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:
uses:
GeometricSimplicialComplex:
A geometric simplicial complex, i.e., a simplicial complex with a geometric realization. Scalar is the numeric data type used for the coordinates.
GrassPluckerCertificate:
This object encodes a decorated graph that certifies that a given simplicial complex is not realizable as the boundary of a convex polytope. The simplicial complex itself should either be a simplicial sphere, or an oriented simplicial complex with boundary. In the latter case, each component of the boundary will be coned over by a new point, and the resulting complex should then be a simplicial sphere. (This is important when handling Criado & Santos's topological prismatoids, for example.) The nodes of the graph are decorated with Grassmann-Plücker polynomials, and the edges with “undetermined solids”, ie, solids whose orientation can vary according to the realization. The point of the certificate is that no matter how these undetermined solids are oriented, there will always be some Grassmann-Plücker polynomial in the tree all of whose terms are positive. But this contradicts realizability, because the matrix of homogeneous coordinates of any putative convex realization of a d-sphere on n vertices determines a point in the Grassmann manifold G(d,n), which means that all GP-polynomials should vanish – but the special one can't, because all its terms are positive.
HyperbolicSurface:
A hyperbolic surface (noncompact, finite area) is given by a triangulation DCEL (the topological data) and PENNER_COORDINATES (the metric data).
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:SimplicialComplex |greedy]]). The alternative is to use a canceling algorithm due to Forman ([[.:topaz:SimplicialComplex |cancel]]) or both ([[.:topaz: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:SimplicialComplex#HASSE_DIAGRAM |HASSE_DIAGRAM]]'' of a simplicial complex as a multi-layer graph.
These functions compare two SimplicialComplex
find_facet_vertex_permutations(SimplicialComplex complex1, SimplicialComplex complex2)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.
SimplicialComplex complex1
SimplicialComplex complex2
In the example below, two complexes are isomorphic, and the output shows first the facet permutation and then the vertex permutation of the isomorphism.
> $s1 = new SimplicialComplex(FACETS => [[0, 1], [0, 2], [1, 2], [2, 3]]); > $s2 = new SimplicialComplex(FACETS => [[0, 1], [0, 3], [1, 3], [2, 3]]); > print find_facet_vertex_permutations($s1, $s2); <0 1 2 3> <0 1 3 2>
isomorphic(SimplicialComplex complex1, SimplicialComplex complex2)Determine whether two given complexes are combinatorially isomorphic. The problem is reduced to graph isomorphism of the vertex-facet incidence graphs.
SimplicialComplex complex1: (or Polytope)
SimplicialComplex complex2: (or Polytope)
A minimal example of two complexes with the same f-vector, which are not isomorphic:
> $s1 = new SimplicialComplex(FACETS=>[[0,1],[0,2],[0,3]]); > $s2 = new SimplicialComplex(FACETS=>[[0,1],[1,2],[2,3]]); > print isomorphic($s1,$s2); false
> print isomorphic($s1,$s1); true
pl_homeomorphic(SimplicialComplex complex1, SimplicialComplex complex2)
Tries to determine whether two complexes are pl-homeomorphic by using bistellar flips and a simulated annealing strategy. You may specify the maximal number of rounds, how often the system may relax before heating up and how much heat should be applied. The function stops computing, once the size of the triangulation has not decreased for rounds iterations. If the abs flag is set, the function stops after rounds iterations regardless of when the last improvement took place. Additionally, you may set the threshold min_n_facets for the number of facets when the simplification ought to stop. Default is d+2 in the CLOSED_PSEUDO_MANIFOLD case and 1 otherwise. If you want to influence the distribution of the dimension of the moves when warming up you may do so by specifying a distribution. The number of values in distribution determines the dimensions used for heating up. The heating and relaxing parameters decrease dynamically unless the constant flag is set. The function prohibits to execute the reversed move of a move directly after the move itself unless the allow_rev_move flag is set. Setting the allow_rev_move flag might help solve a particular resilient problem. If you are interested in how the process is coming along, try the verbose option. It specifies after how many rounds the current best result is displayed. The obj determines the objective function used for the optimization. If obj is set to 0, the function searches for the triangulation with the lexicographically smallest f-vector, if obj is set to 1, the function searches for the triangulation with the reversed-lexicographically smallest f-vector and if obj is set to 2 the sum of the f-vector entries is used. The default is 1.
These functions construct a new SimplicialComplex from other objects of the same type.
alexander_dual(SimplicialComplex complex)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.
SimplicialComplex complex
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The following creates the alexander dual of a particular graph on 5 vertices.
> $s = new SimplicialComplex(INPUT_FACES=>[[0, 1], [1, 2], [0, 2], [0, 3], [4]]); > $t = alexander_dual($s); > print $t -> F_VECTOR; 5 10 6
barycentric_subdivision(SimplicialComplex complex)Computes the barycentric subdivision of complex.
