# application: topaz

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

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

**imports from:**common, graph

**Objects**

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.

**derived from:**graph::Graph<Undirected>#### Properties of MixedGraph

**EDGE_WEIGHTS**: common::EdgeMap<Undirected, Float>Associated edge labels (weights) of the MixedGraph.

An abstract simplicial complex represented by its facets.

#### Properties of SimplicialComplex

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

**DIM**: common::IntMaximal 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

**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.

**HASSE_DIAGRAM**: graph::FaceLatticeThe 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.

**INPUT_FACES**: common::Array<Set<Int>>Any description of the faces of a simplicial complex. Redundancy allowed.

**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.**F_VECTOR**: common::Array<Int>f

_{k}is the number of k-faces, for k = 0,... , d, where d is the dimension.**MINIMAL_NON_FACES**: common::Array<Set<Int>>Inclusion minimal non-faces (vertex subsets which are not faces of the simplicial complex).

**ODD_SUBCOMPLEX**: SimplicialComplexSubcomplex 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.

**SIGNATURE**: common::IntSignature of a geometric simplicial complex embedded in the integer lattice. Like DUAL_GRAPH_SIGNATURE, but only simplices with odd normalized volume are counted.

**VOLUME**: common::RationalVolume of a geometric simplicial complex.

**FIXME:**introduce type RealizedSimplicialComplex<Scalar=Rational> : SimplicialComplex

The following properties are topological invariants.

**BALL**: common::BoolHeuristically true if the topological space homeomorphic to a ball. Similar remarks hold as for SPHERE.

**CLOSED_PSEUDO_MANIFOLD**: common::BoolTrue if this is a PURE simplicial complex with the property that each ridge is contained in exactly two facets.

**COCYCLES**: Array<CycleGroup>Representatives of cocycle groups, listed in increasing codimension order. Encoding similar to CYCLES.

**COHOMOLOGY**: Array<HomologyGroup>Reduced cohomology groups, listed in increasing codimension order. Encoding similar to HOMOLOGY.

**CYCLES**: Array<CycleGroup>Representatives of cycle groups, listed in increasing dimension order.

The first component in each dimension is a matrix of integer coefficients, the second component is a vector of faces. To obtain the chains, one must (symbolically) multiply both components.

**EULER_CHARACTERISTIC**: common::IntReduced Euler characteristic. Alternating sum of the F_VECTOR minus 1.

**FUNDAMENTAL_GROUP**: common::Tuple<Int, List<List<Tuple<Int, Int>>>>A finite representation of the fundamental group. The fundamental group is represented as a pair of an integer, the number of generators, and a list of relations. The generators are numbered consecutively starting with zero. A relation is encoded as a list of pairs, each pair consisting of a generator and its exponent.

You may use the fundamental2gap method to produce a

`GAP`

file.**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}.**HOMOLOGY**: Array<HomologyGroup>Reduced simplicial homology groups H

_{0}, ..., H_{d}(integer coefficients), listed in increasing dimension order.Each group G is encoded as a sequence ( { (t

_{1}m_{1}) ... (t_{n}m_{n}) } f) of non-negative integers, with t_{1}> t_{2}> ... >_{n}> 1, plus an extra non-negative integer f. The group G is isomorphic to (Z/t_{1})^{m1}× ... × (Z/t_{n})^{mn}× Z^{f}, where Z^{0}is the trivial group.**INTERSECTION_FORM**: IntersectionFormParity 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.

**MANIFOLD**: common::BoolTrue if this is a compact simplicial manifold with boundary.

Depends on heuristic SPHERE recognition.

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

Robin Forman: Morse Theory for Cell-Complexes,Advances in Math., 134 (1998), pp. 90-145.This property is computed by one of two heuristics. The default heuristic is a simple greedy algorithm (greedy). The alternative is to use a canceling algorithm due to Forman (cancel). Note that the computation of a Morse matching of largest size is NP-hard. See

Michael Joswig, Marc E. Pfetsch: Computing Optimal Morse MatchingsSIAM J. Discrete Math., 2006, to appear**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::IntNumber of critical faces of the computed Morse matching. FIXME: temporary

**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::BoolTrue if this is a PSEUDO_MANIFOLD with top level homology isomorphic to Z.

**PSEUDO_MANIFOLD**: common::BoolTrue if this is a PURE simplicial complex with the property that each ridge is contained in either one or two facets.

