# 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: group, 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

#### User Methods of GeometricSimplicialComplex

•
VISUAL ()

TODO: consider all inherited user methods; then: user_method ... : COORDINATES ...

##### Options
 option list: Visual::Polygon::decorations option list: Visual::Graph::decorations
•

### HyperbolicSurface

A hyperbolic surface (noncompact, finite area) is given by a triangulation DCEL_DATA (the topological data) and PENNER_COORDINATES (the metric data).

#### Properties of HyperbolicSurface

•
DCEL_DATA: common::Array<Array<Int>>

The triangulation of the surface is encoded by a half edge data structure. For each edge i of the triangulation there are two half edges 2i and 2i+1, one for each orientation. Each row reads as follows: [ 2i.head , (2i+1).head , 2i.next , (2i+1).next]

Example:
• A triangulated sphere with three punctures can be given by the following DCEL_DATA:`> \$S3 = new Array<Array<Int>>([ [1,0,2,5],[2,1,4,1],[0,2,0,3] ]);``> \$s = new HyperbolicSurface(DCEL_DATA=>\$S3);`
•
FLIP_WORDS: common::Array<List<Int>>

Each Delaunay triangulations of the surface can be obtained by successively flipping edges of the triangulation DCEL_DATA. The k-th flip word is a list of integers (the indices of the edges) that describe which edge flips produce the k-th Delaunay triangulation. Note that the k-th Delaunay triangulation also corresponds to the k-th maximal cone of the SECONDARY_FAN.

•
PENNER_COORDINATES: common::Vector<Rational>

Penners lambda lengths, sometimes called Penner coordinates, of the hyperbolic surface. Robert C. Penner. Decorated Teichmüller Theory. QGM Master Class Series. European Mathematical Society, Zürich, 2012. They are given by one positive rational for each edge of the triangulation, ordered in the sense of the triangulation DCEL_DATA.

•
SECONDARY_FAN: fan::PolyhedralFan<Rational>

The secondary fan of the hyperbolic surface. The k-th maximal cone corresponds to the Delaunay triangulation obtained by applying the k-th flip word of FLIP_WORDS. See M. Joswig, R. Löwe, and B. Springborn. Secondary fans and secondary polyhedra of punctured Riemann surfaces. arXiv:1708.08714.

•
SPECIAL_POINT: common::Pair<Rational, Rational>

In order to compute GKZ_VECTORS (or a secondary_polyhedron) one needs to specify a point on the surface, see M. Joswig, R. Löwe, and B. Springborn. Secondary fans and secondary polyhedra of punctured Riemann surfaces. arXiv:1708.08714. This point is specified by choosing a pair of rationals (p,x) that determine how the (decorated) 0th half edge is lifted to a geodesic in H^2. The covering is chosen in such a way that the horocycle at infinity is the vertical line at height p^2 and the lifted 0th half edge goes from infinity to the point x at the ideal boundary.

•

### Other

Special purpose methods.

•
gkz_dome (k, depth) → fan::PolyhedralComplex<Rational>

Computes the GKZ dome of the k-th Delaunay trianglation up to a given depth. Note that k is also the index of the corresponding flip word in FLIP_WORDS. Projection to the disc yields (a part of) the covering triangulation of the Klein disc.

##### Parameters
 Int k index of the flip word Int depth
##### Returns
 fan::PolyhedralComplex

Example:
• `> \$T = new Array<Array<Int>>([[0,0,6,5],[0,0,1,10],[0,0,8,2],[1,0,11,4],[1,0,7,3],[1,0,9,0]]);``> \$s = new HyperbolicSurface(DCEL_DATA=>\$T, PENNER_COORDINATES=>[1,1,1,1,1,1], SPECIAL_POINT=>[1,0]);``> \$d = \$s->gkz_dome(0,3);``> \$d->VISUAL;`
•
GKZ_VECTORS (depth) → Matrix<Rational>

Computes an approximation of the GKZ vectors of a hyperbolic surface. The approximation depends on the parameter depth that restricts the depth of the (covering) triangles that are summed over in the definition of the GKZ vectors.