SimplicialComplex complex
String pin_hasse_section: default: HASSE_DIAGRAM
String label_section: default: VERTEX_LABELS
String coord_section: default: VERTICES
Bool geometric_realization: set to 1 to obtain a GeometricSimplicialComplex, default: 0
To subdivide a triangle into six new triangles, do this:
> $b = barycentric_subdivision(simplex(2));
bistellar_simplification(SimplicialComplex complex)
Heuristic for simplifying the triangulation of the given manifold without changing its PL-type. The function uses bistellar flips and a simulated annealing strategy. You may specify the maximal number of rounds, how often the system may relax before heating up and how much heat should be applied. The function stops computing, once the size of the triangulation has not decreased for rounds iterations. If the abs flag is set, the function stops after rounds iterations regardless of when the last improvement took place. Additionally, you may set the threshold min_n_facets for the number of facets when the simplification ought to stop. Default is d+2 in the CLOSED_PSEUDO_MANIFOLD case and 1 otherwise. If you want to influence the distribution of the dimension of the moves when warming up you may do so by specifying a distribution. The number of values in distribution determines the dimensions used for heating up. The heating and relaxing parameters decrease dynamically unless the constant flag is set. The function prohibits to execute the reversed move of a move directly after the move itself unless the allow_rev_move flag is set. Setting the allow_rev_move flag might help solve a particular resilient problem. If you are interested in how the process is coming along, try the verbose option. It specifies after how many rounds the current best result is displayed. The obj determines the objective function used for the optimization. If obj is set to 0, the function searches for the triangulation with the lexicographically smallest f-vector, if obj is set to any other value the sum of the f-vector entries is used. The default is 1.
SimplicialComplex complex
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
The following example applies bistellar simplification to the second barycentric subdivision of the boundary of the 4-simplex to recover the boundary of the 4-simplex itself.
> $s = iterated_barycentric_subdivision(simplex(4) -> BOUNDARY, 2); > print bistellar_simplification($s) -> F_VECTOR; 5 10 10 5
bs2quotient_by_equivalence(Polytope P)Create a simplicial complex from a given complex by identifying vertices on the boundary of the second barycentric subdivision of the original complex according to some equivalence relation on faces.
Polytope P: the underlying polytope
bs2quotient_by_equivalence(Polytope P, SimplicialComplex complex)Create a simplicial complex from a simplicial subdivision of a given complex by identifying vertices on the boundary of the original complex according to some equivalence relation on faces.
Polytope P: the underlying polytope
SimplicialComplex complex: a sufficiently fine subdivision of P, for example the second barycentric subdivision
bs2quotient_by_group(Polytope P)Create a simplicial complex from a simplicial subdivision of a given complex by identifying vertices on the boundary of the second barycentric subdivision of the original complex according to a group that acts on vertices.
Polytope P: the underlying polytope
colored_ball_from_colored_sphere(SimplicialComplex complex)
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.
SimplicialComplex complex
Bool no_lables
cone(SimplicialComplex complex, Int k)Produce the k-cone over a given simplicial complex.
SimplicialComplex complex
Int k: default is 1
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The following creates the cone with two apices over the triangle, with custom apex labels. The resulting complex is the 4-simplex.
> $c = cone(simplex(2),2,apex_labels=>['foo','bar']); > print $c->FACETS; {0 1 2 3 4}
> print $c->VERTEX_LABELS; 0 1 2 foo bar
connected_sum(SimplicialComplex complex1, SimplicialComplex complex2, Int f1, Int f2)
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 one 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.
SimplicialComplex complex1
SimplicialComplex complex2
Int f1: default: 0
Int f2: default: 0
Bool no_labels
Glueing together two tori to make a genus 2 double torus, rotating the second one clockwise:
> $cs = connected_sum(torus(),torus(),permutation=>[1,2,0]); > print $cs->SURFACE.','.$cs->GENUS; 1,2
deletion(SimplicialComplex complex, Set<Int> face)Remove the given face and all the faces containing it.
SimplicialComplex complex
Set<Int> face: specified by vertex indices. Please use labeled_vertices if you want to specify the face by vertex labels.
Bool no_labels: Do not create VERTEX_LABELS. default: 0
Deleting any face of the 3-simplex yields a pure 2-dimensional complex with 3 facets:
> $s = deletion(simplex(3),[0,1,2]); > print $s->PURE, ', ', $s->DIM, ', ', $s->N_FACETS; true, 2, 3
disjoint_union(SimplicialComplex complex1, SimplicialComplex complex2)Produce the disjoint union of the two given complexes.
SimplicialComplex complex1
SimplicialComplex complex2
Bool no_labels: Do not create VERTEX_LABELS. default: 0 The vertex labels are built from the original labels with a suffix _1 or _2 appended.
The following creates the disjoint union of a triangle and an edge.
> $s = disjoint_union(simplex(2), simplex(1)); > print $s -> F_VECTOR; 5 4 1
edge_contraction(SimplicialComplex complex)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.
SimplicialComplex complex
Int seed
The following takes the second barycentric subdivision of the tetrahedron and performs edge contraction with a random order of the vertices. In the first instance the tetrahedron is recovered in the second it is not.
