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

GeometricSimplicialComplex
A geometric simplicial complex, i.e., a simplicial complex with a geometric realization. Scalar is the numeric data type used for the coordinates.
derived from: SimplicialComplex
Type Parameters
Scalar
default: Rational
Properties of GeometricSimplicialComplex
- COORDINATES: common::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: common::Int
  Dimension e of the space in which the COORDINATES of the complex is embedded.
- 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.
- VOLUME: GeometricSimplicialComplex::Scalar
  Volume of a geometric simplicial complex.
SimplicialComplex
An abstract simplicial complex represented by its facets.
Properties of SimplicialComplex

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.

BOUNDARY: SimplicialComplex
Codimension-1-faces of a PSEUDO_MANIFOLD which are contained in one facet only.

COLORING: common::Array<Int>
A coloring of the vertices.

DIM: common::Int
Maximal dimension of the FACETS, where the dimension of a facet is defined as the number of its vertices minus one.

DUAL_GRAPH: graph::Graph<Undirected>
The graph of facet neighborhood. Two FACETS are neighbors if they share a (d-1)-dimensional face.
Properties of DUAL_GRAPH

COLORING: common::NodeMap<Undirected, Int>
UNDOCUMENTED

F2_VECTOR: common::Matrix<Int, NonSymmetric>
f_ik is the number of incident pairs of i-faces and k-faces; the main diagonal contains the F_VECTOR.

FACETS: common::Array<Set<Int>>
Faces which are maximal with respect to inclusion, encoded as their ordered set of vertices. The vertices must be numbered 0, ..., n-1.

FOLDABLE: common::Bool
True if GRAPH is (DIM + 1)-colorable.

F_VECTOR: common::Array<Int>
f_k is the number of k-faces, for k = 0,... , d, where d is the dimension.

GRAPH: graph::Graph<Undirected>
The subcomplex consisting of all 1-faces.

HASSE_DIAGRAM: graph::FaceLattice
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.
Properties of HASSE_DIAGRAM

MORSE_MATCHING: common::EdgeMap<Directed, Int>
UNDOCUMENTED

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

MORSE_MATCHING_CRITICAL_FACE_VECTOR: common::Array<Int>
The vector of critical faces in each dimension. FIXME: temporary

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

MORSE_MATCHING_SIZE: common::Int
Size of the computed Morse matching. FIXME: temporary

H_VECTOR: common::Array<Int>
The h-vector of the simplicial complex.

MINIMAL_NON_FACES: common::Array<Set<Int>>
Inclusion minimal non-faces (vertex subsets which are not faces of the simplicial complex).

N_FACETS: common::Int
Number of FACETS.

N_MINIMAL_NON_FACES: common::Int
Number of MINIMAL_NON_FACES.

N_VERTICES: common::Int
Number of vertices.

ODD_SUBCOMPLEX: SimplicialComplex
Subcomplex generated by faces of codimension 2 that are contained in an odd number of faces of codimension 1.

PROJ_DICTIONARY: common::Array<Int>
For each vertex the corresponding vertex of facet 0 with respect to the action of the group of projectivities.

PROJ_ORBITS: common::PowerSet<Int>
Orbit decomposition of the group of projectivities acting on the set of vertices of facet 0.

PURE: common::Bool
A simplicial complex is pure if all its facets have the same dimension.

SHELLABLE: common::Bool
True if this is shellable.

SHELLING: common::Array<Set<Int>>
An ordered list of facets constituting a shelling.

VERTEX_INDICES: common::Array<Int>
Indices of the vertices from INPUT_FACES. That is, the map i \mapsto v_i.

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

INPUT_FACES: common::Array<Set<Int>>
Any description of the faces of a simplicial complex with vertices v_0 < v_1 < v_2 < ... arbitrary. Redundant faces allowed.

Topology
The following properties are topological invariants.

BALL: common::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).

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.

COCYCLES: common::Array<CycleGroup<Integer>>
Representatives of cocycle groups, listed in increasing codimension order. See CycleGroup for explanation of encoding of each group.

COHOMOLOGY: common::Array<HomologyGroup<Integer>>
Reduced cohomology groups, listed in increasing codimension order. See HomologyGroup for explanation of encoding of each group.