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

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

**GEOMETRIC_REALIZATION**: common::Matrix<Rational, NonSymmetric>Coordinates for the vertices of the simplicial complex, such that the complex is embedded without crossings in some R

^{e}. Vector (x_{1}, .... x_{e}) represents a point in Euclidean e-space.**G_DIM**: common::IntDimension e of the space in which the GEOMETRIC_REALIZATION of the complex is embedded.

**MIXED_GRAPH**: MixedGraphThe nodes of the mixed graph are the nodes of the primal GRAPH and the DUAL_GRAPH. Additional to the primal and dual edges, there is an edge between a primal and a dual node iff the primal node represents a vertex of the corresponding facet of the dual node.

**FIXME:**temporary

#### User Methods of SimplicialComplex

**CONNECTED_COMPONENTS**()The connected components of the GRAPH, encoded as node sets.

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

**DUAL_BIPARTITE**()True if DUAL_GRAPH is a

*bipartite*.**DUAL_CONNECTED**()True if the DUAL_GRAPH is a connected graph.

**DUAL_CONNECTED_COMPONENTS**()The connected components of the DUAL_GRAPH, encoded as node sets.

**DUAL_CONNECTIVITY**()Node connectivity of the DUAL_GRAPH. Dual to CONNECTIVITY.

**DUAL_GRAPH_SIGNATURE**()Difference of the black and white nodes if the DUAL_GRAPH is BIPARTITE. Otherwise -1.

**DUAL_MAX_CLIQUES**()The maximal cliques of the DUAL_GRAPH, encoded as node sets.

**labeled_vertices**(label ...) → Set<Int>**MAX_CLIQUES**()The maximal cliques of the GRAPH, encoded as node sets.

**N_CONNECTED_COMPONENTS**()Number of connected components of the GRAPH.

**VERTEX_DEGREES**()Degrees of vertices in the GRAPH.

**fundamental2gap**()Writes the FUNDAMENTAL_GROUP using FUNDAMENTAL_GROUP_GEN_LABELS to the given file in GAP input format.

**VISUAL**() → Visual::SimplicialComplexVisualizes the complex.

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

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 embedderInt seed random seed value for the string embedderoption list: Visual::Polygon::decorations option list: Visual::Graph::decorations ##### Returns

Visual::SimplicialComplex **VISUAL_DUAL_GRAPH**() → Visual::SimplicialComplexUses the spring embedder to visualize the DUAL_GRAPH.

##### Options

Int seed random seed value for the string embedderoption list: Visual::Graph::decorations ##### Returns

Visual::SimplicialComplex **VISUAL_FACE_LATTICE**() → Visual::SimplicialComplexLatticeVisualize the HASSE_DIAGRAM of a simplicial complex as a multi-layer graph.

##### Options

Int seed random seed value for the node placementoption list: Visual::Lattice::decorations ##### Returns

Visual::SimplicialComplexLattice **VISUAL_GRAPH**() → Visual::SimplicialComplexUses the spring embedder to visualize the GRAPH.

##### Options

Int seed random seed value for the string embedderoption 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 embedderoption list: Visual::Graph::decorations

#### Permutations of SimplicialComplex

**FacetPerm**UNDOCUMENTED

#### Properties of FacetPerm

**VertexPerm**UNDOCUMENTED

#### Properties of VertexPerm

Visualization of the simplicial complex.

#### User Methods of Visual::SimplicialComplex

**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

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

**cap_product**()Compute and print all cap products of cohomology and homology cycles. args: SimplicialComplex

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

##### Parameters

Array<Set<int>> complex Bool co ##### Options

Int dim_low narrows the dimension range of interest, with negative values being treated as co-dimensionsInt 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 ##### Options

Int dim_low narrows the dimension range of interest, with negative values being treated as co-dimensionsInt dim_high see*dim_low***is_vertex_decomposition**()Check whether a given ordered subset of the vertex set is a vertex decomposition. Works for 1-, 2- and 3-manifolds only!

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

**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.

**isomorphic**(complex1, complex2) → BoolDetermine whether two given complexes are combinatorially isomorphic. The problem is reduced to graph isomorphism of the vertex-facet incidence graphs.

**pl_homeomorphic**(complex1, complex2) → BoolTries to determine wheter 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

**alexander_dual**(complex) → SimplicialComplexComputes the Alexander dual complex, that is, the complements of all non-faces. The vertex labels are preserved unless the

*nol*flag is specified.**balanced_prism**(complex) → SimplicialComplexProduce a

*prism*over a given SimplicialComplex.**barycentric_subdivision**(complex)Create a simplicial complex as a barycentric subdivision of a given complex. Each vertex in the new complex corresponds to a face in the old complex.

