 # 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
uses: ideal

## Objects

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### GeometricSimplicialComplex

UNDOCUMENTED

derived from: SimplicialComplex

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### Geometric Complexes

UNDOCUMENTED
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### Visualization

UNDOCUMENTED
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### SimplicialComplex

An abstract simplicial complex represented by its facets.

#### User Methods of SimplicialComplex

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BIPARTITE ()

True if GRAPH is a bipartite.

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CONNECTED ()

True if the GRAPH is a connected graph.

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CONNECTED_COMPONENTS ()

The connected components of the GRAPH, encoded as node sets.

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CONNECTIVITY ()

Node connectivity of the GRAPH, that is, the minimal number of nodes to be removed from the graph such that the result is disconnected.

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DUAL_BIPARTITE ()

True if DUAL_GRAPH is a bipartite.

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DUAL_CONNECTED ()

True if the DUAL_GRAPH is a connected graph.

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DUAL_CONNECTED_COMPONENTS ()

The connected components of the DUAL_GRAPH, encoded as node sets.

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DUAL_CONNECTIVITY ()

Node connectivity of the DUAL_GRAPH. Dual to CONNECTIVITY.

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DUAL_GRAPH_SIGNATURE ()

Difference of the black and white nodes if the DUAL_GRAPH is BIPARTITE. Otherwise -1.

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DUAL_MAX_CLIQUES ()

The maximal cliques of the DUAL_GRAPH, encoded as node sets.

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GRAPH_SIGNATURE ()

Difference of the black and white nodes if the GRAPH is BIPARTITE. Otherwise -1.

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labeled_vertices (label ...) → Set<Int>

Find the vertices by given labels.

##### Parameters
 String label ... vertex labels
##### Returns
 Set vertex indices
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MAX_CLIQUES ()

The maximal cliques of the GRAPH, encoded as node sets.

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N_CONNECTED_COMPONENTS ()

Number of connected components of the GRAPH.

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VERTEX_DEGREES ()

Degrees of vertices in the GRAPH.

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### Topology

UNDOCUMENTED
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### Visualization

UNDOCUMENTED
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VISUAL () → Visual::SimplicialComplex

Visualizes the complex.

If G_DIM < 4, the GRAPH and the facets are visualized using the COORDINATES.

Otherwise, the spring embedder and the GRAPH are used to produce coordinates for the visualization.

If JavaView is used to visualize the complex, 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.

##### Options
 Bool mixed_graph use the MIXED_GRAPH for the spring embedder Int seed random seed value for the string embedder option list: Visual::Polygon::decorations option list: Visual::Graph::decorations
##### Returns
 Visual::SimplicialComplex
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VISUAL_DUAL_GRAPH () → Visual::SimplicialComplex

Uses the spring embedder to visualize the DUAL_GRAPH.

##### Options
 Int seed random seed value for the string embedder option list: Visual::Graph::decorations
##### Returns
 Visual::SimplicialComplex
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VISUAL_FACE_LATTICE () → Visual::SimplicialComplexLattice

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

##### Options
 Int seed random seed value for the node placement option list: Visual::Lattice::decorations
##### Returns
 Visual::SimplicialComplexLattice
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VISUAL_GRAPH () → Visual::SimplicialComplex

Uses the spring embedder to visualize the GRAPH.

##### Options
 Int seed random seed value for the string embedder option list: Visual::Graph::decorations
##### Returns
 Visual::SimplicialComplex
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VISUAL_MIXED_GRAPH ()

Uses the spring embedder to visualize the MIXED_GRAPH.

##### Options
 Int seed random seed value for the string embedder option list: Visual::Graph::decorations

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

## User Functions

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barycentric_subdivision ()

UNDOCUMENTED

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cap_product ()

Compute and print all cap products of cohomology and homology cycles. args: SimplicialComplex

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homology (complex, co)

Calculate the (co-)homology groups of a simplicial complex.

##### Parameters
 Array> complex Bool co
##### Options
 Int dim_low narrows the dimension range of interest, with negative values being treated as co-dimensions Int dim_high see dim_low
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homology_and_cycles (complex, co)

Calculate the (co-)homology groups and cycle representatives of a simplicial complex.

##### Parameters
 Array> complex Bool co
##### Options
 Int dim_low narrows the dimension range of interest, with negative values being treated as co-dimensions Int dim_high see dim_low
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is_vertex_decomposition ()

Check whether a given ordered subset of the vertex set is a vertex decomposition. Works for 1-, 2- and 3-manifolds only!

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mixed_graph ()

Produces the mixed graph of a simplicial @a complex. args: complex [ edge_weight => VALUE ]

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stiefel_whitney (facets)

A narrower dimension range of interest can be specified. Negative values are treated as co-dimension - 1

##### Parameters
 Array> facets the facets of the simplicial complex
##### Options
 high_dim int low_dim int verbose Bool
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triang_neighborhood ()

If each connected component of the link has an even number of facets, then the following holds: An edge of the resulting complex is contained in an odd number of facets iff it corresponds to one of the edges of the link. args: Complex width

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### Comparing

UNDOCUMENTED
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find_facet_vertex_permutations (complex1, complex2) → Pair<Array<Int>, Array<int>>

Find the permutations of facets and vertices which maps the first complex to the second one. The facet permutation is the first component of the return value. If the complexes are not isomorphic, an exception is thrown.