CYCLES: common::Array<CycleGroup<Integer>>
Representatives of cycle groups, listed in increasing dimension order. See CycleGroup for explanation of encoding of each group.

EULER_CHARACTERISTIC: common::Int
Reduced Euler characteristic. Alternating sum of the F_VECTOR minus 1.

FUNDAMENTAL_GROUP: common::Pair<Int, List<List<Pair<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.

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}.

GENUS: common::Int
The genus of a surface.

HOMOLOGY: common::Array<HomologyGroup<Integer>>
Reduced simplicial homology groups H₀, ..., H_d (integer coefficients), listed in increasing dimension order. See HomologyGroup for explanation of encoding of each group.

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

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.

LOCALLY_STRONGLY_CONNECTED: common::Bool
True if the vertex star of each vertex is DUAL_CONNECTED.

MANIFOLD: common::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).

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.

ORIENTED_PSEUDO_MANIFOLD: common::Bool
True if this is a PSEUDO_MANIFOLD with top level homology isomorphic to Z.

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.

SPHERE: common::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).

STIEFEL_WHITNEY: common::Array<PowerSet<Int>>
Mod 2 cycle representation of Stiefel-Whitney classes. Each cycle is represented as a set of simplices.

SURFACE: common::Bool
True if this is a CONNECTED MANIFOLD of dimension 2.

Visualization
These properties are for visualization.

MIXED_GRAPH: graph::EdgeWeightedGraph
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

VERTEX_LABELS: common::Array<String>
Labels of the vertices.
User Methods of SimplicialComplex

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

BIPARTITE () → Bool

True if GRAPH is a bipartite.
Returns
Bool

CONNECTED_COMPONENTS () → PowerSet<Int>

The connected components of the GRAPH, encoded as node sets.
Returns
PowerSet<Int>

CONNECTIVITY () → Int

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

DUAL_BIPARTITE () → Bool

True if DUAL_GRAPH is a bipartite.
Returns
Bool

DUAL_CONNECTED_COMPONENTS () → PowerSet<Int>

The connected components of the DUAL_GRAPH, encoded as node sets.
Returns
PowerSet<Int>

DUAL_CONNECTIVITY () → PowerSet<Int>

Node connectivity of the DUAL_GRAPH. Dual to CONNECTIVITY.
Returns
PowerSet<Int>

DUAL_GRAPH_SIGNATURE () → Int

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

DUAL_MAX_CLIQUES () → PowerSet<Int>

The maximal cliques of the DUAL_GRAPH, encoded as node sets.
Returns
PowerSet<Int>

GRAPH_SIGNATURE () → Int

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

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

Find the vertices by given labels.
Parameters
String label ...
vertex labels
Returns
Set<Int>
vertex indices

MAX_CLIQUES () → PowerSet<Int>

The maximal cliques of the GRAPH, encoded as node sets.
Returns
PowerSet<Int>

VERTEX_DEGREES () → Array<Int>

Degrees of the vertices in the GRAPH.
Returns
Array<Int>

Topology
The following methods compute topological invariants.

CONNECTED () → Bool

True if the GRAPH is a connected graph.
Returns
Bool

DUAL_CONNECTED () → Bool

True if the DUAL_GRAPH is a connected graph.
Returns
Bool

fundamental2gap (filename)

Writes the FUNDAMENTAL_GROUP using FUNDAMENTAL_GROUP_GEN_LABELS to the given file in GAP input format.
Parameters
String filename

N_CONNECTED_COMPONENTS () → Int

Number of connected components of the GRAPH.
Returns
Int

Visualization
These methods are for visualization.

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

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

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

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

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
Permutations of SimplicialComplex

FacetPerm
UNDOCUMENTED
Properties of FacetPerm

PERMUTATION: common::Array<Int>
Transforming FACETS from this into basic object

VertexPerm
UNDOCUMENTED
Properties of VertexPerm

PERMUTATION: common::Array<Int>
Transforming vertices from this into basic object.
Visual::SimplicialComplex
Visualization of the simplicial complex.
User Methods of Visual::SimplicialComplex

FACES (PROPERTY_NAME)

Add faces with optional different graphical attributes.
Parameters
String PROPERTY_NAME
or [ Faces ]
Options
option list: Visual::Polygon::decorations

MORSE_MATCHING ()