##### Parameters

SimplicialComplex complex ##### Options

Bool relabel generate vertex labels from the faces of the original complex.Bool geom_real read GEOMETRIC_REALIZATION of the input complex, compute the coordinates of the new vertices and store them as GEOMETRIC_REALIZATION of the produced complex.**bistellar_simplification**(complex) → SimplicialComplexHeuristic 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 **cone**(complex, k) → SimplicialComplexProduce 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, f_1, f_2)Compute the connected sum of two complexes.

Parameters

*f_1 and*f_2// specify which facet of the first and second complex correspondingly are glued together. Default is the 0-th facet of both.The vertices in the selected facets are identified with each other according to their order in the facet (that is, in icreasing index order). The option

*permutation*allows to get an alternative identification. It should specify a permutation of the vertices of the second facet.The vertices of the new complex get the original labels with

`_1`

or`_2`

appended, according to the input complex they came from. If you set the*no_labels*flag, the label generation will be omitted.##### Parameters

SimplicialComplex complex1 SimplicialComplex complex2 int f_1 default is 0int f_2 default is 0##### Options

Array<int> permutation Bool no_lables **deletion**(complex, face) → SimplicialComplexRemove 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)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.**edge_contraction**(complex) → SimplicialComplexHeuristic 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.

**h_induced_quotient**(C, vertices) → SimplicialComplexLet

*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)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 GEOMETRIC_REALIZATION.**join_complexes**(complex1, complex2) → SimplicialComplexCreates 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) → SimplicialComplexProduce the

*k*-skeleton.##### Parameters

SimplicialComplex complex int k ##### Options

Bool nol suppresses creation of VERTEX_LABELS##### Returns

SimplicialComplex **link_complex**(complex, face)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.**simplicial_product**(complex1, complex2)Computes the simplicial product of two complexes. Vertex orderings may be given as options.

##### Parameters

SimplicialComplex complex1 SimplicialComplex complex2 ##### Options

Array<Int> vertex_order1 Array<Int> vertex_order2 Bool geom_real Bool color_cons Bool no_labels **star**(complex, face) → SimplicialComplexProduce 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) → SimplicialComplexRemove 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) → SimplicialComplexComputes 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 geom_real ##### Returns

SimplicialComplex **stellar_subdivision**(complex, faces) → SimplicialComplexComputes 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 geom_real ##### Returns

SimplicialComplex **suspension**(complex, k)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.**union**(complex1, complex2) → SimplicialComplexProduce 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

**clique_complex**(graph) → SimplicialComplexProduce the

*clique complex*of a given graph. If*no_labels*is set to 1, the labels are not copied.**crosscut_complex**(p) → SimplicialComplexProduce the

*crosscut complex*of the boundary of a polytope.##### Parameters

polytope::Polytope p ##### Options

Bool noc suppresses copying the vertex coordinates to GEOMETRIC_REALIZATION##### Returns

SimplicialComplex

**ball**(d) → 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**projective_plane**() → SimplicialComplex**rand_knot**(n_edges) → SimplicialComplexProduce 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.**simplex**(d) → SimplicialComplex**sphere**(d) → SimplicialComplexThe

*d*-dimansional sphere, realized as the boundary of the (*d*+1)-simplex.**surface**(g) → SimplicialComplexProduce a surface of genus

*g*. For*g*>= 0 the client produces an orientable surface, otherwise it produces a non-orientable one.**torus**() → SimplicialComplexThe Császár Torus. Geometric realization by Frank Lutz, Electronic Geometry Model No. 2001.02.069

##### Returns

SimplicialComplex **unknot**(m, n)Produces a triangulated 3-sphere with the particular NASTY embedding of the unknot in its 1-skeleton. The parameters

*m*>= 2 and*n*>= 1 determine how entangled the unknot is.*eps*determines the GEOMETRIC_REALIZATION.

## Property Types

**CycleGroup**fields: coeff, faces (defined in homology.cc)

**derived from:**common::Tuple<SparseMatrix<Integer, NonSymmetric>, Array<Set<Int>>>**HomologyGroup**fields: torsion, betti (defined in homology.cc)

**derived from:**common::Tuple<List<Pair<Integer, Int>>, Int>**IntersectionForm**fields: parity, positive, negative (defined in intersection_form.cc)

**derived from:**common::Tuple<Int, Int, Int>