##### Parameters
 Int depth
##### Returns
 Matrix

Example:
• `> \$T = new Array<Array<Int>>([[0,0,6,5],[0,0,1,10],[0,0,8,2],[1,0,11,4],[1,0,7,3],[1,0,9,0]]);``> \$s = new HyperbolicSurface(DCEL_DATA=>\$T, PENNER_COORDINATES=>[1,1,1,1,1,1], SPECIAL_POINT=>[1,0]);``> print \$s->GKZ_VECTORS(2);`` 1 240509/380250 517/1950`` 1 98473/694950 915006978873/1469257962050`
•

### MorseMatching

Category: Topology

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) or both (both) together. 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

•

### SimplicialComplex

An abstract simplicial complex represented by its facets.

•

### 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
•
boundary_matrix (d) → SparseMatrix<Integer>

Output the boundary matrix of dimension d. Indexing is according to the face indices in the HASSE_DIAGRAM of the complex. The matrix is a map via multiplying it to a vector from the left. Beware, this computes the whole face lattice of your complex, which is expensive.

##### Parameters
 Int d Dimension of the boundary matrix.
##### Returns
 SparseMatrix

Example:
• This prints the boundary matrix of the 3-simplex:`> print simplex(3)->boundary_matrix(1);`` -1 1 0 0`` -1 0 1 0`` 0 -1 1 0`` -1 0 0 1`` 0 -1 0 1`` 0 0 -1 1` The output can be interpreted like this: the zeroth column of the matrix corresponds to the facet with index 0, which contains the edges with indices 0,1 and 3.
•
CONNECTED_COMPONENTS () → PowerSet<Int>

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

##### Returns
 PowerSet
•
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
•
DUAL_CONNECTIVITY () → PowerSet<Int>

Node connectivity of the DUAL_GRAPH. Dual to CONNECTIVITY.

##### Returns
 PowerSet
•
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
•
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 vertex indices
•
MAX_CLIQUES () → PowerSet<Int>

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

##### Returns
 PowerSet
•
VERTEX_DEGREES () → Array<Int>

Degrees of the vertices in the GRAPH.

##### Returns
 Array
•

### Topology

The following methods compute topological invariants.

•

### 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 () → Visual::Container

Uses the spring embedder to visualize the MIXED_GRAPH.

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

•

### 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.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) → Visual::SimplicialComplexLattice

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
##### Returns
 Visual::SimplicialComplexLattice
•
MORSE_MATCHING () → Visual::SimplicialComplexLattice

Add the MORSE_MATCHING.MATCHING to the visualization of the face lattice of the simplicial complex. Decoration options @c EdgeColor and @c EdgeStyle apply to the matched edges only.

##### Options
 option list: Visual::Lattice::decorations
##### Returns
 Visual::SimplicialComplexLattice
•
SUBCOMPLEX (property) → Visual::SimplicialComplexLattice

Add a subcomplex with different graphical attributes.

##### Parameters
 String property name of the subcomplex property (to be changed in the near future)
##### Options
 Bool show_filter containment relationship between the subcomplex and the lattice faces option list: Visual::Lattice::decorations
##### Returns
 Visual::SimplicialComplexLattice

## User Functions

•

### Combinatorics

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

•
n_poset_homomorphisms (P, Q) → Int

Count all order preserving maps from one poset to another. They are in fact enumerated, but only the count is kept track of using constant memory.

##### Parameters
 Graph P Graph Q
##### Options
 Array prescribed_map A vector of length P.nodes() with those images in Q that should be fixed. Negative entries will be enumerated over.
##### Returns
 Int
•

### 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>
•
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 distribution
##### Returns
 Bool
•

### Other

Special purpose functions.

•
dualOutitudes (dcel_data) → Array<Polynomial<Rational,Int>>

Given a triangulation of a punctured surface this calculates all the outitude polynomials of the dual structure. The first e = {oriented edges} monomials correspond to A-coordinates of the oriented edges of the primal structure , labeled as in the input. The last t = {triangles} monomials correspond to A-coordinates of the triangles of the primal structure.

##### Parameters
 Array> dcel_data the data for the doubly connected edge list representing the triangulation.
##### Returns
 Array> an array containing the dual outitudes in order of the input.
•
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
•
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 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
•
outitudes (dcel_data) → Array<Polynomial<Rational,Int>>

Given a triangulation of a punctured surface this calculates all the outitude polynomials. The first e = {oriented edges} monomials correspond to A-coordinates of the oriented edges, labeled as in the input. The last t = {triangles} monomials correspond to A-coordinates of the triangles.

##### Parameters
 Array> dcel_data the data for the doubly connected edge list representing the triangulation.
##### Returns
 Array> an array containing the outitudes in order of the input.