##### Parameters
 SimplicialComplex complex1 SimplicialComplex complex2
##### Returns
 Pair, Array>
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isomorphic (complex1, complex2) → Bool

Determine whether two given complexes are combinatorially isomorphic. The problem is reduced to graph isomorphism of the vertex-facet incidence graphs.

##### Parameters
 SimplicialComplex complex1 SimplicialComplex complex2
##### Returns
 Bool
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pl_homeomorphic (complex1, complex2) → Bool

Tries to determine whether two complexes are pl-homeomorphic by using 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
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 Int rounds Bool abs Int obj Int relax Int heat Bool constant Bool allow_rev_move Int min_n_facets Int verbose Int seed Bool quiet Array distribution
##### Returns
 Bool
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### Computing properties

UNDOCUMENTED
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morse_matching (SimplicialComplex) → EdgeMap

Compute a Morse matching. Two heuristics are implemented:

- A simple greedy algorithm: The arcs are visited in lexicographical order, i.e.:

we proceed by levels from top to bottom, visit the faces in each dimension in lexicographical order, and visited the faces covered by these faces in lexicographical order.

This heuristic is used by default and with heuristic => 1.

- A Morse matching can be improved by canceling critical cells along unique alternating paths, see function processAlternatingPaths() in file morse_matching_tools.h . This idea is due to Robin Forman:

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

This heuristic is used by default and with heuristic => 2.

The default setting is to use both, i.e., to run the greedy algorithm and then improve the result by the canceling algorithm.

Morse matchings for the bottom level can be found optimally by spanning tree techniques. This can be enabled by the option levels => 1. If the complex is a pseudo-manifold the same can be done for the top level (option levels => 2). By specifying option levels => 0, both levels can be computed by spanning trees. For 2-dim pseudo-manifolds this computes an optimal Morse matching.

##### Parameters
 p SimplicialComplex given by its Hasse diagram
##### Options
 Int heuristic (1=greedy, 2=cancel, 0=both (default)) Int levels (1=bottom, 2=top, 0=both (default))
##### Returns
 EdgeMap matching a labelling of the edges of the Hasse diagram with integer values, where 1 means that the edge is in the matching
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morse_matching_size (a) → Int

Compute the number of edges in a Morse matching.

##### Parameters
 SimplicialComplex a complex with a Morse matching
##### Returns
 Int the number of edges in the matching.
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### Other

UNDOCUMENTED
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is_generalized_shelling (FaceList) → Bool

Check if a given sequence of faces of a simplicial complex is a generalized shelling.

##### Parameters
 Array FaceList
##### Options
 Bool verbose
##### Returns
 Bool
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### Producing a new simplicial complex from others

UNDOCUMENTED
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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
 SimplicialComplex complex
##### Options
 Bool nol
##### Returns
 SimplicialComplex
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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 any other value the sum of the f-vector entries is used. The default is 1.

##### Parameters
 SimplicialComplex complex
##### Options
 Int rounds Bool abs Int obj Int relax Int heat Bool constant Bool allow_rev_move Int min_n_facets Int verbose Int seed Bool quiet Array distribution
##### Returns
 SimplicialComplex
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bs2quotient (P, S)

Create a simplicial complex from a simplicial subdivision of a given complex by identifying vertices on the boundary of the original complex according to a group that acts on vertices.

##### Parameters
 Polytope P the underlying polytope SimplicialComplex S a sufficiently fine subdivision of P, for example the second barycentric subdivision
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colored_ball_from_colored_sphere (complex) → SimplicialComplex

Extends the triangulation and coloring of a k-colored (d-1)-sphere to a max{k,d+1}-colored triangulation of a d-ball. The colors are integer numbers.

The old vertex labels are preserved unless the no_labels flag is specified. The new vertices get labeled `new_i` (i=0, 1, 2, ...). If a new label is not unique, `_j` is added for the smallest integer j which makes the label unique.

##### Parameters
 SimplicialComplex complex
##### Options
 Bool no_lables
##### Returns
 SimplicialComplex
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cone (complex, k) → SimplicialComplex

Produce the k-cone over a given simplicial complex.

##### Parameters
 SimplicialComplex complex int k default is 1
##### Options
 Array 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. Bool nol 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
 SimplicialComplex complex1 SimplicialComplex complex2 int f_1 default is 0 int f_2 default is 0
##### Options
 Array permutation Bool no_lables
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deletion (complex, face) → SimplicialComplex

Remove the given face and all the faces containing it.

##### Parameters
 SimplicialComplex complex Set face specified by vertex indices. Please use labeled_vertices if you want to specify the face by vertex labels.
##### Options
 Bool no_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
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 labels creates 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
 SimplicialComplex complex
##### Options
 Int seed
##### Returns
 SimplicialComplex
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foldable_prism (complex) → GeometricSimplicialComplex

Produce a prism over a given SimplicialComplex.