> $s = iterated_barycentric_subdivision(simplex(3) -> BOUNDARY, 2); > $t = edge_contraction($s, seed=>100); > $t1 = edge_contraction($s, seed => 101); > print $t -> F_VECTOR; 4 6 4
> print $t1 -> F_VECTOR; 6 12 8
foldable_prism(GeometricSimplicialComplex complex)
Produce a prism over a given SimplicialComplex.
GeometricSimplicialComplex complex
Bool geometric_realization
h_induced_quotient(SimplicialComplex C, Set<Int> vertices)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.
Bool no_labels: Do not create VERTEX_LABELS. default: 0
String apex
The following takes C to be the suspension over a triangle and A to be the set of vertices of that triangle. The quotient induced is homotopy equivalent to a wedge of spheres.
> $C = suspension(simplex(2) -> BOUNDARY); > $t = h_induced_quotient($C, [0, 1, 2]); > print $t -> HOMOLOGY; ({} 0) ({} 0) ({} 2)
induced_subcomplex(SimplicialComplex complex, Set<Int> vertices)Produce the subcomplex consisting of all faces which are contained in the given set of vertices.
SimplicialComplex complex
Bool no_labels: Do not create VERTEX_LABELS. default: 0
Bool geom_real: tells the client to inherit the COORDINATES.
The following takes C to be the suspension over a triangle and the vertices to be the vertices of that triangle.
> $C = suspension(simplex(2) -> BOUNDARY); > $t = induced_subcomplex($C, [0, 1, 2]); > print $t -> F_VECTOR; 3 3
iterated_barycentric_subdivision(SimplicialComplex complex, Int k)
Computes the k-th barycentric subdivision of complex by iteratively calling barycentric_subdivision.
SimplicialComplex complex
Int k
String pin_hasse_section: default: HASSE_DIAGRAM
String label_section: default: VERTEX_LABELS
String coord_section: default: VERTICES
Bool geometric_realization: set to 1 to obtain a GeometricSimplicialComplex, default: 0
The following applies barycentric subdivision to the triangle twice.
> $b = iterated_barycentric_subdivision(simplex(2), 2); > print $b -> F_VECTOR; 25 60 36
join_complexes(SimplicialComplex complex1, SimplicialComplex complex2)Creates the join of complex1 and complex2.
SimplicialComplex complex1
SimplicialComplex complex2
Bool no_labels: Do not create VERTEX_LABELS. default: 0 The vertex labels are built from the original labels with a suffix _1 or _2 appended.
The following constructs the tetrahedron as the join of two edges.
> $s = join_complexes(simplex(1), simplex(1)); > print $s -> F_VECTOR; 4 6 4 1
k_skeleton(SimplicialComplex complex, Int k)Produce the k-skeleton.
SimplicialComplex complex
Int k
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The 2-skeleton of the 3-simplex is its boundary, a 2-sphere:
> print isomorphic(k_skeleton(simplex(3),2), simplex(3)->BOUNDARY); true
k_skeleton(GeometricSimplicialComplex complex, Int k)Produce the k-skeleton.
GeometricSimplicialComplex complex
Int k
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The 2-skeleton of the 3-ball is its boundary, a 2-sphere:
> print isomorphic(k_skeleton(ball(3),2), ball(3)->BOUNDARY); true
link_subcomplex(SimplicialComplex complex, Set<Int> face)Produce the link of a face of the complex
SimplicialComplex complex
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The following returns the 4-cycle obtained as the link of vertex 0 in the suspension over the triangle.
> $s = suspension(simplex(2)->BOUNDARY); > $t = link_subcomplex($s, [0]); > print $t->F_VECTOR; 4 4
VERTEX_INDICES keep track of the embedding:
> $K = new SimplicialComplex(FACETS=>[[0,1,2,3],[1,2,3,4]]); > $lk_12 = link_subcomplex($K,[1,2]); > print $lk_12->FACETS->[0]; {0 1}
simplicial_product(SimplicialComplex complex1, SimplicialComplex complex2)Computes the simplicial product of two complexes. Vertex orderings may be given as options.
SimplicialComplex complex1
SimplicialComplex complex2
Bool geometric_realization: default 0
Bool color_cons
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The following returns the product of two edges.
> $s = simplicial_product(simplex(1), simplex(1)); > print $s -> F_VECTOR; 4 5 2
simplicial_product<Scalar>(GeometricSimplicialComplex complex1, GeometricSimplicialComplex complex2)Computes the simplicial product of two complexes. Vertex orderings may be given as options.
Scalar
GeometricSimplicialComplex complex1
GeometricSimplicialComplex complex2
Bool geometric_realization: default 1
Bool color_cons
Bool no_labels: Do not create VERTEX_LABELS. default: 0
GeometricSimplicialComplex<Scalar>
The following returns the product of the edges (0, 0)–(1, 0) and (0, 0) – (2, 0).
