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| ====== BigObject Cone ====== | ====== application topaz ====== |
| An abstract simplicial complex represented by its facets. | The __TOP__ology __A__pplication __Z__oo deals with abstract simplicial complexes.\\ A complex is given as a list of facets. You can ask for its global properties ([[playground/playground/SimplicialComplex#MANIFOLD |manifold recognition]],\\ [[playground/playground/SimplicialComplex#HOMOLOGY |homology groups]], etc.), explore the local vertex environment ([[playground/playground#star |stars]], [[playground/playground#link |links]], etc.), and make a lot\\ of constructions.\\ The visualization means are constrained, as they are mostly based on the ''[[playground/playground/SimplicialComplex#GRAPH |GRAPH]]'' (1-skeleton) of a complex. |
| ===== Properties ===== | imports from: |
| | * [[graph|application graph]] |
| | * [[common|application common]] |
| | uses: |
| | * [[ideal|application ideal]] |
| | * [[group|application group]] |
| | ===== Objects ===== |
| | * [[topaz:GeometricSimplicialComplex|GeometricSimplicialComplex]]:\\ A geometric simplicial complex, i.e., a simplicial complex with a geometric realization. //Scalar// is the numeric data type used for the coordinates. |
| | * [[topaz:HyperbolicSurface|HyperbolicSurface]]:\\ A hyperbolic surface (noncompact, finite area) is given by a triangulation [[/topaz/objects/HyperbolicSurface/properties/DCEL_DATA]] (the topological data) and [[/topaz/objects/HyperbolicSurface/properties/PENNER_COORDINATES]] (the metric data). |
| | * [[topaz:MorseMatching|MorseMatching]]:\\ 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 ([[/topaz/objects/SimplicialComplex|greedy]]). The alternative is to use a canceling algorithm due to Forman ([[/topaz/objects/SimplicialComplex|cancel]]) or both ([[/topaz/objects/SimplicialComplex|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 |
| | * [[topaz:SimplicialComplex|SimplicialComplex]]:\\ An abstract simplicial complex represented by its facets. |
| | * [[topaz:Visual::SimplicialComplex|Visual::SimplicialComplex]]:\\ Visualization of the simplicial complex. |
| | * [[topaz:Visual::SimplicialComplexLattice|Visual::SimplicialComplexLattice]]:\\ Visualization of the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/HASSE_DIAGRAM]] of a simplicial complex as a multi-layer graph. |
| | ===== Functions ===== |
| |
| ==== Input Properties ==== | ---- |
| These properties are for input only. They allow redundant information. | ==== 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. |
| == INPUT_FACES == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Any description of the faces of a simplicial complex with vertices v_0 < v_1 < v_2 < ... arbitrary. Redundant faces allowed. | {{anchor:n_poset_homomorphisms:}} **''n_poset_homomorphisms([[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]> P, [[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]> Q)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]>'' ''P'' : |
| | * ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]>'' ''Q'' : |
| | * //Returns:// ''[[playground/playground#Int |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. |
| |
| ---- | ---- |
| ==== Visualization ==== | ==== Comparing ==== |
| These properties are for visualization. | These functions compare two [[/topaz/objects/SimplicialComplex]] |
| | \\ |
| | {{anchor:isomorphic:}} **''isomorphic([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex1, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex2)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex1'' : |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex2'' : |
| | * //Returns:// ''[[playground/playground#Bool |Bool]]'' |
| | * Determine whether two given complexes are combinatorially isomorphic.\\ The problem is reduced to graph isomorphism of the vertex-facet incidence graphs. |
| |
| == MIXED_GRAPH == | \\ |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Visualization/MIXED_GRAPH\\ The nodes of the mixed graph are the nodes of the primal [[/topaz/objects/SimplicialComplex/properties/Combinatorics/GRAPH]] and | {{anchor:find_facet_vertex_permutations:}} **''find_facet_vertex_permutations([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex1, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex2)''** |
| the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/DUAL_GRAPH]]. Additional to the primal and dual edges, there is | * //Parameters:// |
| an edge between a primal and a dual node iff the primal node represents | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex1'' : |
| a vertex of the corresponding facet of the dual node. | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex2'' : |
| \\ __Properties of MIXED_GRAPH:__\\ | * //Returns:// ''[[playground/playground#Pair |Pair]]<[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>,[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>>'' |
| * **EDGE_WEIGHTS**\\ ''Type:'' /common/property_types/Graph Types/EdgeMap\\ Associated edge weights. | * 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. |
| |
| == VERTEX_LABELS == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Labels of the vertices. | {{anchor:pl_homeomorphic:}} **''pl_homeomorphic([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex1, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex2)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex1'' : |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex2'' : |
| | * //Returns:// ''[[playground/playground#Bool |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 ''[[playground/playground/SimplicialComplex#CLOSED_PSEUDO_MANIFOLD |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. |