##### Parameters
 GeometricSimplicialComplex complex
##### Options
 Bool geometric_realization
##### Returns
 GeometricSimplicialComplex
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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
 SimplicialComplex C Set vertices
##### Options
 Bool no_labels tells the client not to write any labels. String apex
##### 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
 SimplicialComplex complex Set vertices
##### Options
 Bool no_label tells the client not to create any labels. Bool geom_real tells the client to inherit the COORDINATES.
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join_complexes (complex1, complex2) → SimplicialComplex

Creates the join of complex1 and complex2.

##### Parameters
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 Bool labels 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
 SimplicialComplex complex int k
##### Options
 Bool vertex_labels whether to create VERTEX_LABELS
##### Returns
 SimplicialComplex
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k_skeleton (complex, k) → GeometricSimplicialComplex

Produce the k-skeleton.

##### Parameters
 GeometricSimplicialComplex complex int k
##### Options
 Bool vertex_labels whether to create VERTEX_LABELS
##### Returns
 GeometricSimplicialComplex
• link_complex (complex, face)
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simplicial_product (complex1, complex2)

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

##### Parameters
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 Array vertex_order1 Array vertex_order2 Bool geometric_realization default 0 Bool color_cons Bool no_labels
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simplicial_product <Scalar> (complex1, complex2)

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

##### Type Parameters
 Scalar
##### Parameters
 GeometricSimplicialComplex complex1 GeometricSimplicialComplex complex2
##### Options
 Array vertex_order1 Array vertex_order2 Bool geometric_realization default 1 Bool color_cons Bool no_labels
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star (complex, face) → SimplicialComplex

Produce the star of the face of the complex.

##### Parameters
 SimplicialComplex complex Set face
##### Options
 Bool labels creates VERTEX_LABELS.
##### Returns
 SimplicialComplex
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star_deletion (complex, face) → SimplicialComplex

Remove the star of a given face.

##### Parameters
 SimplicialComplex complex Set face specified by vertex indices. Please use labeled_vertices if you want to specify the face by vertex labels.
##### Options
 Bool no_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
 SimplicialComplex complex Array> faces
##### Options
 Bool no_labels Bool geometric_realization default 0
##### Returns
 SimplicialComplex
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stellar_subdivision (complex, face) → SimplicialComplex

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

##### Parameters
 SimplicialComplex complex Set face
##### Options
 Bool no_labels Bool geometric_realization default 0
##### Returns
 SimplicialComplex
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suspension (complex, k)

Produce the k-suspension over a given simplicial complex.

##### Parameters
 SimplicialComplex complex Int k default value is 1
##### Options
 Array 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. Bool nol 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
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 Bool labels creates VERTEX_LABELS.
##### Returns
 SimplicialComplex
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### Producing a simplicial complex from other objects

UNDOCUMENTED
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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
 Graph graph
##### Options
 Bool no_labels
##### Returns
 SimplicialComplex
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### Producing from other

UNDOCUMENTED
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stanley_reisner (C) → ideal::Ideal

Creates the Stanley-Reisner ideal of a simplicial complex C. Optional ring parameter is required to have precisely as many variables as C has vertices.

##### Parameters
 SimplicialComplex C
##### Options
 Ring ring
##### Returns
 ideal::Ideal
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### Producing from scratch

UNDOCUMENTED
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ball (d) → GeometricSimplicialComplex

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

##### Parameters
 int d dimension
##### Returns
 GeometricSimplicialComplex
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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
 Int n_edges
##### Options
 Int n_comp number of components, default is 1. Bool on_sphere Bool brownian Int seed
##### Returns
 SimplicialComplex
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simplex (d) → SimplicialComplex

A simplex of dimension d.

##### Parameters
 int d dimension
##### Returns
 SimplicialComplex
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sphere (d) → GeometricSimplicialComplex

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

##### Parameters
 int d dimension
##### Returns
 GeometricSimplicialComplex
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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
 int g 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 COORDINATES.

##### Parameters
 int m int n
##### Options
 Rational eps

## Property Types

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### Topology

UNDOCUMENTED
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CycleGroup <Scalar>

A group is encoded as a pair of an integer matrix and a vector of faces. The elements of the group can be obtained by symbolic multiplication of both.

Access methods: coeff delivers the integer matrix, faces the vector of faces.

##### Type Parameters
 Scalar integer type of matrix elements
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HomologyGroup

A group is encoded as a sequence ( { (t1 m1) ... (tn mn) } f) of non-negative integers, with t1 > t2 > ... > tn > 1, plus an extra non-negative integer f.

The group is isomorphic to (Z/t1)m1 × ... × (Z/tn)mn × Zf, where Z0 is the trivial group.

Access methods: torsion delivers the list of Z-groups, betti_number the number f.

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IntersectionForm

TODO: explain the meaning of the fields: parity, positive, negative