> $C = new GeometricSimplicialComplex(COORDINATES => [[0, 0], [1, 0]], FACETS => [[0, 1]]); > $C1 = new GeometricSimplicialComplex(COORDINATES => [[0, 2], [0, 0]], FACETS => [[0, 1]]); > $s = simplicial_product($C, $C1); > print $s -> COORDINATES; 0 0 0 2 1 0 0 2 0 0 0 0 1 0 0 0
star_deletion(SimplicialComplex complex, Set<Int> face)Remove the star of a given face.
SimplicialComplex complex
Set<Int> face: specified by vertex indices. Please use labeled_vertices if you want to specify the face by vertex labels.
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The following removes the star of the vertex 0 from the suspension over a triangle.
> $s = suspension(simplex(2) -> BOUNDARY); > $t = star_deletion($s, [0]); > print $t -> F_VECTOR; 4 5 2
star_subcomplex(SimplicialComplex complex, Set<Int> face)Produce the star of the face of the complex.
SimplicialComplex complex
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The following returns the cone over the 4-cycle obtained as the star of vertex 0 in the suspension over the triangle.
> $s = suspension(simplex(2) -> BOUNDARY); > $t = star_subcomplex($s, [0]); > print $t -> F_VECTOR; 5 8 4
stellar_subdivision(SimplicialComplex complex, Array<Set<Int>> faces)Computes the complex obtained by stellar subdivision of the given faces of the complex.
SimplicialComplex complex
Bool no_labels: Do not create VERTEX_LABELS. default: 0
Bool geometric_realization: default 0
stellar_subdivision(SimplicialComplex complex, Set<Int> face)Computes the complex obtained by stellar subdivision of the given face of the complex.
SimplicialComplex complex
Bool no_labels: Do not create VERTEX_LABELS. default: 0
Bool geometric_realization: default 0
sum_triangulation(GeometricSimplicialComplex P, GeometricSimplicialComplex Q, IncidenceMatrix WebOfStars)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
GeometricSimplicialComplex P: first complex
GeometricSimplicialComplex Q: second complex
IncidenceMatrix WebOfStars: Every row corresponds to a full dimensional simplex in P and every column to a full dimensional simplex in Q.
Bool origin_first: decides if the origin should be the first point in the resulting complex. Default=0
suspension(SimplicialComplex complex, Int k)Produce the k-suspension over a given simplicial complex.
SimplicialComplex complex
Int k: default value is 1
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The suspension of a 1-sphere is a 2-sphere:
> $s = new SimplicialComplex(FACETS=>[[0,1],[1,2],[2,0]]); > print suspension($s)->HOMOLOGY; ({} 0) ({} 0) ({} 1)
triang_neighborhood(SimplicialComplex complex, Rational width)
Create a triangulated tubular neighborhood of a pure 2-complex. If the complex is a link with the property that each vertex and its two neighbours are in general position after projection to the x,y-plane, then one might specify a rational number width to tell the client to compute COORDINATES of the triangulated tubular neighborhood. If the width/ is chosen too large, the computed realization will be self intersecting. If each connected component of the link has an even number of facets, then the following holds: An edge of the resulting complex is contained in an odd number of facets iff it corresponds to one of the edges of the link.
SimplicialComplex complex
Rational width: default: 0
union(SimplicialComplex complex1, SimplicialComplex complex2)Produce the union of the two given complexes, identifying vertices with equal labels.
SimplicialComplex complex1
SimplicialComplex complex2
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The union of two 3-simplices with the same labels on vertices produces the 3-simplex again.
> print union(simplex(3), simplex(3)) -> F_VECTOR; 4 6 4 1
web_of_stars(Array<Int> poset_hom, Array<Set<Set<Int>>> star_shaped_balls, Array<Set<Int>> triang)Produce a web of stars from two given triangulations and a map between them.
These functions construct a new SimplicialComplex from other objects.
broken_circuit_complex(Matroid M, Array<Int> O)Compute the broken circuit complex of a matroid. Given an ordering on the ground set of the matroid, a broken circuit is simply C{c}, where C is a circuit and c its maximal element under this ordering. The broken circuit complex of a matroid is then the abstract simplicial complex generated by those subsets of its ground set, which do not contain a broken circuit. Every such set must be a basis of the matroid.
Matroid M: the input matroid
A matroid with 3 bases {0,1}, {0,2}, and {1,2}. The only circuit is {0,1,2}, hence the only broken circuit (with the standard ordering) is {0,1}.
> $m = new matroid::Matroid(VECTORS=>[[1,0],[0,1],[1,1]]); > print broken_circuit_complex($m)->FACETS; {0 2} {1 2}
The same matroid, but with a different ordering on its ground set.
> $m = new matroid::Matroid(VECTORS=>[[1,0],[0,1],[1,1]]); > $ord = new Array<Int>(0,2,1); > print broken_circuit_complex($m, $ord)->FACETS; {0 1} {1 2}
clique_complex(Graph graph)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.