| |
| ---- | ---- |
| ==== Symmetry ==== | ==== Other ==== |
| These properties capture information of the object that is concerned with the action of permutation groups. | Special purpose functions. |
| | \\ |
| | {{anchor:poset_by_inclusion:}} **''poset_by_inclusion([[playground/playground#Array |Array]]<[[playground/playground#poset |poset]] P)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#poset |poset]]'' ''P'' : |
| | * //Returns:// ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |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 |
| | |
| | \\ |
| | {{anchor:is_vertex_decomposition:}} **''is_vertex_decomposition([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]> vertices)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>'' ''vertices'' : shedding vertices |
| | * //Returns:// ''[[playground/playground#Bool |Bool]]'' |
| | * Check whether a given ordered subset of the vertex set is a __vertex decomposition__.\\ Works for 1-, 2- and 3-manifolds only! |
| | |
| | \\ |
| | {{anchor:stiefel_whitney:}} **''stiefel_whitney([[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>> facets)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>>'' ''facets'' : the facets of the simplicial complex |
| | * //Returns:// ''[[playground/playground#Array |Array]]<[[playground/playground#PowerSet |PowerSet]]<[[playground/playground#Int |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 |
| | |
| | \\ |
| | {{anchor:star_of_zero:}} **''star_of_zero([[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]] C)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' ''C'' : |
| | * //Returns:// ''[[playground/playground#Set |Set]]<[[playground/playground#Set |Set]]<[[playground/playground#Int |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 |
| | |
| | \\ |
| | {{anchor:mixed_graph:}} **''mixed_graph([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * Produces the mixed graph of a //complex//. |
| | |
| | \\ |
| | {{anchor:star_shaped_balls:}} **''star_shaped_balls([[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]] P)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' ''P'' : |
| | * //Returns:// ''[[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]<[[playground/playground#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 |
| | |
| | \\ |
| | {{anchor:stabbing_order:}} **''stabbing_order([[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]] P)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' ''P'' : |
| | * //Returns:// ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |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 |
| | |
| | \\ |
| | {{anchor:stanley_reisner:}} **''stanley_reisner([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * //Returns:// ''[[playground/playground/Ideal# |Ideal]]'' |
| | * Creates the __Stanley-Reisner ideal__ of a simplicial complex. |
| | |
| | \\ |
| | {{anchor:is_generalized_shelling:}} **''is_generalized_shelling([[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]> FaceList)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]>'' ''FaceList'' : |
| | * //Returns:// ''[[playground/playground#Bool |Bool]]'' |
| | * Check if a given sequence of faces of a simplicial complex is a generalized shelling. |
| | |
| | \\ |
| | {{anchor:persistent_homology:}} **''persistent_homology''** |
| | |
| | \\ |
| | {{anchor:random_discrete_morse:}} **''random_discrete_morse([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * //Returns:// ''[[playground/playground#Map |Map]]<[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>,[[playground/playground#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) |
| | |
| | \\ |
| | {{anchor:vietoris_rips_filtration:}} **''vietoris_rips_filtration<Coeff>([[playground/playground#Matrix |Matrix]] D, [[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]> deg, [[playground/playground#Float |Float]] step_size, [[playground/playground#Int |Int]] k)''** |
| | * //Template Parameters:// |
| | * ''Coeff'' : desired coefficient type of the filtration |
| | * //Parameters:// |
| | * ''[[playground/playground#Matrix |Matrix]]'' ''D'' : the "distance matrix" of the point set (can be upper triangular) |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>'' ''deg'' : the degrees of input points |
| | * ''[[playground/playground#Float |Float]]'' ''step_size'' : |
| | * ''[[playground/playground#Int |Int]]'' ''k'' : dimension of the resulting filtration |
| | * //Returns:// ''[[playground/playground#Filtration |Filtration]]<[[playground/playground#SparseMatrix |SparseMatrix]]<[[playground/playground#vietoris |vietoris]]'' |
| | * 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. |
| |
| == GROUP == | |
| ''Type:'' /group/objects/Group | |
| ---- | ---- |
| ==== Combinatorics ==== | ==== Producing a new simplicial complex from others ==== |
| These properties capture combinatorial information of the object. | These functions construct a new [[/topaz/objects/SimplicialComplex]] from other objects of the same type. |
| Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice. | \\ |
| | {{anchor:star_deletion:}} **''star_deletion([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]> face)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>'' ''face'' : specified by vertex indices.\\ Please use ''[[playground/playground/SimplicialComplex#labeled_vertices |labeled_vertices]]'' if you want to specify the face by vertex labels. |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Remove the star of a given //face//. |
| |
