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application: topaz

The TOPology Application Zoo deals with abstract simplicial complexes. A complex is given as a list of facets. You can ask for its global properties (manifold recognition, homology groups, etc.), explore the local vertex environment (stars, links, etc.), and make a lot of constructions.

The visualization means are constrained, as they are mostly based on the GRAPH (1-skeleton) of a complex.


imports from: common, graph

Objects

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    Geometric Complexes

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    SIGNATURE: common::Int

    Signature of a geometric simplicial complex embedded in the integer lattice. Like DUAL_GRAPH_SIGNATURE, but only simplices with odd normalized volume are counted.

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    VOLUME: common::Rational

    Volume of a geometric simplicial complex. FIXME: introduce type RealizedSimplicialComplex<Scalar=Rational> : SimplicialComplex

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    Topology

    The following properties are topological invariants.

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    BALL: common::Bool

    Heuristically true if the topological space homeomorphic to a ball. Similar remarks hold as for SPHERE.

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    CLOSED_PSEUDO_MANIFOLD: common::Bool

    True if this is a PURE simplicial complex with the property that each ridge is contained in exactly two facets.

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    COCYCLES: Array<CycleGroup>

    Representatives of cocycle groups, listed in increasing codimension order. Encoding similar to CYCLES.

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    COHOMOLOGY: Array<HomologyGroup>

    Reduced cohomology groups, listed in increasing codimension order. Encoding similar to HOMOLOGY.

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    CYCLES: Array<CycleGroup>

    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 (symbolically) multiply both components.

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    EULER_CHARACTERISTIC: common::Int

    Reduced Euler characteristic. Alternating sum of the F_VECTOR minus 1.

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    FUNDAMENTAL_GROUP: common::Tuple<Int, List<List<Tuple<Int, Int>>>>

    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.

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    FUNDAMENTAL_GROUP_GEN_LABELS: common::Array<String>

    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}.

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    GENUS: common::Int

    The genus of a surface.

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    HOMOLOGY: Array<HomologyGroup>

    Reduced simplicial homology groups H0, ..., Hd (integer coefficients), listed in increasing dimension order.

    Each group G is encoded as a sequence ( { (t1 m1) ... (tn mn) } f) of non-negative integers, with t1 > t2 > ... > n > 1, plus an extra non-negative integer f. The group G is isomorphic to (Z/t1)m1 × ... × (Z/tn)mn × Zf, where Z0 is the trivial group.

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    INTERSECTION_FORM: IntersectionForm

    Parity and signature of the intersection form of a closed oriented 4-manifold.

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    KNOT: common::Array<Set<Int>>

    Edge-subset of a 3-sphere which is a knot or link, that is, a collection of pairwise disjoint cycles.

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    LOCALLY_STRONGLY_CONNECTED: common::Bool

    True if the vertex star of each vertex is DUAL_CONNECTED.

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    MANIFOLD: common::Bool

    True if this is a compact simplicial manifold with boundary.

    Depends on heuristic SPHERE recognition.

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    MORSE_MATCHING: graph::Graph<Undirected>

    A Morse matching is a reorientation of the arcs in the Hasse diagram of a simplicial complex such that at most one arc incident to each face is reoriented (matching condition) and the resulting orientation is acyclic (acyclicity condition). Morse matchings capture the main structure of discrete Morse functions, see

    Robin Forman: Morse Theory for Cell-Complexes,
    Advances in Math., 134 (1998), pp. 90-145.