Graph graph
Bool no_labels: Do not create VERTEX_LABELS. default: 0
Create the clique complex of a simple graph with one 3-clique and one 2-clique, not creating labels.
> $g = graph_from_edges([[0,1],[0,2],[1,2],[2,3]]); > $c = clique_complex($g,no_labels=>1); > print $c->FACETS; {0 1 2} {2 3}
independence_complex(Matroid matroid)Produce the independence complex of a given matroid. If no_labels is set to 1, the labels are not copied.
Matroid matroid
Bool no_labels: Do not create VERTEX_LABELS. default: 0
The following example constructs the independence complex from a rank 3 matroid on 4 elements.
> $M = new matroid::Matroid(VECTORS=>[[1, 0, 0], [1, 0, 1], [1, 1, 0], [1, 0, 2]]); > print independence_complex($M) -> F_VECTOR; 4 6 3
vietoris_rips_complex(Matrix D, Rational delta)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.
Matrix D: the “distance matrix” of the point set (can be upper triangular)
Rational delta
The VR-complex from 3 points (vertices of a triangle with side lengths 3, 3, and 5) for different delta:
> print vietoris_rips_complex(new Matrix([[0,3,3],[0,0,5],[0,0,0]]), 2)->FACETS; {0} {1} {2}
> print vietoris_rips_complex(new Matrix([[0,3,3],[0,0,5],[0,0,0]]), 4)->FACETS; {0 1} {0 2}
> print vietoris_rips_complex(new Matrix([[0,3,3],[0,0,5],[0,0,0]]), 6)->FACETS; {0 1 2}
With these clients you can create special examples of simplicial complexes and complexes belonging to parameterized families.
ball(Int d)A d-dimensional ball, realized as the d-simplex.
complex_projective_plane()The complex projective plane with the vertex-minimal triangulation by Kühnel and Brehm.
Construct the complex projective plane, store it in the variable $p2c, and print its homology group.
> $p2c = complex_projective_plane(); > print $p2c->HOMOLOGY; ({} 0) ({} 0) ({} 1) ({} 0) ({} 1)
cube_complex(Array<Int> x)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.
Arrange four triangulated 3-cubes to form a big 2 by 2 cube:
> $cc = cube_complex([2,2,2]); > print $cc->description; 2x2x2 Pile of 3-dimensional triangulated cubes.
jockusch_3_ball(Int n)Create the ball B^{3,1}_n contained in Jockusch's centrally symmetric 3-sphere Delta^3_n on 2n vertices see Lemma 3.1 in arxiv.org/abs/2005.01155
jockusch_3_sphere(Int n)Create Jockusch's centrally symmetric 3-sphere Delta^3_n on 2n vertices see Lemma 3.1 in arxiv.org/abs/2005.01155
klein_bottle()The Klein bottle.
multi_associahedron_sphere(Int n, Int k)Produce the simplicial sphere Δ(n,k) of (k+1)-crossing free multitriangulations of an n-gon P, along with the group action on the diagonals induced from D_{2n}. Δ(n,k) is the simplicial complex on the set of relevant diagonals of P whose faces are those sets of diagonals such that no k+1 of them mutually cross. A diagonal is relevant if it leaves k or more vertices of P on both sides. (Any diagonal having less than k vertices on one side trivially cannot participate in a (k+1)-crossing, so is irrelevant. The corresponding complex on all diagonals is therefore the simplicial join of this one with the simplex of irrelevant diagonals.)
> Jakob Jonsson, “Generalized triangulations and diagonal-free subsets of stack polyominoes”,
> J. Combin. Theory Ser. A, 112(1):117–142, 2005 .. //Δ(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. ? Parameters: :: ''[[.:common#Int |Int]]'' ''n'': the number of vertices of the polygon :: ''[[.:common#Int |Int]]'' ''k'': the number of diagonals that are allowed to mutually cross ? Options: : :: ''[[.:common#Bool |Bool]]'' ''no_facets'': don't calculate the facets (for large examples)? Default 0 :: ''[[.:common#Bool |Bool]]'' ''no_crossings'': don't calculate the crossings? Default 0 ? Returns: :''[[.:topaz:SimplicialComplex |SimplicialComplex]]'' ? Example: :: The f-vector of Δ(9,3) is that of a neighborly polytope, since ν=2: :: <code perl> > print multi_associahedron_sphere(9,3)->F_VECTOR;
9 36 84 117 90 30 </code>
The option no_facets⇒1 still leaves useful information:
> $s = multi_associahedron_sphere(8,2, no_facets=>1); > print $s->VERTEX_LABELS; (0 3) (1 4) (2 5) (3 6) (4 7) (0 5) (1 6) (2 7) (0 4) (1 5) (2 6) (3 7)
> print $s->GROUP->PERMUTATION_ACTION->GENERATORS; 7 0 1 2 3 4 5 6 11 8 9 10 4 3 2 1 0 7 6 5 11 10 9 8
> print $s->get_attachment("K_PLUS_1_CROSSINGS")->size(); 28
nz_4_ball(Int n)Create the ball B^{4,1}_n contained in Novik & Zheng's centrally symmetric 4-sphere Delta^4_n on 2n vertices see arxiv.org/abs/2005.01155
nz_4_sphere(Int n)Create Novik & Zheng's centrally symmetric 4-sphere Delta^4_n on 2n vertices see arxiv.org/abs/2005.01155
poincare_sphere()The 16-vertex triangulation of the Poincaré homology 3-sphere by Björner and Lutz, Experimental Mathematics, Vol. 9 (2000), No. 2.