Add the MORSE_MATCHING to the visualization of the SimplicialComplex.
Options
option list: Visual::Graph::decorations

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
Visual::SimplicialComplexLattice
Visualization of the HASSE_DIAGRAM of a simplicial complex as a multi-layer graph.
User Methods of Visual::SimplicialComplexLattice

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

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

Comparing
These functions compare two SimplicialComplex
- 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<Int>, Array<int>>
- 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
- 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<Int> distribution
  Returns
  Bool
Other
Special purpose functions.
- is_generalized_shelling (FaceList) → Bool
  
  Check if a given sequence of faces of a simplicial complex is a generalized shelling.
  Parameters
  Array<Set> FaceList
  Options
  Bool verbose
  Returns
  Bool
- is_vertex_decomposition (complex, vertices) → Bool
  
  Check whether a given ordered subset of the vertex set is a vertex decomposition. Works for 1-, 2- and 3-manifolds only!
  Parameters
  SimplicialComplex complex
  Array<Int> vertices
  shedding vertices
  Options
  Bool verbose
  Returns
  Bool
- mixed_graph (complex)
  
  Produces the mixed graph of a complex.
  Parameters
  SimplicialComplex complex
  Options
  Float edge_weight
- morse_matching (complex) → EdgeMap
  
  Compute a Morse matching. Two heuristics are implemented:
  (1) 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.
  (2) 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
  SimplicialComplex complex
  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
- morse_matching_size (complex) → Int
  
  Compute the number of edges in a Morse matching.
  Parameters
  SimplicialComplex complex
  a complex with a Morse matching
  Returns
  Int
  the number of edges in the matching.
- stanley_reisner (complex) → ideal::Ideal
  
  Creates the Stanley-Reisner ideal of a simplicial complex. Optional ring parameter is required to have precisely as many variables as C has vertices.
  Parameters
  SimplicialComplex complex
  Options
  Ring<Rational,int> ring
  Returns
  ideal::Ideal
- stiefel_whitney (facets) → Array<PowerSet<Int>>
  
  Computes Stiefel-Whitney classes of mod 2 Euler space (in particular, closed manifold). Use option verbose to show regular pairs and cycles. A narrower dimension range of interest can be specified. Negative values are treated as co-dimension - 1
  Parameters
  Array<Set<Int>> facets
  the facets of the simplicial complex
  Options
  Int high_dim
  Int low_dim
  Bool verbose
  Returns
  Array<PowerSet<Int>>

Producing a new simplicial complex from others

These functions construct a new SimplicialComplex from other objects of the same type.

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

barycentric_subdivision (complex) → SimplicialComplex

Computes the barycentric subdivision of complex.

Parameters

SimplicialComplex

complex

Options

String	pin_hasse_section	default: HASSE_DIAGRAM
String	label_section	default: VERTEX_LABELS
String	coord_section	default: VERTICES
Bool	geometric_realization	set to 1 to obtain a GeometricSimplicialComplex, default: 0

Returns

SimplicialComplex

(or GeometricSimplicialComplex)

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<Int> distribution
Returns
SimplicialComplex
bs2quotient (P, complex) → SimplicialComplex

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.
Contained in extension bundled:group.
Parameters
Polytope P
the underlying polytope
SimplicialComplex complex
a sufficiently fine subdivision of P, for example the second barycentric subdivision
Returns
SimplicialComplex

cone (complex, k) → SimplicialComplex

Produce the k-cone over a given simplicial complex.

Parameters

SimplicialComplex	complex
int	k	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.
Bool	nol	don't generate any vertex labels.

Returns

SimplicialComplex

connected_sum (complex1, complex2, f1, f2) → SimplicialComplex

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 f1
default: 0
int f2
default: 0
Options
Array<int> permutation
Bool no_lables
Returns
SimplicialComplex
deletion (complex, face) → SimplicialComplex

Remove the given face and all the faces containing it.
Parameters
SimplicialComplex complex
Set<Int> 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
disjoint_union (complex1, complex2) → SimplicialComplex

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.
Returns
SimplicialComplex
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
foldable_prism (complex) → GeometricSimplicialComplex

Produce a prism over a given SimplicialComplex.
Parameters
GeometricSimplicialComplex complex
Options
Bool geometric_realization
Returns
GeometricSimplicialComplex
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<Int> vertices
Options
Bool no_labels
tells the client not to write any labels.
String apex
Returns
SimplicialComplex
induced_subcomplex (complex, vertices) → SimplicialComplex

Produce the subcomplex consisting of all faces which are contained in the given set of vertices.
Parameters
SimplicialComplex complex
Set<Int> vertices
Options
Bool no_label
tells the client not to create any labels.
Bool geom_real
tells the client to inherit the COORDINATES.
Returns
SimplicialComplex

iterated_barycentric_subdivision (complex, k) → SimplicialComplex

Computes the k-th barycentric subdivision of complex by iteratively calling barycentric_subdivision.