| == PURE == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ A simplicial complex is __pure__ if all its facets have the same dimension. | {{anchor:bistellar_simplification:}} **''bistellar_simplification([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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 ''[[playground/playground/SimplicialComplex#CLOSED_PSEUDO_MANIFOLD |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. |
| |
| == N_MINIMAL_NON_FACES == | \\ |
| ''Type:'' /common/property_types/Basic Types/Int\\ Number of [[/topaz/objects/SimplicialComplex/properties/Combinatorics/MINIMAL_NON_FACES]]. | {{anchor:simplicial_product:}} **''simplicial_product''** |
| |
| == F_VECTOR == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ f<sub>k</sub> is the number of k-faces, for k = 0,... , d, where d is the dimension. | {{anchor:connected_sum:}} **''connected_sum([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex1, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex2, [[playground/playground#Int |Int]] f1, [[playground/playground#Int |Int]] f2)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex1'' : |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex2'' : |
| | * ''[[playground/playground#Int |Int]]'' ''f1'' : default: 0 |
| | * ''[[playground/playground#Int |Int]]'' ''f2'' : default: 0 |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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. |
| | * //Example://\\ Glueing together two tori to make a genus 2 double torus, rotating the second one clockwise: \\ <code> > $cs = connected_sum(torus(),torus(),permutation=>[1,2,0]); |
| | > print $cs->SURFACE.','.$cs->GENUS; |
| | 1,2 |
| | </code> |
| |
| == MINIMAL_NON_FACES == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Inclusion minimal non-faces (vertex subsets which are not faces of the simplicial complex). | {{anchor:triang_neighborhood:}} **''triang_neighborhood([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Rational |Rational]] width)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Rational |Rational]]'' ''width'' : default: 0 |
| | * Create a triangulated tubular neighborhood of a [[playground/playground/SimplicialComplex#PURE |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 ''[[playground/playground/GeometricSimplicialComplex#COORDINATES |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. |
| |
| == H_VECTOR == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ The h-vector of the simplicial complex. | {{anchor:hom_poset:}} **''hom_poset''** |
| |
| == FOLDABLE == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if [[/topaz/objects/SimplicialComplex/properties/Combinatorics/GRAPH]] is ([[/topaz/objects/SimplicialComplex/properties/Combinatorics/DIM]] + 1)-colorable. | {{anchor:covering_relations:}} **''covering_relations([[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]> P)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]>'' ''P'' : |
| | * //Returns:// ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]>'' |
| | * Construct the covering relations of a poset |
| |
| == GRAPH == | \\ |
| ''Type:'' /graph/objects/Combinatorics/Graph\\ The subcomplex consisting of all 1-faces. | {{anchor:h_induced_quotient:}} **''h_induced_quotient([[playground/playground/SimplicialComplex# |SimplicialComplex]] C, [[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]> vertices)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''C'' : |
| | * ''[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>'' ''vertices'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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//. |
| |
| == VERTEX_INDICES == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Indices of the vertices from [[/topaz/objects/SimplicialComplex/properties/Input property/INPUT_FACES]]. That is, the map i \mapsto v_i. | {{anchor:sum_triangulation:}} **''sum_triangulation([[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]] P, [[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]] Q, [[playground/playground#IncidenceMatrix |IncidenceMatrix]] WebOfStars)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' ''P'' : first complex |
| | * ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' ''Q'' : second complex |
| | * ''[[playground/playground#IncidenceMatrix |IncidenceMatrix]]'' ''WebOfStars'' : Every row corresponds to a full dimensional simplex in P and every column to a full dimensional simplex in Q. |
| | * //Returns:// ''[[playground/playground/GeometricSimplicialComplex# |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 |
| |
| == BOUNDARY == | \\ |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Combinatorics/BOUNDARY\\ Codimension-1-faces of a [[/topaz/objects/SimplicialComplex/properties/Topology/PSEUDO_MANIFOLD]] which are contained in one facet only. | {{anchor:induced_subcomplex:}} **''induced_subcomplex([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]> vertices)''** |
| \\ __Properties of BOUNDARY:__\\ These properties capture combinatorial information of the object. | * //Parameters:// |
| Combinatorial properties only depend on combinatorial data of the object like, e.g., the face lattice. | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>'' ''vertices'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the subcomplex consisting of all faces which are contained in the given set of //vertices//. |
| |
| * **VERTEX_MAP**\\ ''Type:'' /common/property_types/Basic Types/Array\\ Maps vertices of the boundary complex to the corresponding ones in the supercomplex | \\ |
| | {{anchor:foldable_prism:}} **''foldable_prism([[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' ''complex'' : |
| | * //Returns:// ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' |