Example:
• We may calculate the outitude polynomials of a once punctured torus by typing:`> \$T1 = new Array<Array<Int>>([[0,0,2,3],[0,0,4,5],[0,0,0,1]]);``> print outitudes(\$T1);`` - x_0*x_1*x_6 - x_0*x_1*x_7 + x_0*x_2*x_7 + x_0*x_4*x_6 + x_1*x_3*x_6 + x_1*x_5*x_7 x_0*x_2*x_6 + x_1*x_3*x_7 - x_2*x_3*x_6 - x_2*x_3*x_7 + x_2*x_4*x_7 + x_3*x_5*x_6 x_0*x_4*x_7 + x_1*x_5*x_6 + x_2*x_4*x_6 + x_3*x_5*x_7 - x_4*x_5*x_6 - x_4*x_5*x_7`
•
outitudes (surface) → Array<Polynomial<Rational,Int>>

Given a punctured surface by a string from the list below, this calculates all the outitude polynomials. Choose among: S3, S4 (ipunctured spheres) and T1, T2, T3 (punctured tori) and D1 (punctured double torus). A triangulation of the surface will be chosen for you. The first e = {oriented edges} monomials correspond to A-coordinates of the oriented edges. The last t = {triangles} monomials correspond to A-coordinates of the triangles.

##### Parameters
 String surface the surface name.
##### Returns
 Array> an array containing the outitudes.
•
persistent_homology <MatrixType> (F, i, p, k) → Pair<SparseMatrix<SCALAR>, List< Pair<SCALAR, SparseMatrix<SCALAR> > > >

Given a Filtration and three indices i,p and k, this computes the p-persistent k-th homology group of the i-th frame of the filtration for coefficients from any PID. Returns a basis for the free part and a list of torsion coefficients with bases.

##### Type Parameters
 MatrixType type of boundary matrices
##### Parameters
 Filtration F Int i the filtration frame Int p the number of frames to consider Int k the dimension in which to compute
##### Returns
 Pair, List< Pair > > >
•
persistent_homology <MatrixType> (F) → Array<List<Pair<Int, Int> > >

Given a Filtration, this computes its persistence barcodes in all dimension, using the algorithm described in the 2005 paper 'Computing Persistent Homology' by Afra Zomorodian and Gunnar Carlsson. It only works for field coefficients.

##### Type Parameters
 MatrixType type of the boundary matrices
##### Parameters
 Filtration F
##### Returns
 Array > >
•
poset_by_inclusion (P) → Graph<Directed>

Construct the inclusion poset from a given container. The elements of the container are interpreted as sets. They define a poset by inclusion. The function returns this poset encoded as a directed graph. The direction is towards to larger sets. All relations are encoded, not only the covering relations. For details see Assarf, Joswig & Pfeifle: Webs of stars or how to triangulate sums of polytopes, to appear

##### Parameters
 Array P
##### Returns
 Graph
•
random_discrete_morse (complex) → Map< Array<Int>, Int >

Implementation of random discrete Morse algorithms by Lutz and Benedetti Returns a map of the number of occurrences of different reduction results indexed by the corresponding discrete Morse vectors (containing the number of critical cells per dimension)

##### Parameters
 SimplicialComplex complex
##### Options
 Int rounds Run for r rounds Int seed Set seed number for random number generator Int strategy Set strategy=>0 (default) for random-random: uniformly random selecting of a face to collapse or as critical face Set strategy=>1 for random-lex-first: uniformly random relabeling of vertices, then selecting lexicographically first face for collapse or as a critical face Set strategy=>2 for random-lex-last: uniformly random relabeling of vertices, then selecting lexicographically last face for collapse or as a critical face Int verbose v Prints message after running every v rounds Array try_until_reached Used together with rounds=>r; When try_until_reached=>[a,...,b], runs for r rounds or until [a,...,b] is found Array try_until_exception Used together with rounds=>r; When try_until_exception=>[a,...,b], runs for r rounds or until anything other than [a,...,b] is found String save_collapsed Save all facets that remain after initial collapse to an XML file of a Simplicial Complex. Rounds that have Morse vector [1,0,...,0] or [1,0,...,0,1] will save nothing. Filename must have quotation marks: save_collapsed=>"path/to/filename". The XML files are saved as "path/to/filename_currentround.top".
##### Returns
 Map< Array, Int >
•
stabbing_order (P) → graph::Graph<Directed>

Determine the stabbing partial order of a simplicial ball with respect to the origin. The origin may be a vertex or not. For details see Assarf, Joswig & Pfeifle: Webs of stars or how to triangulate sums of polytopes, to appear

##### Parameters
 GeometricSimplicialComplex P
##### Returns
 graph::Graph
•
stanley_reisner (complex) → ideal::Ideal

Creates the Stanley-Reisner ideal of a simplicial complex.