    This property is computed by one of two heuristics. The default heuristic is a simple greedy algorithm (greedy). The alternative is to use a canceling algorithm due to Forman (cancel). Note that the computation of a Morse matching of largest size is NP-hard. See

    Michael Joswig, Marc E. Pfetsch: Computing Optimal Morse Matchings
    SIAM J. Discrete Math., 2006, to appear
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    MORSE_MATCHING_CRITICAL_FACES: common::Array<Set<Int>>

    The critical faces of the computed Morse matching, i.e., the faces not incident to any reoriented arc (not matched). FIXME: temporary

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    MORSE_MATCHING_CRITICAL_FACE_VECTOR: common::Array<Int>

    The vector of critical faces in each dimension. FIXME: temporary

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    MORSE_MATCHING_N_CRITICAL_FACES: common::Int

    Number of critical faces of the computed Morse matching. FIXME: temporary

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    MORSE_MATCHING_SIZE: common::Int

    Size of the computed Morse matching. FIXME: temporary

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    ORIENTATION: common::Array<Bool>

    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 is given by the increasing vertex ordering and is marked with false if the orientation is obtained from the increasing vertex ordering by a transposition.

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    ORIENTED_PSEUDO_MANIFOLD: common::Bool

    True if this is a PSEUDO_MANIFOLD with top level homology isomorphic to Z.

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    PSEUDO_MANIFOLD: common::Bool

    True if this is a PURE simplicial complex with the property that each ridge is contained in either one or two facets.

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    SPHERE: common::Bool

    If true then the space is homeomorphic to a sphere. However, recognizing spheres is undecidable in dimensions >= 5, and all known algorithms in dimension 3 are of outrageous complexity. We therefore use the flip-heuristics by Bjoerner and Lutz. This means that if this property is false then the space still might be a sphere.

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    STIEFEL_WHITNEY: common::Array<PowerSet<Int>>

    Mod 2 cycle representation of Stiefel-Whitney classes. Each cycle is represented as a set of simplices.

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    SURFACE: common::Bool

    True if this is a CONNECTED MANIFOLD of dimension 2.

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    Visualization

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    GEOMETRIC_REALIZATION: common::Matrix<Rational>

    Coordinates for the vertices of the simplicial complex, such that the complex is embedded without crossings in some Re. Vector (x1, .... xe) represents a point in Euclidean e-space.

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    G_DIM: common::Int

    Dimension e of the space in which the GEOMETRIC_REALIZATION of the complex is embedded.

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    MIXED_GRAPH: MixedGraph

    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. FIXME: temporary

    • User Methods of SimplicialComplex

    Permutations of SimplicialComplex

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    Visual::SimplicialComplex

    Visualization of the simplicial complex.

    User Methods of Visual::SimplicialComplex

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      FACES (PROPERTY_NAME)

      Add faces with optional different graphical attributes.

      Parameters
      StringPROPERTY_NAME
      or [ Faces ]
      Options
      option list:Visual::Polygon::decorations
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      MORSE_MATCHING ()

      Add the MORSE_MATCHING to the visualization of the SimplicialComplex.

      Options
      option list:Visual::Graph::decorations
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      SUBCOMPLEX (PROPERTY_NAME)

      Add a subcomplex with optional different graphical attributes.

      Parameters
      StringPROPERTY_NAME
      or [ Facets ]
      Options
      option list:Visual::Polygon::decorations
      option list:Visual::Graph::decorations
      option list:Visual::PointSet::decorations
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    Visual::SimplicialComplexLattice

    Visualization of the HASSE_DIAGRAM of a simplicial complex as a multi-layer graph.

    User Methods of Visual::SimplicialComplexLattice

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      FACES (faces)

      Add distinguished faces with different graphical attributes NodeColor and NodeStyle.

      Parameters
      Array<Set>faces
      (to be changed in the near future)
      Options
      option list:Visual::Lattice::decorations
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      SUBCOMPLEX (property)

      Add a subcomplex with different graphical attributes.

      Parameters
      Stringproperty
      name of the subcomplex property (to be changed in the near future)
      Options
      option list:Visual::Lattice::decorations

    • User Functions

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    Producing a new simplicial complex from others

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    alexander_dual (complex) → SimplicialComplex

    Computes the Alexander dual complex, that is, the complements of all non-faces. The vertex labels are preserved unless the nol flag is specified.