| | * Produce a __prism__ over a given ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]''. |
| |
| == HASSE_DIAGRAM == | \\ |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Combinatorics/HASSE_DIAGRAM\\ The face lattice of the simplical complex | {{anchor:edge_contraction:}} **''edge_contraction([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| organized as a directed graph. Each node corresponds to some face | * //Parameters:// |
| of the simplical complex. It is represented as the list of vertices | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| comprising the face. The outgoing arcs point to the containing faces | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| of the next dimension. An artificial top node is added to represent | * 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. |
| the entire complex. | |
| |
| == COLORING == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ A coloring of the vertices. | {{anchor:disjoint_union:}} **''disjoint_union([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex1, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex2)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex1'' : |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex2'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the __disjoint union__ of the two given complexes. |
| |
| == DUAL_GRAPH == | \\ |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Combinatorics/DUAL_GRAPH\\ The graph of facet neighborhood. | {{anchor:stellar_subdivision:}} **''stellar_subdivision''** |
| Two [[/topaz/objects/SimplicialComplex/properties/Combinatorics/FACETS]] are neighbors if they share a (d-1)-dimensional face. | |
| \\ __Properties of DUAL_GRAPH:__\\ | |
| * **COLORING**\\ ''Type:'' /common/property_types/Graph Types/NodeMap\\ UNDOCUMENTED | |
| |
| == PROJ_DICTIONARY == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ For each vertex the corresponding vertex of facet 0 with respect to the action of the group of projectivities. | {{anchor:web_of_stars:}} **''web_of_stars([[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]> poset_hom, [[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]<[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>>> star_shaped_balls, [[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>> triang)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>'' ''poset_hom'' : the poset homomorphism from stabbing order to star-shaped balls |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]<[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>>>'' ''star_shaped_balls'' : the collection of star-shaped balls of T |
| | * ''[[playground/playground#Array |Array]]<[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>>'' ''triang'' : the facets of the underlying triangulation of Q |
| | * //Returns:// ''[[playground/playground#IncidenceMatrix |IncidenceMatrix]]'' |
| | * Produce a web of stars from two given triangulations\\ and a map between them. |
| |
| == PROJ_ORBITS == | \\ |
| ''Type:'' /common/property_types/Set Types/PowerSet\\ Orbit decomposition of the group of projectivities acting on the set of vertices of facet 0. | {{anchor:barycentric_subdivision:}} **''barycentric_subdivision([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Computes the __barycentric subdivision__ of //complex//. |
| | * //Example://\\ To subdivide a triangle into six new triangles, do this: \\ <code> > $b = barycentric_subdivision(simplex(2)); |
| | </code> |
| |
| == FACETS == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Faces which are maximal with respect to inclusion, encoded as their ordered set of vertices. | {{anchor:union:}} **''union([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex1, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex2)''** |
| The vertices must be numbered 0, ..., n-1. | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex1'' : |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex2'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the union of the two given complexes, identifying\\ vertices with equal labels. |
| |
| == DIM == | \\ |
| ''Type:'' /common/property_types/Basic Types/Int\\ Maximal dimension of the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/FACETS]], where the dimension of a facet is defined as | {{anchor:cone:}} **''cone([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Int |Int]] k)''** |
| the number of its vertices minus one. | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Int |Int]]'' ''k'' : default is 1 |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the //k//-cone over a given simplicial complex. |
| | * //Example://\\ The following creates the cone with two apices over the triangle,\\ with custom apex labels. The resulting complex is the 4-simplex. \\ <code> > $c = cone(simplex(2),2,apex_labels=>['foo','bar']); |
| | > print $c->FACETS; |
| | {0 1 2 3 4} |
| | </code>\\ \\ <code> > print $c->VERTEX_LABELS; |
| | 0 1 2 foo bar |
| | </code> |
| |
| == SHELLABLE == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is shellable. | {{anchor:join_complexes:}} **''join_complexes([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex1, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex2)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex1'' : |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex2'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Creates the join of //complex1// and //complex2//. |
| |
| == N_FACETS == | \\ |
| ''Type:'' /common/property_types/Basic Types/Int\\ Number of [[/topaz/objects/SimplicialComplex/properties/Combinatorics/FACETS]]. | {{anchor:alexander_dual:}} **''alexander_dual([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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. |