##### Parameters
 SimplicialComplex complex
##### Returns
 ideal::Ideal
•
star_of_zero (C) → Set<Set<Int>>

Find the facets of the star of the origin in the simplicial complex. The origin may be a vertex or not. For details see Assarf, Joswig & Pfeifle: Webs of stars or how to triangulate sums of polytopes, to appear

##### Parameters
 GeometricSimplicialComplex C
##### Returns
 Set>
•
star_shaped_balls (P) → Array<Set<Set>>

Enumerate all balls formed by the simplices of a geometric simplicial complex that are strictly star-shaped with respect to the origin. The origin may be a vertex or not. For details see Assarf, Joswig & Pfeifle: Webs of stars or how to triangulate sums of polytopes, to appear

##### Parameters
 GeometricSimplicialComplex P
##### Returns
 Array>
•
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> facets the facets of the simplicial complex
##### Options
 Int high_dim Int low_dim Bool verbose
##### Returns
 Array>
•
vietoris_rips_filtration <Coeff> (D, deg, step_size, k) → Filtration<SparseMatrix<Coeff, NonSymmetric> >

Constructs the k-skeleton of the Vietrois Rips filtration of a point set. The set is passed as its so-called "distance matrix", whose (i,j)-entry is the distance between point i and j. This matrix can e.g. be computed using the distance_matrix function. The other inputs are an integer array containing the degree of each point, the desired distance step size between frames, and the dimension up to which to compute the skeleton. Redundant points will appear as seperate vertices of the complex. Setting k to |S| will compute the entire VR-Complex for each frame.

##### Type Parameters
 Coeff desired coefficient type of the filtration
##### Parameters
 Matrix D the "distance matrix" of the point set (can be upper triangular) Array deg the degrees of input points Float step_size Int k dimension of the resulting filtration
##### Returns
 Filtration >
•

### 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 no_labels flag is specified.

##### Parameters
 SimplicialComplex complex
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0
##### 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

Example:
• To subdivide a triangle into six new triangles, do this:`> \$b = barycentric_subdivision(simplex(2));`
•
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
•
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.

##### Parameters
 polytope::Polytope P the underlying polytope SimplicialComplex complex a sufficiently fine subdivision of P, for example the second barycentric subdivision
##### Returns
 SimplicialComplex
•
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.

Contained in extension `local`.
##### Parameters
 SimplicialComplex complex
##### Options
 Bool no_lables
##### 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 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 no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex

Example:
• The following creates the cone with two apices over the triangle, with custom apex labels. The resulting complex is the 4-simplex.`> \$c = cone(simplex(2),2,apex_labels=>['foo','bar']);``> print \$c->FACETS;`` {0 1 2 3 4}``> print \$c->VERTEX_LABELS;`` 0 1 2 foo bar`
•
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 glueing facet iteself is not included in the connected sum. 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 permutation Bool no_labels
##### Returns
 SimplicialComplex

Example:
• Glueing together two tori to make a genus 2 double torus, rotating the second one clockwise:`> \$cs = connected_sum(torus(),torus(),permutation=>[1,2,0]);``> print \$cs->SURFACE.','.\$cs->GENUS;`` 1,2`
•
covering_relations (P) → Graph<Directed>

Construct the covering relations of a poset

##### Parameters
 Graph P
##### Returns
 Graph
•
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 create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex
•
disjoint_union (complex1, complex2) → SimplicialComplex

Produce the disjoint union of the two given complexes.

##### Parameters
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0 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
•
hom_poset (P, Q) → Graph<Directed>

Construct the poset of order preserving maps from one poset to another

##### Parameters
 Graph P Graph Q
##### Returns
 Graph
•
hom_poset (homs, Q) → Graph<Directed>

Construct the poset of order preserving maps from one poset to another

##### Parameters
 Array> homs Graph Q
##### Returns
 Graph
•
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 Do not create VERTEX_LABELS. default: 0 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 vertices
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0 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
•
join_complexes (complex1, complex2) → SimplicialComplex

Creates the join of complex1 and complex2.