    Parameters
    SimplicialComplexcomplex
    Options
    Boolnol
    Returns
    SimplicialComplex
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    balanced_prism (complex) → SimplicialComplex

    Produce a prism over a given SimplicialComplex.

    Parameters
    SimplicialComplexcomplex
    Options
    Boolgeom_real
    Returns
    SimplicialComplex
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    barycentric_subdivision (complex)

    Create a simplicial complex as a barycentric subdivision of a given complex. Each vertex in the new complex corresponds to a face in the old complex.

    Parameters
    SimplicialComplexcomplex
    Options
    Boolrelabel
    generate vertex labels from the faces of the original complex.
    Boolgeom_real
    read GEOMETRIC_REALIZATION of the input complex, compute the coordinates of the new vertices and store them as GEOMETRIC_REALIZATION of the produced complex.
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    bistellar_simplification (complex) → SimplicialComplex

    Heuristic for simplifying the triangulation of the given manifold without changing its PL-type. The function uses bistellar flips and a simulated annealing strategy.

    You may specify the maximal number of rounds, how often the system may relax before heating up and how much heat should be applied. The function stops computing, once the size of the triangulation has not decreased for rounds iterations. If the abs flag is set, the function stops after rounds iterations regardless of when the last improvement took place. Additionally, you may set the threshold min_n_facets for the number of facets when the simplification ought to stop. Default is d+2 in the CLOSED_PSEUDO_MANIFOLD case and 1 otherwise.

    If you want to influence the distribution of the dimension of the moves when warming up you may do so by specifying a distribution. The number of values in distribution determines the dimensions used for heating up. The heating and relaxing parameters decrease dynamically unless the constant flag is set. The function prohibits to execute the reversed move of a move directly after the move itself unless the allow_rev_move flag is set. Setting the allow_rev_move flag might help solve a particular resilient problem.

    If you are interested in how the process is coming along, try the verbose option. It specifies after how many rounds the current best result is displayed.

    The obj determines the objective function used for the optimization. If obj is set to 0, the function searches for the triangulation with the lexicographically smallest f-vector, if obj is set to 1, the function searches for the triangulation with the reversed-lexicographically smallest f-vector and if obj is set to 2 the sum of the f-vector entries is used. The default is 1.

    Parameters
    SimplicialComplexcomplex
    Options
    Introunds
    Boolabs
    Intobj
    Intrelax
    Intheat
    Boolconstant
    Boolallow_rev_move
    Intmin_n_facets
    Intverbose
    Intseed
    Boolquiet
    Array<Int>distribution
    Returns
    SimplicialComplex
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    cone (complex, k) → SimplicialComplex

    Produce the k-cone over a given simplicial complex.

    Parameters
    SimplicialComplexcomplex
    intk
    default is 1
    Options
    Array<String>apex_labels
    labels of the apex vertices. Default labels have the form apex_0, apex_1, .... In the case the input complex has already vertex labels of this kind, the duplicates are avoided.
    Boolnol
    don't generate any vertex labels.
    Returns
    SimplicialComplex
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    connected_sum (complex1, complex2, f_1, f_2)

    Compute the connected sum of two complexes.

    Parameters f_1 and f_2// specify which facet of the first and second complex correspondingly are glued together. Default is the 0-th facet of both.

    The vertices in the selected facets are identified with each other according to their order in the facet (that is, in icreasing index order). The option permutation allows to get an alternative identification. It should specify a permutation of the vertices of the second facet.

    The vertices of the new complex get the original labels with _1 or _2 appended, according to the input complex they came from. If you set the no_labels flag, the label generation will be omitted.

    Parameters
    SimplicialComplexcomplex1
    SimplicialComplexcomplex2
    intf_1
    default is 0
    intf_2
    default is 0
    Options
    Array<int>permutation
    Boolno_lables
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    deletion (complex, face) → SimplicialComplex

    Remove the given face and all the faces containing it.