Parameters

SimplicialComplex	complex
Int	k

Options

String	pin_hasse_section	default: HASSE_DIAGRAM
String	label_section	default: VERTEX_LABELS
String	coord_section	default: VERTICES
Bool	geometric_realization	set to 1 to obtain a GeometricSimplicialComplex, default: 0

Returns

SimplicialComplex

(or GeometricSimplicialComplex)

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
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
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) → SimplicialComplex

Produce the link of a face of the complex
Parameters
SimplicialComplex complex
Set<int> face
Options
Bool no_labels
tells the client not to create any labels.
Returns
SimplicialComplex

p_sum_triangulation (P, Q, WebOfStars) → GeometricSimplicialComplex

Produce a specific P-sum-triangulation of two given triangulations and a WebOfStars.

Parameters

GeometricSimplicialComplex	P	First complex which will be favoured
GeometricSimplicialComplex	Q	second complex
IncidenceMatrix	WebOfStars	Every row corresponds to a full dimensional simplex in P and every column to a full dimensional simplex in Q.

Returns

GeometricSimplicialComplex

simplicial_product (complex1, complex2) → SimplicialComplex

Computes the simplicial product of two complexes. Vertex orderings may be given as options.
Parameters
SimplicialComplex complex1
SimplicialComplex complex2
Options
Array<Int> vertex_order1
Array<Int> vertex_order2
Bool geometric_realization
default 0
Bool color_cons
Bool no_labels
Returns
SimplicialComplex
simplicial_product <Scalar> (complex1, complex2) → GeometricSimplicialComplex<Scalar>

Computes the simplicial product of two complexes. Vertex orderings may be given as options.
Type Parameters
Scalar
Parameters
GeometricSimplicialComplex complex1
GeometricSimplicialComplex complex2
Options
Array<Int> vertex_order1
Array<Int> vertex_order2
Bool geometric_realization
default 1
Bool color_cons
Bool no_labels
Returns
GeometricSimplicialComplex<Scalar>
star (complex, face) → SimplicialComplex

Produce the star of the face of the complex.
Parameters
SimplicialComplex complex
Set<int> face
Options
Bool labels
creates VERTEX_LABELS.
Returns
SimplicialComplex
star_deletion (complex, face) → SimplicialComplex

Remove the star of a given face.
Parameters
SimplicialComplex complex
Set<Int> 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
stellar_subdivision (complex, faces) → SimplicialComplex

Computes the complex obtained by stellar subdivision of the given faces of the complex.
Parameters
SimplicialComplex complex
Array<Set<Int>> faces
Options
Bool no_labels
Bool geometric_realization
default 0
Returns
SimplicialComplex
stellar_subdivision (complex, face) → SimplicialComplex

Computes the complex obtained by stellar subdivision of the given face of the complex.
Parameters
SimplicialComplex complex
Set<Int> face
Options
Bool no_labels
Bool geometric_realization
default 0
Returns
SimplicialComplex

suspension (complex, k) → SimplicialComplex

Produce the k-suspension over a given simplicial complex.

Parameters

SimplicialComplex	complex
Int	k	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.
Bool	nol	do not produce any labels.