##### Parameters
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0 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 no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex
•
k_skeleton (complex, k) → GeometricSimplicialComplex

Produce the k-skeleton.

##### Parameters
 GeometricSimplicialComplex complex Int k
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 GeometricSimplicialComplex
• link_complex (complex, face) → SimplicialComplex
•
poset_homomorphisms (P, Q) → Array<Array<Int>>

Enumerate all order preserving maps from one poset to another

##### Parameters
 Graph P Graph Q
##### Options
 Array prescribed_map A vector of length P.nodes() with those images in Q that should be fixed. Negative entries will be enumerated over.
##### Returns
 Array>
•
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 vertex_order1 Array vertex_order2 Bool geometric_realization default 0 Bool color_cons Bool no_labels Do not create VERTEX_LABELS. default: 0
##### 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 vertex_order1 Array vertex_order2 Bool geometric_realization default 1 Bool color_cons Bool no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 GeometricSimplicialComplex
•
star (complex, face) → SimplicialComplex

Produce the star of the face of the complex.

##### Parameters
 SimplicialComplex complex Set face
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex
•
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 create VERTEX_LABELS. default: 0
##### 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> faces
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0 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 face
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0 Bool geometric_realization default 0
##### Returns
 SimplicialComplex
•
sum_triangulation (P, Q, WebOfStars) → GeometricSimplicialComplex

Produce a specific sum-triangulation of two given triangulations. and a WebOfStars. There are P-sum-triangulations and Q-sum-triangulations. If the image of the star of the origin of P is empty then we have a Q-sum-triangulation; otherwise it is a P-sum-triangulation. For details see Assarf, Joswig & Pfeifle: Webs of stars or how to triangulate sums of polytopes, to appear

##### Parameters
 GeometricSimplicialComplex P first complex 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.
##### Options
 Bool origin_first decides if the origin should be the first point in the resulting complex. Default=0
##### Returns
 GeometricSimplicialComplex
•
suspension (complex, k) → SimplicialComplex

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 no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex
•
triang_neighborhood (complex, width)

Create a triangulated tubular neighborhood of a pure 2-complex. If the complex is a link with the property that each vertex and its two neighbours are in general position after projection to the x,y-plane, then one might specify a rational number width to tell the client to compute COORDINATES of the triangulated tubular neighborhood. If the width/ is chosen too large, the computed realization will be self intersecting. 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.

Contained in extension `local`.
##### Parameters
 SimplicialComplex complex Rational width default: 0
•
union (complex1, complex2) → SimplicialComplex

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

##### Parameters
 SimplicialComplex complex1 SimplicialComplex complex2
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex
•
web_of_stars (poset_hom, star_shaped_balls, triang) → IncidenceMatrix

Produce a web of stars from two given triangulations and a map between them.

##### Parameters
 Array poset_hom the poset homomorphism from stabbing order to star-shaped balls Array>> star_shaped_balls the collection of star-shaped balls of T Array> triang the facets of the underlying triangulation of Q
##### Returns
 IncidenceMatrix WebOfStars Every row corresponds to a full dimensional simplex in P and every column to a full dimensional simplex in Q.
•

### 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, that is, the simplicial complex that has an n-dimensional facet for each n+1-clique. If no_labels is set to 1, the labels are not copied.

##### Parameters
 Graph graph
##### Options
 Bool no_labels Do not create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex

Example:
• Create the clique complex of a simple graph with one 3-clique and one 2-clique, not creating labels.`> \$g = graph_from_edges([[0,1],[0,2],[1,2],[2,3]]);``> \$c = clique_complex(\$g,no_labels=>1);``> print \$c->FACETS;`` {0 1 2}`` {2 3}`
•
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 Do not create VERTEX_LABELS. default: 0
##### Returns
 SimplicialComplex
•
vietoris_rips_complex (D, delta) → SimplicialComplex

Computes the Vietoris Rips complex of a point set. The set is passed as its so-called "distance matrix", whose (i,j)-entry is the distance between point i and j. This matrix can e.g. be computed using the distance_matrix function. The points corresponding to vertices of a common simplex will all have a distance less than delta from each other.

##### Parameters
 Matrix D the "distance matrix" of the point set (can be upper triangular) Rational delta
##### 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

Example:
• The following produces the 3-ball and stores it in the variable \$b:`> \$b = ball(3);` You can print the facets of the resulting simplicial complex like so:`> print \$b->FACETS;`` {0 1 2 3}`
•
complex_projective_plane () → SimplicialComplex

The complex projective plane with the vertex-minimal triangulation by Kühnel and Brehm.