| |
| == F2_VECTOR == | \\ |
| ''Type:'' /common/property_types/Algebraic Types/Matrix\\ f<sub>ik</sub> is the number of incident pairs of i-faces and k-faces; the main | {{anchor:deletion:}} **''deletion([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]> face)''** |
| diagonal contains the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/F_VECTOR]]. | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>'' ''face'' : specified by vertex indices.\\ Please use ''[[playground/playground/SimplicialComplex#labeled_vertices |labeled_vertices]]'' if you want to specify the face by vertex labels. |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Remove the given //face// and all the faces containing it. |
| |
| == ODD_SUBCOMPLEX == | \\ |
| ''Type:'' /topaz/objects/SimplicialComplex\\ Subcomplex generated by faces of codimension 2 that are contained in an odd | {{anchor:link_complex:}} **''link_complex([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]> face)''** |
| number of faces of codimension 1. | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>'' ''face'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the __link__ of a //face// of the //complex// |
| |
| == N_VERTICES == | \\ |
| ''Type:'' /common/property_types/Basic Types/Int\\ Number of vertices. | {{anchor:star:}} **''star([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]> face)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Set |Set]]<[[playground/playground#Int |Int]]>'' ''face'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the __star__ of the //face// of the //complex//. |
| |
| == SHELLING == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ An ordered list of facets constituting a shelling. | {{anchor:colored_ball_from_colored_sphere:}} **''colored_ball_from_colored_sphere([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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. |
| | |
| | \\ |
| | {{anchor:poset_homomorphisms:}} **''poset_homomorphisms([[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]> P, [[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]> Q)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]>'' ''P'' : |
| | * ''[[playground/playground/Graph# |Graph]]<[[playground/playground#Directed |Directed]]>'' ''Q'' : |
| | * //Returns:// ''[[playground/playground#Array |Array]]<[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>>'' |
| | * Enumerate all order preserving maps from one poset to another |
| | |
| | \\ |
| | {{anchor:iterated_barycentric_subdivision:}} **''iterated_barycentric_subdivision([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Int |Int]] k)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Int |Int]]'' ''k'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Computes the //k//-th __barycentric subdivision__ of //complex// by iteratively calling ''[[playground/playground#barycentric |barycentric]]''. |
| | |
| | \\ |
| | {{anchor:k_skeleton:}} **''k_skeleton''** |
| | |
| | \\ |
| | {{anchor:suspension:}} **''suspension([[playground/playground/SimplicialComplex# |SimplicialComplex]] complex, [[playground/playground#Int |Int]] k)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : |
| | * ''[[playground/playground#Int |Int]]'' ''k'' : default value is 1 |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the __//k//-suspension__ over a given simplicial complex. |
| | |
| | \\ |
| | {{anchor:bs2quotient:}} **''bs2quotient([[playground/playground/Polytope# |Polytope]] P, [[playground/playground/SimplicialComplex# |SimplicialComplex]] complex)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/Polytope# |Polytope]]'' ''P'' : the underlying polytope |
| | * ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' ''complex'' : a sufficiently fine subdivision of P, for example the second barycentric subdivision |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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. |
| |
| ---- | ---- |
| ==== Topology ==== | ==== Producing a simplicial complex from other objects ==== |
| The following properties are topological invariants. | These functions construct a new [[/topaz/objects/SimplicialComplex]] from other objects. |
| | \\ |
| | {{anchor:vietoris_rips_complex:}} **''vietoris_rips_complex([[playground/playground#Matrix |Matrix]] D, [[playground/playground#Rational |Rational]] delta)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Matrix |Matrix]]'' ''D'' : the "distance matrix" of the point set (can be upper triangular) |
| | * ''[[playground/playground#Rational |Rational]]'' ''delta'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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. |
| |
| == HOMOLOGY == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Reduced simplicial homology groups H<sub>0</sub>, ..., H<sub>d</sub> (integer coefficients), listed in increasing dimension order. | {{anchor:independence_complex:}} **''independence_complex([[playground/playground/Matroid# |Matroid]] matroid)''** |
| See [[/topaz/property_types/Topology/HomologyGroup]] for explanation of encoding of each group. | * //Parameters:// |
| | * ''[[playground/playground/Matroid# |Matroid]]'' ''matroid'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * Produce the __independence complex__ of a given matroid.\\ If //no_labels// is set to 1, the labels are not copied. |
| |