Print the face numbers.
> print poincare_sphere()->F_VECTOR; 16 106 180 90
rand_knot(Int n_edges)Produce a random knot (or link) as a polygonal closed curve in 3-space. The knot (or each connected component of the link) has n_edges edges. The vertices are uniformly distributed in [-1,1]3, unless the on_sphere option is set. In the latter case the vertices are uniformly distributed on the 3-sphere. Alternatively the brownian option produces a knot by connecting the ends of a simulated brownian motion.
real_projective_plane()The real projective plane with its unique minimal triangulation on six vertices.
sphere(Int d)The d-dimensional sphere, realized as the boundary of the (d+1)-simplex.
Int d: dimension
surface(Int g)Produce a surface of genus g. For g >= 0 the client produces an orientable surface, otherwise it produces a non-orientable one.
Int g: genus
torus()The Császár Torus. Geometric realization by Frank Lutz, Electronic Geometry Model No. 2001.02.069
unknot(Int m, Int n)
Produces a triangulated 3-sphere with the particularly NASTY embedding of the unknot in its 1-skeleton. The parameters m >= 2 and n >= 1 determine how entangled the unknot is. eps determines the COORDINATES.
Functions producing big objects which are not contained in application topaz.
outitudes(Matrix<Int> DCEL_data, Vector<Rational> A_coords)Computes the outitudes along edges.
In the following example only edge 2 has negative outitude.
> $T1 = new Matrix<Int>([[0,0,2,3,0,1],[0,0,4,5,0,1],[0,0,0,1,0,1]]); > print outitudes($T1,[1,2,3,4,5,6,1,2]); 37 37 -5
projective_potato(Matrix<Int> DCEL_data, Vector<Rational> A_coords, Matrix<Rational> first_two_vertices, Int depth)Computes the triangulated convex projective set that covers the convex RP^2 surface. The latter is given by the DCEL data for the triangulation of the surface along with A-coordinates (one positive Rational for each oriented edge and each triangle). Obviously, we only can compute a finite part of the infinite covering triangulation
Int depth
Bool lifted: for producing the lifted triangulation in R^3 with vertices in the light cone
The following computes a covering triangulation of a once punctured torus up to depth 5:
> $T1 = new Matrix<Int>([[0,0,2,3,0,1],[0,0,4,5,0,1],[0,0,0,1,0,1]]); > $p = projective_potato($T1,new Vector([1,1,1,1,1,1,2,2]),new Matrix([[1,0,0],[0,1,0]]),1); > print $p->VERTICES; 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 -1 1 -1/5 2/5 4/5 1 2/5 -1/5 4/5
secondary_polyhedron(HyperbolicSurface s, Int depth)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.
Int depth
These functions capture information of the object that is concerned with the action of permutation groups.
combinatorial_symmetries(SimplicialComplex sigma)Compute the combinatorial symmetries (i.e., automorphisms of the face lattice) of a given simplicial complex. They are stored in terms of a GROUP.RAYS_ACTION and a GROUP.FACETS_ACTION property, and the GROUP.RAYS_ACTION is also returned.
SimplicialComplex sigma
To get the vertex symmetry group of the square and print its generators, type the following:
> $sigma = new SimplicialComplex(FACETS=>[[0,1],[1,2],[2,3],[0,3]]); > print combinatorial_symmetries($sigma)->GENERATORS; 3 2 1 0 0 3 2 1
> print $sigma->GROUP->RAYS_ACTION->GENERATORS; 0 3 2 1 1 0 3 2
> print $sigma->GROUP->FACETS_ACTION->GENERATORS; 3 2 1 0 0 3 2 1
The following functions compute topological invariants.
betti_numbers<Coeff>(ChainComplex C)Calculate the betti numbers of a general chain complex over a field.
Coeff: The coefficient field type for homology computation. Defaults to Rational
The following constructs a simple chain complex with only one non-empty differential:
> $cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));
You can print its betti numbers like this:
> print betti_numbers($cc); 1 0
betti_numbers<Coeff>(SimplicialComplex S)Calculate the reduced betti numbers of a simplicial complex over a field.
Coeff: The coefficient field type for homology computation. Defaults to Rational
To print the betti numbers for the torus, do this:
> $t = torus(); > print betti_numbers($t); 0 2 1
cap_product(CycleGroup<E> cocycles, CycleGroup<E> cycles)Compute all cap products of cohomology and homology cycles in two given groups.