##### Returns
 SimplicialComplex

Example:
• Construct the complex projective plane, store it in the variable \$p2c, and print its homology group.`> \$p2c = complex_projective_plane();``> print \$p2c->HOMOLOGY;`` ({} 0)`` ({} 0)`` ({} 1)`` ({} 0)`` ({} 1)`
•
cube_complex (x) → GeometricSimplicialComplex<Rational>

Produces a triangulated pile of hypercubes, arranged in a d-dimensional array. Each cube is split into d! tetrahedra, and the tetrahedra are all grouped around one of the diagonal axes of the cube.

##### Parameters
 Array x specifies the shape of the pile: d=x.size is the dimension of the cubes to be stacked, and the stack will be x_1 by x_2 by ... by x_d cubes.
##### Returns
 GeometricSimplicialComplex

Example:
• Arrange four triangulated 3-cubes to form a big 2 by 2 cube:`> \$cc = cube_complex([2,2,2]);``> print \$cc->description;`` 2x2x2 Pile of 3-dimensional triangulated cubes.`
•
klein_bottle () → SimplicialComplex

The Klein bottle.

##### Returns
 SimplicialComplex
•
multi_associahedron_sphere (n, k) → SimplicialComplex

Produce the simplicial sphere //Δ(n,k) of (k+1)-crossing free multitriangulations of an n-gon P, along with the group action on the diagonals induced from D_{2n}. Δ(n,k) is the simplicial complex on the set of relevant diagonals of P whose faces are those sets of diagonals such that no k+1 of them mutually cross. A diagonal is relevant if it leaves k or more vertices of P on both sides. (Any diagonal having less than k vertices on one side trivially cannot participate in a (k+1)-crossing, so is irrelevant. The corresponding complex on all// diagonals is therefore the simplicial join of this one with the simplex of irrelevant diagonals.)

Jakob Jonsson, "Generalized triangulations and diagonal-free subsets of stack polyominoes",
J. Combin. Theory Ser. A, 112(1):117–142, 2005

Delta(n,k) is known to be a k-neighborly vertex-decomposable sphere of dimension kν-1, where the parameter ν=n-2k-1 measures the complexity of this construction. For ν=0, the complex is a point; for ν=1 a k-simplex; for ν=2 the boundary of a cyclic polytope. Setting k=1 yields the boundary of the simplicial associahedron. The list of (k+1)-crossings in the n-gon is included as the attachment K_PLUS_1_CROSSINGS. It can also be obtained as the property MINIMAL_NON_FACES, but this requires the HASSE_DIAGRAM to be computed.

##### Parameters
 Int n the number of vertices of the polygon Int k the number of diagonals that are allowed to mutually cross
##### Options
 Bool no_facets don't calculate the facets (for large examples)? Default 0 Bool no_crossings don't calculate the crossings? Default 0
##### Returns
 SimplicialComplex

Examples:
• The f-vector of Δ(9,3) is that of a neighborly polytope, since ν=2:`> print multi_associahedron_sphere(9,3)->F_VECTOR;`` 9 36 84 117 90 30`
• The option no_facets=>1 still leaves useful information:`> \$s = multi_associahedron_sphere(8,2, no_facets=>1);``> print \$s->VERTEX_LABELS;`` (0 3) (1 4) (2 5) (3 6) (4 7) (0 5) (1 6) (2 7) (0 4) (1 5) (2 6) (3 7)``> print \$s->GROUP->PERMUTATION_ACTION->GENERATORS;`` 7 0 1 2 3 4 5 6 11 8 9 10`` 4 3 2 1 0 7 6 5 11 10 9 8``> print \$s->get_attachment("K_PLUS_1_CROSSINGS")->size();`` 28`
•
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
•

### Producing other objects

Functions producing big objects which are not contained in application topaz.

•
secondary_polyhedron (s, depth) → polytope::Polytope<Float>

Computes the secondary polyhedron of a hyperbolic surface up to a given depth of the spanning tree of the covering triangluation of the hypoerbolic plane.

##### Parameters
 HyperbolicSurface s Int depth
##### Returns
 polytope::Polytope
•

### Topology

The following functions compute topological invariants.

•
betti_numbers <Coeff> (C) → Array<Int>

Calculate the betti numbers of a general chain complex over a field.