| == KNOT == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Edge-subset of a 3-sphere which is a knot or link, that is, a collection of pairwise disjoint cycles. | {{anchor:clique_complex:}} **''clique_complex([[playground/playground/Graph# |Graph]] graph)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/Graph# |Graph]]'' ''graph'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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. |
| | * //Example://\\ Create the clique complex of a simple graph with one 3-clique and\\ one 2-clique, not creating labels. \\ <code> > $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} |
| | </code> |
| |
| == COCYCLES == | ---- |
| ''Type:'' /common/property_types/Basic Types/Array\\ Representatives of cocycle groups, listed in increasing codimension order. | ==== Producing from scratch ==== |
| See [[/topaz/property_types/Topology/CycleGroup]] for explanation of encoding of each group. | With these clients you can create special examples of simplicial complexes and complexes belonging to parameterized families. |
| | \\ |
| | {{anchor:torus:}} **''torus''** |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * The Császár Torus. Geometric realization by Frank Lutz,\\ Electronic Geometry Model No. 2001.02.069 |
| |
| == CYCLES == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Representatives of cycle groups, listed in increasing dimension order. | {{anchor:simplex:}} **''simplex([[playground/playground#Int |Int]] d)''** |
| See [[/topaz/property_types/Topology/CycleGroup]] for explanation of encoding of each group. | * //Parameters:// |
| | * ''[[playground/playground#Int |Int]]'' ''d'' : dimension |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * A __simplex__ of dimension //d//. |
| |
| == EULER_CHARACTERISTIC == | \\ |
| ''Type:'' /common/property_types/Basic Types/Int\\ Reduced Euler characteristic. Alternating sum of the [[/topaz/objects/SimplicialComplex/properties/Combinatorics/F_VECTOR]] minus 1. | {{anchor:complex_projective_plane:}} **''complex_projective_plane''** |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * The complex projective plane with the vertex-minimal triangulation by Kühnel and Brehm. |
| | * //Example://\\ Construct the complex projective plane, store it in the variable $p2c, and print its homology group. \\ <code> > $p2c = complex_projective_plane(); |
| | > print $p2c->HOMOLOGY; |
| | ({} 0) |
| | ({} 0) |
| | ({} 1) |
| | ({} 0) |
| | ({} 1) |
| | </code> |
| |
| == SPHERE == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ Determines if this is homeomorphic to a sphere. | {{anchor:ball:}} **''ball([[playground/playground#Int |Int]] d)''** |
| In general, this is undecidable; therefore, the implementation depends on heuristics. | * //Parameters:// |
| May be true or false or undef (if heuristic does not succeed). | * ''[[playground/playground#Int |Int]]'' ''d'' : dimension |
| | * //Returns:// ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' |
| | * A //d//-dimensional __ball__, realized as the //d//-simplex. |
| | * //Example://\\ The following produces the 3-ball and stores it in the variable $b: \\ <code> > $b = ball(3); |
| | </code>\\ You can print the facets of the resulting simplicial complex like so: \\ <code> > print $b->FACETS; |
| | {0 1 2 3} |
| | </code> |
| |
| == INTERSECTION_FORM == | \\ |
| ''Type:'' /topaz/property_types/Topology/IntersectionForm\\ The integral quadratic form obtained from restricting the multiplication of the cohomology of a closed | {{anchor:cube_complex:}} **''cube_complex([[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]> x)''** |
| 4k-manifold to H^{2k} x H^{2k} -> H^{4k} = Z. As a quadratic form over the reals it is characterized | * //Parameters:// |
| by its dimension and its index of inertia (or, equivalenty, by the number of positive and negative ones | * ''[[playground/playground#Array |Array]]<[[playground/playground#Int |Int]]>'' ''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. |
| in its canonical form). An integral quadratic form is even if it takes values in 2Z. | * //Returns:// ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]<[[playground/playground#Rational |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. |
| | * //Example://\\ Arrange four triangulated 3-cubes to form a big 2 by 2 cube: \\ <code> > $cc = cube_complex([2,2,2]); |
| | > print $cc->description; |
| | 2x2x2 Pile of 3-dimensional triangulated cubes. |
| | </code> |
| |
| == FUNDAMENTAL_GROUP == | \\ |
| ''Type:'' /common/property_types/Basic Types/Pair\\ A finite representation of the fundamental group. | {{anchor:surface:}} **''surface([[playground/playground#Int |Int]] g)''** |
| The fundamental group is represented as a pair of an integer, | * //Parameters:// |
| the number of generators, and a list of relations. The generators are numbered | * ''[[playground/playground#Int |Int]]'' ''g'' : genus |
| consecutively starting with zero. A relation is encoded as a list of pairs, | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| each pair consisting of a generator and its exponent. | * Produce a __surface of genus //g//__. For //g// >= 0\\ the client produces an orientable surface, otherwise\\ it produces a non-orientable one. |
| |
| You may use the [[/topaz/objects/SimplicialComplex/methods/Topology/fundamental2gap]] method to produce a ''GAP'' file. | \\ |
| | {{anchor:rand_knot:}} **''rand_knot([[playground/playground#Int |Int]] n_edges)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Int |Int]]'' ''n_edges'' : |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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]<sup>3</sup>, 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. |
| |
| == PSEUDO_MANIFOLD == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/properties/Combinatorics/PURE]] simplicial complex with the property that each ridge is | {{anchor:klein_bottle:}} **''klein_bottle''** |