CycleGroup<E> cocycles
CycleGroup<E> cycles
The following stores all cap products of the cocycles and cycles of the homology group of the oriented surface of genus 1 in the variable $cp.
> $s = surface(1); > $cp = cap_product($s->COCYCLES->[1],$s->CYCLES->[1]);
homology(Array<Set<Int>> complex, Bool co)Calculate the reduced (co-)homology groups of a simplicial complex.
homology(ChainComplex CC, Bool co)Calculate the (co-)homology groups of a chain complex.
ChainComplex CC: The chain complex for which to compute homology.
Bool co: set to true for cohomology
Int dim_low: narrows the dimension range of interest, with negative values being treated as co-dimensions
Int dim_high: see dim_low
To construct a small chain complex with only one non-zero differential:
> $cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));
This prints its homology groups.
> print homology($cc,0); ({(2 1)} 1) ({} 0)
The output means that the zeroth homology group has 2-torsion with multiplicity one, and betti number one. The first homology group is empty.
homology_and_cycles(Array<Set<Int>> complex, Bool co)Calculate the reduced (co-)homology groups and cycle representatives of a simplicial complex.
homology_and_cycles(ChainComplex<SparseMatrix<Integer>> CC, Bool co)Calculate the (co-)homology groups and cycle coefficient matrices of a chain complex.
ChainComplex<SparseMatrix<Integer>> CC: The chain complex for which to compute homology.
Bool co: set to true for cohomology
Int dim_low: narrows the dimension range of interest, with negative values being treated as co-dimensions
Int dim_high: see dim_low
To construct a small chain complex with only one non-zero differential:
> $cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));
This prints its homology groups and corresponding generators.
> print homology_and_cycles($cc,0); (({(2 1)} 1) <1 0 0 1 > ) (({} 0) <> )
The output means that the zeroth homology group has 2-torsion with multiplicity one generated by the first elemen of the chain group, and free part of rank one generated by the second element. The first homology group is empty.
homology_flint(Array<Set<Int>> complex, Bool co)Calculate the reduced (co-)homology groups of a simplicial complex.
homology_flint(ChainComplex CC, Bool co)Calculate the (co-)homology groups of a chain complex.
ChainComplex CC: The chain complex for which to compute homology.
Bool co: set to true for cohomology
Int dim_low: narrows the dimension range of interest, with negative values being treated as co-dimensions
Int dim_high: see dim_low
To construct a small chain complex with only one non-zero differential:
> $cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));
This prints its homology groups.
> print homology_flint($cc,0); ({(2 1)} 1) ({} 0)
The output means that the zeroth homology group has 2-torsion with multiplicity one, and betti number one. The first homology group is empty.
Special purpose functions.
dualOutitudePolynomials(Matrix<Int> dcel_data)Given a triangulation of a punctured surface this calculates all the outitude polynomials of the dual structure. The first e = {oriented edges} monomials correspond to A-coordinates of the oriented edges of the primal structure , labeled as in the input. The last t = {triangles} monomials correspond to A-coordinates of the triangles of the primal structure.
flips_to_canonical_triangulation(Matrix<Int> DCEL_data, Vector<Rational> A_coords)Computes a flip sequence to a canonical triangulation (first list). The second output is a list of flat edges in the canonical triangulation.
In the following example only edge 2 has negative outitude, so the flip sequence should start with 2. After performing this flip, the triangulation thus obtained is canonical.
> $T1 = new Matrix<Int>([[0,0,2,3,0,1],[0,0,4,5,0,1],[0,0,0,1,0,1]]); > print flips_to_canonical_triangulation($T1,[1,2,3,4,5,6,1,2]); {2} {}
grass_plucker(SimplicialComplex S)Combinatorial search for Grassmann-Plucker tree certificates as described in J. Pfeifle, Positive Plücker tree certificates for non-realizability, Experimental Math. 2022 https://doi.org/10.1080/10586458.2021.1994487 and J. Pfeifle, A polymake implementation of Positive Plücker tree certificates for non-realizability, MEGA 2022
Int verbosity: default 1, max 2
Int max_n_undetermined: the maximal allowed number of undetermined solids in a GP relation. Default 3
Int abort_after: stop the computation after how many trees. Default 10000000
Int tree_log_interval: after how many trees a log message occurs. Default 10000
Int cube_log_interval: after how many cubes a log message occurs. Default 100
is_generalized_shelling(Array<Set> FaceList)Check if a given sequence of faces of a simplicial complex is a generalized shelling.
Bool verbose
is_vertex_decomposition(SimplicialComplex complex, Array<Int> vertices)Check whether a given ordered subset of the vertex set is a vertex decomposition. Works for 1-, 2- and 3-manifolds only!
SimplicialComplex complex
Bool verbose
mixed_graph(SimplicialComplex complex)Produces the mixed graph of a complex.
SimplicialComplex complex
Float edge_weight
outitudePolynomials(Matrix<Int> dcel_data)Given a triangulation of a punctured surface this calculates all the outitude polynomials. The first e = {oriented edges} monomials correspond to A-coordinates of the oriented edges, labeled as in the input. The last t = {triangles} monomials correspond to A-coordinates of the triangles.