    Parameters
    SimplicialComplexcomplex
    Set<Int>face
    specified by vertex indices. Please use labeled_vertices if you want to specify the face by vertex labels.
    Options
    Boolno_labels
    do not write vertex labels.
    Returns
    SimplicialComplex
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    disjoint_union (complex1, complex2)

    Produce the disjoint union of the two given complexes.

    Parameters
    SimplicialComplexcomplex1
    SimplicialComplexcomplex2
    Options
    labelscreates
    VERTEX_LABELS. The vertex labels are built from the original labels with a suffix _1 or _2 appended.
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    edge_contraction (complex) → SimplicialComplex

    Heuristic for simplifying the triangulation of the given manifold without changing its PL-type. Choosing a random order of the vertices, the function tries to contract all incident edges.

    Parameters
    SimplicialComplexcomplex
    Options
    Intseed
    Returns
    SimplicialComplex
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    h_induced_quotient (C, vertices) → SimplicialComplex

    Let C be the given simplicial and A the subcomplex induced by the given vertices. Then this function produces a simplicial complex homotopy equivalent to C mod A by adding the cone over A with apex a to C. The label of the apex my be specified via the option apex.

    Parameters
    SimplicialComplexC
    Set<Int>vertices
    Options
    Boolno_labels
    tells the client not to write any labels.
    Stringapex
    Returns
    SimplicialComplex
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    induced_subcomplex (complex, vertices)

    Produce the subcomplex consisting of all faces which are contained in the given set of vertices.

    Parameters
    SimplicialComplexcomplex
    Set<Int>vertices
    Options
    Boolno_label
    tells the client not to create any labels.
    Boolgeom_real
    tells the client to inherit the GEOMETRIC_REALIZATION.
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    join_complexes (complex1, complex2) → SimplicialComplex

    Creates the join of complex1 and complex2.

    Parameters
    SimplicialComplexcomplex1
    SimplicialComplexcomplex2
    Options
    Boollabels
    creates VERTEX_LABELS. The vertex labels are built from the original labels with a suffix _1 or _2 appended.
    Returns
    SimplicialComplex
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    k_skeleton (complex, k) → SimplicialComplex

    Produce the k-skeleton.

    Parameters
    SimplicialComplexcomplex
    intk
    Options
    Boolnol
    suppresses creation of VERTEX_LABELS
    Returns
    SimplicialComplex
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    link_complex (complex, face)

    Produce the link of a face of the complex

    Parameters
    SimplicialComplexcomplex
    Set<int>face
    Options
    Boolno_labels
    tells the client not to create any labels.
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    simplicial_product (complex1, complex2)

    Computes the simplicial product of two complexes. Vertex orderings may be given as options.

    Parameters
    SimplicialComplexcomplex1
    SimplicialComplexcomplex2
    Options
    Array<Int>vertex_order1
    Array<Int>vertex_order2
    Boolgeom_real
    Boolcolor_cons
    Boolno_labels
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    star (complex, face) → SimplicialComplex

    Produce the star of the face of the complex.

    Parameters
    SimplicialComplexcomplex
    Set<int>face
    Options
    Boollabels
    creates VERTEX_LABELS.
    Returns
    SimplicialComplex
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    star_deletion (complex, face) → SimplicialComplex

    Remove the star of a given face.

    Parameters
    SimplicialComplexcomplex
    Set<Int>face
    specified by vertex indices. Please use labeled_vertices if you want to specify the face by vertex labels.
    Options
    Boolno_labels
    do not write vertex labels.
    Returns
    SimplicialComplex
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    stellar_subdivision (complex, faces) → SimplicialComplex

    Computes the complex obtained by stellar subdivision of the given faces of the complex.

    Parameters
    SimplicialComplexcomplex
    Array<Set<int>>faces
    Options
    Boolno_labels
    Boolgeom_real
    Returns
    SimplicialComplex
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    stellar_subdivision (complex, faces) → SimplicialComplex

    Computes the complex obtained by stellar subdivision of the given faces of the complex.