| contained in either one or two facets. | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * The Klein bottle. |
| |
| == LOCALLY_STRONGLY_CONNECTED == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if the vertex star of each vertex is [[/topaz/objects/SimplicialComplex/methods/Topology/DUAL_CONNECTED]]. | {{anchor:multi_associahedron_sphere:}} **''multi_associahedron_sphere([[playground/playground#Int |Int]] n, [[playground/playground#Int |Int]] k)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Int |Int]]'' ''n'' : the number of vertices of the polygon |
| | * ''[[playground/playground#Int |Int]]'' ''k'' : the number of diagonals that are allowed to mutually cross |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |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//_{2//n//}.\\ //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 //k//ν-1,\\ where the parameter ν=//n//-2//k//-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. |
| | * //Example://\\ The f-vector of Δ(9,3) is that of a neighborly polytope, since ν=2: \\ <code> > print multi_associahedron_sphere(9,3)->F_VECTOR; |
| | 9 36 84 117 90 30 |
| | </code> |
| | * //Example://\\ The option no_facets=>1 still leaves useful information: \\ <code> > $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) |
| | </code>\\ \\ <code> > 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 |
| | </code>\\ \\ <code> > print $s->get_attachment("K_PLUS_1_CROSSINGS")->size(); |
| | 28 |
| | </code> |
| |
| == STIEFEL_WHITNEY == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Mod 2 cycle representation of Stiefel-Whitney classes. Each cycle is represented as a set of simplices. | {{anchor:real_projective_plane:}} **''real_projective_plane''** |
| | * //Returns:// ''[[playground/playground/SimplicialComplex# |SimplicialComplex]]'' |
| | * The real projective plane with its unique minimal triangulation on six vertices. |
| |
| == ORIENTATION == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ An orientation of the facets of an [[/topaz/objects/SimplicialComplex/properties/Topology/ORIENTED_PSEUDO_MANIFOLD]], such that the induced orientations | {{anchor:unknot:}} **''unknot([[playground/playground#Int |Int]] m, [[playground/playground#Int |Int]] n)''** |
| of a common ridge of two neighboring facets cancel each other out. Each facet is marked with //true// | * //Parameters:// |
| if the orientation agrees with the (chosen) orientation of the first facet, and is marked with //false// otherwise. | * ''[[playground/playground#Int |Int]]'' ''m'' : |
| | * ''[[playground/playground#Int |Int]]'' ''n'' : |
| | * //Returns:// ''[[playground/playground/GeometricSimplicialComplex# |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 ''[[playground/playground/GeometricSimplicialComplex#COORDINATES |COORDINATES]]''. |
| |
| == COHOMOLOGY == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Reduced cohomology groups, listed in increasing codimension order. | {{anchor:sphere:}} **''sphere([[playground/playground#Int |Int]] d)''** |
| See [[/topaz/property_types/Topology/HomologyGroup]] for explanation of encoding of each group. | * //Parameters:// |
| | * ''[[playground/playground#Int |Int]]'' ''d'' : dimension |
| | * //Returns:// ''[[playground/playground/GeometricSimplicialComplex# |GeometricSimplicialComplex]]'' |
| | * The //d//-dimensional __sphere__, realized as the boundary of the (//d//+1)-simplex. |
| |
| == BALL == | ---- |
| ''Type:'' /common/property_types/Basic Types/Bool\\ Determines if this is homeomorphic to a ball. | ==== Producing other objects ==== |
| In general, this is undecidable; therefore, the implementation depends on heuristics. | Functions producing big objects which are not contained in application topaz. |
| May be true or false or undef (if heuristic does not succeed). | \\ |
| | {{anchor:secondary_polyhedron:}} **''secondary_polyhedron([[playground/playground/HyperbolicSurface# |HyperbolicSurface]] s, [[playground/playground#Int |Int]] depth)''** |
| | * //Parameters:// |
| | * ''[[playground/playground/HyperbolicSurface# |HyperbolicSurface]]'' ''s'' : |
| | * ''[[playground/playground#Int |Int]]'' ''depth'' : |
| | * //Returns:// ''[[playground/playground/Polytope# |Polytope]]<[[playground/playground#Float |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. |
| |
| == MANIFOLD == | ---- |
| ''Type:'' /common/property_types/Basic Types/Bool\\ Determines if this is a compact simplicial manifold with boundary. | ==== Topology ==== |
| Depends on heuristic [[/topaz/objects/SimplicialComplex/properties/Topology/SPHERE]] recognition. | The following functions compute topological invariants. |
| May be true or false or undef (if heuristic does not succeed). | \\ |
| | {{anchor:test:}} **''test''** |
| | * //Returns:// ''[[playground/playground#Int |Int]]'' |
| | * Calculate the betti numbers of a general chain complex over a field. |
| |
| == GENUS == | \\ |
| ''Type:'' /common/property_types/Basic Types/Int\\ The __genus__ of a surface. | {{anchor:homology_and_cycles:}} **''homology_and_cycles''** |
| |
| == FUNDAMENTAL_GROUP_GEN_LABELS == | \\ |
| ''Type:'' /common/property_types/Basic Types/Array\\ Labels of the generators of the [[/topaz/objects/SimplicialComplex/properties/Topology/FUNDAMENTAL_GROUP]]. | {{anchor:betti_numbers:}} **''betti_numbers''** |
| The labels can be chosen freely. If the [[/topaz/objects/SimplicialComplex/properties/Topology/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. | {{anchor:homology:}} **''homology''** |
| ''g3_6'' corresponds to the edge {3 6}. | |
| | \\ |