##### Type Parameters
 Coeff The coefficient field type for homology computation. Defaults to Rational
##### Parameters
 ChainComplex C
##### Returns
 Array containing the i-th betti number at entry i

Example:
• The following constructs a simple chain complex with only one non-empty differential:`> \$cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));` You can print its betti numbers like this:`> print betti_numbers(\$cc);`` 1 0`
•
betti_numbers <Coeff> (S) → Array<Int>

Calculate the reduced betti numbers of a simplicial complex over a field.

##### Type Parameters
 Coeff The coefficient field type for homology computation. Defaults to Rational
##### Parameters
 SimplicialComplex S
##### Returns
 Array containing the i-th betti number at entry i

Example:
• To print the betti numbers for the torus, do this:`> \$t = torus();``> print betti_numbers(\$t);`` 0 2 1`
•
cap_product (cocycles, cycles) → Pair<CycleGroup<E>,Map<Pair<Int,Int>,Int>>

Compute all cap products of cohomology and homology cycles in two given groups.

##### Parameters
 CycleGroup cocycles CycleGroup cycles
##### Returns
 Pair,Map,Int>>

Example:
• The following stores all cap products of the cocycles and cycles of the homology group of the oriented surface of genus 1 in the variable \$cp.`> \$s = surface(1);``> \$cp = cap_product(\$s->COCYCLES->[1],\$s->CYCLES->[1]);`
•
homology (complex, co)

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

##### Parameters
 Array> 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 (CC, co) → Array<HomologyGroup<Integer>>

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

##### Parameters
 ChainComplex CC The chain complex for which to compute homology. 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
##### Returns
 Array>

Example:
• To construct a small chain complex with only one non-zero differential:`> \$cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));` This prints its homology groups.`> print homology(\$cc,0);`` ({(2 1)} 1)`` ({} 0)` The output means that the zeroth homology group has 2-torsion with multiplicity one, and betti number one. The first homology group is empty.
•
homology_and_cycles (complex, co)

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

##### Parameters
 Array> 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 (CC, co) → Array<Pair<HomologyGroup, SparseMatrix>>

Calculate the (co-)homology groups and __cycle coefficient matrices_ of a chain complex.

##### Parameters
 ChainComplex> CC The chain complex for which to compute homology. 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
##### Returns
 Array> For each dimension, contains the homology group and corresponding cycle group coefficient matrix where each row of the matrix represents a generator, column indices referring to indices of the chain group elements involved.

Example:
• To construct a small chain complex with only one non-zero differential:`> \$cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));` This prints its homology groups and corresponding generators.`> print homology_and_cycles(\$cc,0);`` (({(2 1)} 1)`` <1 0`` 0 1`` >`` )`` (({} 0)`` <>`` )` The output means that the zeroth homology group has 2-torsion with multiplicity one generated by the first elemen of the chain group, and free part of rank one generated by the second element. The first homology group is empty.

## Property Types

•

### Topology

The following property_types are topological invariants.

•
Cell
UNDOCUMENTED
•
ChainComplex <MatrixType>

A finite chain complex, represented as its boundary matrices. Check out the tutorial on the polymake homepage for examples on constructing ChainComplexes and computing their homology.

##### Type Parameters
 MatrixType The type of the differential matrices. default: SparseMatrix

#### User Methods of ChainComplex

•
boundary_matrix (d) → MatrixType

Returns the d-boundary matrix of the chain complex.

##### Parameters
 Int d
##### Returns
 MatrixType
•
dim () → Int

Returns the number of non-empty modules in the complex.

##### Returns
 Int
•
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
•
Filtration <MatrixType>

A filtration of chain complexes.

##### Type Parameters
 MatrixType

#### User Methods of Filtration

•
boundary_matrix (d, t)

Returns the d-boundary matrix of the t-th frame of the filtration.

##### Parameters
 Int d Int t
•
cells () → Array<Cell>

Returns the cells of the filtration, given as array of 3-tuples containing degree, dimension and boundary matrix row number of the cell.

##### Returns
 Array
•
dim () → Int

Returns the dimension of the maximal cells in the last frame of the filtration.

##### Returns
 Int
•
n_cells () → Int

Returns the number of cells in the last frame of the filtration.

##### Returns
 Int
•
n_frames () → Int

Returns the number of frames in of the filtration.

##### Returns
 Int
•
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.

•
IntersectionForm

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