    Parameters
    SimplicialComplexcomplex
    Array<Set<int>>faces
    Options
    Boolno_labels
    Boolgeom_real
    Returns
    SimplicialComplex
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    suspension (complex, k)

    Produce the k-suspension over a given simplicial complex.

    Parameters
    SimplicialComplexcomplex
    Intk
    default value is 1
    Options
    Array<String>labels
    for the apices. By default apices are labeled with apex_0+, apex_0-, apex_1+, etc. If one of the specified labels already exists, a unique one is made by appending _i where i is some small number.
    Boolnol
    do not produce any labels.
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    union (complex1, complex2) → SimplicialComplex

    Produce the union of the two given complexes, identifying vertices with equal labels.

    Parameters
    SimplicialComplexcomplex1
    SimplicialComplexcomplex2
    Options
    Boollabels
    creates VERTEX_LABELS.
    Returns
    SimplicialComplex
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    Producing a simplicial complex from other objects

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    clique_complex (graph) → SimplicialComplex

    Produce the clique complex of a given graph. If no_labels is set to 1, the labels are not copied.

    Parameters
    Graphgraph
    Options
    Boolno_labels
    Returns
    SimplicialComplex
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    crosscut_complex (p) → SimplicialComplex

    Produce the crosscut complex of the boundary of a polytope.

    Parameters
    polytope::Polytopep
    Options
    Boolnoc
    suppresses copying the vertex coordinates to GEOMETRIC_REALIZATION
    Returns
    SimplicialComplex
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    Producing from scratch

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    ball (d) → SimplicialComplex

    A d-dimensional ball, realized as the d-simplex.

    Parameters
    intd
    dimension
    Returns
    SimplicialComplex
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    cube_complex ()

    Produces a triangulated pile of hypercubes: Each cube is split into d! tetrahedra, and the tetrahedra are all grouped around one of the diagonal axes of the cube. DOC_FIXME args: x_1, ... , x_d

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    klein_bottle () → SimplicialComplex

    The Klein bottle.

    Returns
    SimplicialComplex
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    projective_plane () → SimplicialComplex

    The projective plane.

    Returns
    SimplicialComplex
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    rand_knot (n_edges) → SimplicialComplex

    Produce a random knot (or link) as a polygonal closed curve in 3-space. The knot (or each connected component of the link) has n_edges edges.

    The vertices are uniformly distributed in [-1,1]3, unless the on_sphere option is set. In the latter case the vertices are uniformly distributed on the 3-sphere. Alternatively the brownian option produces a knot by connecting the ends of a simulated brownian motion.

    Parameters
    Intn_edges
    Options
    Intn_comp
    number of components, default is 1.
    Boolon_sphere
    Boolbrownian
    Intseed
    Returns
    SimplicialComplex
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    simplex (d) → SimplicialComplex

    A simplex of dimension d.

    Parameters
    intd
    dimension
    Returns
    SimplicialComplex
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    sphere (d) → SimplicialComplex

    The d-dimansional sphere, realized as the boundary of the (d+1)-simplex.

    Parameters
    intd
    dimension
    Returns
    SimplicialComplex
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    surface (g) → SimplicialComplex

    Produce a surface of genus g. For g >= 0 the client produces an orientable surface, otherwise it produces a non-orientable one.

    Parameters
    intg
    genus
    Returns
    SimplicialComplex
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    torus () → SimplicialComplex

    The Császár Torus. Geometric realization by Frank Lutz, Electronic Geometry Model No. 2001.02.069

    Returns
    SimplicialComplex
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    unknot (m, n)

    Produces a triangulated 3-sphere with the particular NASTY embedding of the unknot in its 1-skeleton. The parameters m >= 2 and n >= 1 determine how entangled the unknot is. eps determines the GEOMETRIC_REALIZATION.

    Parameters
    intm
    intn
    Options
    Rationaleps

    • Property Types