| | {{anchor:cap_product:}} **''cap_product([[playground/playground#CycleGroup |CycleGroup]]<[[playground/playground#cap |cap]] cocycles, [[playground/playground#CycleGroup |CycleGroup]]<[[playground/playground#cap |cap]] cycles)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#CycleGroup |CycleGroup]]<[[playground/playground#cap |cap]]'' ''cocycles'' : |
| | * ''[[playground/playground#CycleGroup |CycleGroup]]<[[playground/playground#cap |cap]]'' ''cycles'' : |
| | * //Returns:// ''[[playground/playground#Pair |Pair]]<[[playground/playground#CycleGroup |CycleGroup]]<[[playground/playground#cap |cap]]'' |
| | * Compute all cap products of cohomology and homology cycles in two given groups. |
| | * //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. \\ <code> > $s = surface(1); |
| | > $cp = cap_product($s->COCYCLES->[1],$s->CYCLES->[1]); |
| | </code> |
| | ===== Small Object Types ===== |
| | |
| | ---- |
| | ==== Topology ==== |
| | The following property_types are topological invariants. |
| | \\ |
| | {{anchor:filtration:}} **''Filtration<MatrixType>''** |
| | * //Template Parameters:// |
| | * ''MatrixType'' : |
| | * A filtration of chain complexes. |
| | * //Methods of Filtration:// |
| | * **''dim''** |
| | * //Returns:// ''[[playground/playground#Int |Int]]'' |
| | * Returns the dimension of the maximal cells in the last frame of the filtration. |
| | * **''n_cells''** |
| | * //Returns:// ''[[playground/playground#Int |Int]]'' |
| | * Returns the number of cells in the last frame of the filtration. |
| | * **''cells''** |
| | * //Returns:// ''[[playground/playground#Array |Array]]<[[playground/playground#Cell |Cell]]>'' |
| | * Returns the cells of the filtration, given as array of 3-tuples containing degree, dimension and\\ boundary matrix row number of the cell. |
| | * **''boundary_matrix([[playground/playground#Int |Int]] d, [[playground/playground#Int |Int]] t)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Int |Int]]'' ''d'' : |
| | * ''[[playground/playground#Int |Int]]'' ''t'' : |
| | * Returns the d-boundary matrix of the t-th frame of the filtration. |
| | * **''n_frames''** |
| | * //Returns:// ''[[playground/playground#Int |Int]]'' |
| | * Returns the number of frames in of the filtration. |
| |
| == ORIENTED_PSEUDO_MANIFOLD == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/properties/Topology/PSEUDO_MANIFOLD]] with top level homology isomorphic to Z. | {{anchor:chaincomplex:}} **''ChainComplex<MatrixType>''** |
| | * //Template Parameters:// |
| | * ''MatrixType'' : The type of the differential matrices. default: SparseMatrix<Integer> |
| | * 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. |
| | * //Example://\\ You can create a new ChainComplex by passing the Array of differential matrices (as maps via _left_ multiplication): \\ <code> > $cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]])); |
| | </code>\\ Note that this creates a ChainComplex consisting three differential matrices -- the trivial zeroth and last ones\\ are omitted in the constructor.\\ You can look at the boundary matrices: \\ <code> > print $cc->boundary_matrix(1); |
| | 2 0 |
| | </code>\\ The functions ''homology'', ''homology_and_cycles'' and ''betti_numbers'' can be used to analyse your complex. \\ <code> > print homology($cc,0); |
| | ({(2 1)} 1) |
| | ({} 0) |
| | </code> |
| | * //Methods of ChainComplex:// |
| | * **''boundary_matrix([[playground/playground#Int |Int]] d)''** |
| | * //Parameters:// |
| | * ''[[playground/playground#Int |Int]]'' ''d'' : |
| | * //Returns:// ''[[playground/playground#ChainComplex |ChainComplex]]'' |
| | * Returns the d-boundary matrix of the chain complex. |
| | * **''dim''** |
| | * //Returns:// ''[[playground/playground#Int |Int]]'' |
| | * Returns the number of non-empty modules in the complex. |
| |
| == MORSE_MATCHING == | \\ |
| ''Type:'' /topaz/objects/SimplicialComplex/properties/Topology/MORSE_MATCHING\\ Morse matching in the Hasse diagram of the simplicial complex | {{anchor:cell:}} **''Cell''** |
| \\ __Properties of MORSE_MATCHING:__\\ | * |
| * **MATCHING**\\ ''Type:'' /common/property_types/Graph Types/EdgeMap\\ The matching in the HasseDiagram of the SimplicialComplex | |
| |
| == CLOSED_PSEUDO_MANIFOLD == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/properties/Combinatorics/PURE]] simplicial complex with the property that each ridge is | {{anchor:cyclegroup:}} **''CycleGroup<Scalar>''** |
| contained in exactly two facets. | * //Template Parameters:// |
| | * ''Scalar'' : integer type of matrix elements |
| | * 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. |
| |
| == SURFACE == | \\ |
| ''Type:'' /common/property_types/Basic Types/Bool\\ True if this is a [[/topaz/objects/SimplicialComplex/methods/Topology/CONNECTED]] [[/topaz/objects/SimplicialComplex/properties/Topology/MANIFOLD]] of dimension 2. | {{anchor:intersectionform:}} **''IntersectionForm''** |
| | * Parity and signature of the intersection form of a closed oriented 4k-manifold.\\ See ''[[playground/playground/SimplicialComplex#INTERSECTION_FORM |INTERSECTION_FORM]]''. |
| |
| | \\ |
| | {{anchor:homologygroup:}} **''HomologyGroup''** |
| | * A group is encoded as a sequence ( { (t<sub>1</sub> m<sub>1</sub>) ... (t<sub>n</sub> m<sub>n</sub>) } f) of non-negative integers,\\ with t<sub>1</sub> > t<sub>2</sub> > ... > t<sub>n</sub> > 1, plus an extra non-negative integer f.\\ The group is isomorphic to (Z/t<sub>1</sub>)<sup>m<sub>1</sub></sup> × ... × (Z/t<sub>n</sub>)<sup>m<sub>n</sub></sup> × Z<sup>f</sup>, \\ where Z<sup>0</sup> is the trivial group.\\ Access methods: //torsion// delivers the list of Z-groups, //betti_number// the number f. |
| |