]. That is, the map i \mapsto v_i. | |
| |
| == BOUNDARY == | |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Combinatorics/BOUNDARY\\ Codimension-1-faces of a [[/topaz/objects/SimplicialComplex/properties/Topology/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 | |
| |
| == HASSE_DIAGRAM == | |
| ''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. | |
| |
| == COLORING == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ A coloring of the vertices. | |
| |
| == DUAL_GRAPH == | |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Combinatorics/DUAL_GRAPH\\ The graph of facet neighborhood. | |
| Two [[/topaz/objects/SimplicialComplex/properties/Combinatorics/FACETS]] are neighbors if they share a (d-1)-dimensional face. | |
| \\ __Properties of DUAL_GRAPH:__\\ | |
| * **COLORING**\\ ''Type:'' /common/property_types/Graph Types/NodeMap\\ UNDOCUMENTED | |
| |
| == PROJ_DICTIONARY == | |
| ''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. | |
| |
| == PROJ_ORBITS == | |
| ''Type:'' /common/property_types/Set Types/PowerSet\\ Orbit decomposition of the group of projectivities acting on the set of vertices of facet 0. | |
| |
| == FACETS == | |
| ''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. | |
| |
| == DIM == | |
| ''Type:'' /common/property_types/Basic Types/Int\\ Maximal dimension of the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/FACETS]], where the dimension of a facet is defined as | |
| the number of its vertices minus one. | |
| |
| == SHELLABLE == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is shellable. | |
| |
| == N_FACETS == | |
| ''Type:'' /common/property_types/Basic Types/Int\\ Number of [[/topaz/objects/SimplicialComplex/properties/Combinatorics/FACETS]]. | |
| |
| == F2_VECTOR == | |
| ''Type:'' /common/property_types/Algebraic Types/Matrix\\ f<sub>ik</sub> is the number of incident pairs of i-faces and k-faces; the main | |
| diagonal contains the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/F_VECTOR]]. | |
| |
| == ODD_SUBCOMPLEX == | |
| ''Type:'' /topaz/objects/SimplicialComplex\\ Subcomplex generated by faces of codimension 2 that are contained in an odd | |
| number of faces of codimension 1. | |
| |
| == N_VERTICES == | |
| ''Type:'' /common/property_types/Basic Types/Int\\ Number of vertices. | |
| |
| == SHELLING == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ An ordered list of facets constituting a shelling. | |
| |
| ---- | |
| ==== Topology ==== | |
| The following properties are topological invariants. | |
| |
| == HOMOLOGY == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Reduced simplicial homology groups H<sub>0</sub>, ..., H<sub>d</sub> (integer coefficients), listed in increasing dimension order. | |
| See [[/topaz/property_types/Topology/HomologyGroup]] for explanation of encoding of each group. | |
| |
| == KNOT == | |
| ''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. | |
| |
| == COCYCLES == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Representatives of cocycle groups, listed in increasing codimension order. | |
| See [[/topaz/property_types/Topology/CycleGroup]] for explanation of encoding of each group. | |
| |
| == CYCLES == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Representatives of cycle groups, listed in increasing dimension order. | |
| See [[/topaz/property_types/Topology/CycleGroup]] for explanation of encoding of each group. | |
| |
| == EULER_CHARACTERISTIC == | |
| ''Type:'' /common/property_types/Basic Types/Int\\ Reduced Euler characteristic. Alternating sum of the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/F_VECTOR]] minus 1. | |
| |
| == SPHERE == | |
| ''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). | |
| |
| == INTERSECTION_FORM == | |
| ''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. | |
| |
| == FUNDAMENTAL_GROUP == | |
| ''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 [[/topaz/objects/SimplicialComplex/methods/Topology/fundamental2gap]] method to produce a ''GAP'' file. | |
| |
| == PSEUDO_MANIFOLD == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/properties/Combinatorics/PURE]] simplicial complex with the property that each ridge is | |
| contained in either one or two facets. | |
| |
| == LOCALLY_STRONGLY_CONNECTED == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if the vertex star of each vertex is [[/topaz/objects/SimplicialComplex/methods/Topology/DUAL_CONNECTED]]. | |
| |
| == STIEFEL_WHITNEY == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Mod 2 cycle representation of Stiefel-Whitney classes. Each cycle is represented as a set of simplices. | |
| |
| == ORIENTATION == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ An orientation of the facets of an [[/topaz/objects/SimplicialComplex/properties/Topology/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. | |
| |
| == COHOMOLOGY == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Reduced cohomology groups, listed in increasing codimension order. | |
| See [[/topaz/property_types/Topology/HomologyGroup]] for explanation of encoding of each group. | |
| |
| == BALL == | |
| ''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). | |
| |
| == MANIFOLD == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ Determines if this is a compact simplicial manifold with boundary. | |
| Depends on heuristic [[/topaz/objects/SimplicialComplex/properties/Topology/SPHERE]] recognition. | |
| May be true or false or undef (if heuristic does not succeed). | |
| |
| == GENUS == | |
| ''Type:'' /common/property_types/Basic Types/Int\\ The __genus__ of a surface. | |
| |
| == FUNDAMENTAL_GROUP_GEN_LABELS == | |
| ''Type:'' /common/property_types/Basic Types/Array\\ Labels of the generators of the [[/topaz/objects/SimplicialComplex/properties/Topology/FUNDAMENTAL_GROUP]]. | |
| The labels can be chosen freely. If the [[/topaz/objects/SimplicialComplex/properties/Topology/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}. | |
| |
| == ORIENTED_PSEUDO_MANIFOLD == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/properties/Topology/PSEUDO_MANIFOLD]] with top level homology isomorphic to Z. | |
| |
| == MORSE_MATCHING == | |
| ''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 | |
| |
| == CLOSED_PSEUDO_MANIFOLD == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/properties/Combinatorics/PURE]] simplicial complex with the property that each ridge is | |
| contained in exactly two facets. | |
| |
| == SURFACE == | |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/methods/Topology/CONNECTED]] [[/topaz/objects/SimplicialComplex/properties/Topology/MANIFOLD]] of dimension 2. | |
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