Returns

SimplicialComplex

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

Producing a simplicial complex from other objects
These functions construct a new SimplicialComplex from other objects.
- 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
- independence_complex (matroid) → SimplicialComplex
  
  Produce the independence complex of a given matroid. If no_labels is set to 1, the labels are not copied.
  Parameters
  matroid::Matroid matroid
  Options
  Bool no_labels
  Returns
  SimplicialComplex
Producing from scratch
With these clients you can create special examples of simplicial complexes and complexes belonging to parameterized families.
- ball (d) → GeometricSimplicialComplex
  
  A d-dimensional ball, realized as the d-simplex.
  Parameters
  Int d
  dimension
  Returns
  GeometricSimplicialComplex
- complex_projective_plane () → SimplicialComplex
  
  The complex projective plane with the vertex-minimal triangulation by Kühnel and Brehm.
  Returns
  SimplicialComplex
- 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
- klein_bottle () → SimplicialComplex
  
  The Klein bottle.
  Returns
  SimplicialComplex
- 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
- real_projective_plane () → SimplicialComplex
  
  The real projective plane with its unique minimal triangulation on six vertices.
  Returns
  SimplicialComplex
- simplex (d) → SimplicialComplex
  
  A simplex of dimension d.
  Parameters
  Int d
  dimension
  Returns
  SimplicialComplex
- sphere (d) → GeometricSimplicialComplex
  
  The d-dimensional sphere, realized as the boundary of the (d+1)-simplex.
  Parameters
  Int d
  dimension
  Returns
  GeometricSimplicialComplex
- 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
- torus () → SimplicialComplex
  
  The Császár Torus. Geometric realization by Frank Lutz, Electronic Geometry Model No. 2001.02.069
  Returns
  SimplicialComplex
- unknot (m, n) → GeometricSimplicialComplex
  
  Produces a triangulated 3-sphere with the particularly 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
  Returns
  GeometricSimplicialComplex
Topology
The following functions compute topological invariants.
- cap_product (complex)
  
  Compute and print all cap products of cohomology and homology cycles.
  Parameters
  SimplicialComplex complex
- homology (complex, co)
  
  Calculate the (co-)homology groups of a simplicial complex.
  Parameters
  Array<Set<int>> complex
  Bool co
  set to true for cohomology
  Options
  Int dim_low
  narrows the dimension range of interest, with negative values being treated as co-dimensions
  Int dim_high
  see dim_low
- homology_and_cycles (complex, co)
  
  Calculate the (co-)homology groups and cycle representatives of a simplicial complex.
  Parameters
  Array<Set<int>> complex
  Bool co
  set to true for cohomology
  Options
  Int dim_low
  narrows the dimension range of interest, with negative values being treated as co-dimensions
  Int dim_high
  see dim_low

Property Types

Topology
The following property_types are topological invariants.
- 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
- HomologyGroup
  A group 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 is isomorphic to (Z/t₁)^m₁ × ... × (Z/t_n)^m_n × Z^f, where Z⁰ is the trivial group.
  Access methods: torsion delivers the list of Z-groups, betti_number the number f.
- IntersectionForm
  TODO: explain the meaning of the fields: parity, positive, negative

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

Int	seed	random seed value for the node placement
option list:	Visual::Lattice::decorations

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<Int>	distribution

Int	heuristic	(1=greedy, 2=cancel, 0=both (default))
Int	levels	(1=bottom, 2=top, 0=both (default))

Polytope	P	the underlying polytope
SimplicialComplex	complex	a sufficiently fine subdivision of P, for example the second barycentric subdivision

SimplicialComplex	complex1
SimplicialComplex	complex2
int	f1	default: 0
int	f2	default: 0

SimplicialComplex	complex
Set<Int>	face	specified by vertex indices. Please use labeled_vertices if you want to specify the face by vertex labels.

Bool	no_labels	tells the client not to write any labels.
String	apex

Bool	no_label	tells the client not to create any labels.
Bool	geom_real	tells the client to inherit the COORDINATES.

Array<Int>	vertex_order1
Array<Int>	vertex_order2
Bool	geometric_realization	default 0
Bool	color_cons
Bool	no_labels

Int	n_comp	number of components, default is 1.
Bool	on_sphere
Bool	brownian
Int	seed

Int	dim_low	narrows the dimension range of interest, with negative values being treated as co-dimensions
Int	dim_high	see dim_low

Navigation

application: topaz

Objects

Type Parameters

Properties of GeometricSimplicialComplex

Properties of SimplicialComplex

Properties of DUAL_GRAPH

Properties of HASSE_DIAGRAM

User Methods of SimplicialComplex

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Permutations of SimplicialComplex

Properties of FacetPerm

Properties of VertexPerm

User Methods of Visual::SimplicialComplex

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User Methods of Visual::SimplicialComplexLattice

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