We may calculate the outitude polynomials of a thrice punctured sphere. Here the first six monomials x_0, … , x_5 are associated to the six oriented edges, x_6 and x_7 are associated to the triangles enclosed by the oriented edges 0,2,4 and 1,3,5 respectively.
> $S3 = new Matrix<Int>([[1,0,2,5,0,1],[2,1,4,1,0,1],[0,2,0,3,0,1]]);; > print outitudePolynomials($S3); - x_0*x_1*x_6 - x_0*x_1*x_7 + x_0*x_2*x_6 + x_0*x_2*x_7 + x_1*x_5*x_6 + x_1*x_5*x_7 x_1*x_3*x_6 + x_1*x_3*x_7 - x_2*x_3*x_6 - x_2*x_3*x_7 + x_2*x_4*x_6 + x_2*x_4*x_7 x_0*x_4*x_6 + x_0*x_4*x_7 + x_3*x_5*x_6 + x_3*x_5*x_7 - x_4*x_5*x_6 - x_4*x_5*x_7
persistent_homology(Filtration<Matrix<Scalar>> F, Int i, Int p, Int k)Given a Filtration and three indices i,p and k, this computes the p-persistent k-th homology group of the i-th frame of the filtration for coefficients from any PID. Returns a basis for the free part and a list of torsion coefficients with bases.
Filtration<Matrix<Scalar>> F
Int i: the filtration frame
Int p: the number of frames to consider
Int k: the dimension in which to compute
Pair<SparseMatrix<Scalar>,List<Pair<Scalar,SparseMatrix<Scalar>>>>
persistent_homology(Filtration F)Given a Filtration, this computes its persistence barcodes in all dimension, using the algorithm described in the 2005 paper 'Computing Persistent Homology' by Afra Zomorodian and Gunnar Carlsson. It only works for field coefficients.
random_discrete_morse(SimplicialComplex complex)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)
SimplicialComplex complex
Int rounds: Run for r rounds
Int seed: Set seed number for random number generator
Int strategy: Set strategy⇒0 (default) for random-random: uniformly random selecting of a face to collapse or as critical face Set strategy⇒1 for random-lex-first: uniformly random relabeling of vertices, then selecting lexicographically first face for collapse or as a critical face Set strategy⇒2 for random-lex-last: uniformly random relabeling of vertices, then selecting lexicographically last face for collapse or as a critical face
Int verbose: v Prints message after running every v rounds
String save_collapsed: In every round, save all facets that remain after initial collapse in a data file as a SimplicialComplex. Rounds that have Morse vector [1,0,…,0] or [1,0,…,0,1] will save nothing. The actual file names are <filename>_<currentround>.top
The example below runs five rounds on the 5-simplex and in all cases returns a discrete Morse function with a single critical vertex.
> print random_discrete_morse(simplex(5), rounds => 5); {(<1 0 0 0 0 0> 5)}
second_barycentric_subdivision(Lattice L)Create the list of faces of the second barycentric subdivision
Lattice L: (for example, a HASSE_DIAGRAM)
second_barycentric_subdivision(Polytope P)Create the list of faces of the second barycentric subdivision
Polytope P: or SimplicialComplex S
stabbing_order(GeometricSimplicialComplex P)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
stanley_reisner(SimplicialComplex complex)Creates the Stanley-Reisner ideal of a simplicial complex.
SimplicialComplex complex
> $s = new SimplicialComplex(INPUT_FACES=>[[0, 1], [0, 2], [1, 2], [2, 3]]); > $i = stanley_reisner($s); > print $i -> GENERATORS; x_0*x_3 x_1*x_3 x_0*x_1*x_2
star_of_zero(GeometricSimplicialComplex C)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
star_shaped_balls(GeometricSimplicialComplex P)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
stiefel_whitney(Array<Set<Int>> facets)Computes Stiefel-Whitney homology classes of mod 2 Euler space (in particular, closed manifold). See Richard Z. Goldstein and Edward C. Turner, Proc. Amer. Math. Soc., 58:339-342 (1976) 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
vietoris_rips_filtration<Coeff>(Matrix D, Array<Int> deg, Float step_size, Int k)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 separate vertices of the complex. Setting k to |S| will compute the entire VR-Complex for each frame.
Coeff: desired coefficient type of the filtration
Matrix D: the “distance matrix” of the point set (can be upper triangular)
Float step_size
Int k: dimension of the resulting filtration
Filtration<SparseMatrix<Coeff,NonSymmetric>>
The following property_types are topological invariants.
CellChainComplex<MatrixType>A finite chain complex, represented as its boundary matrices. Check out the tutorial on the polymake homepage for examples on constructing ChainComplexes and computing their homology.
MatrixType: The type of the differential matrices. default: SparseMatrix<Integer>
You can create a new ChainComplex by passing the Array of differential matrices (as maps via _left_ multiplication):
> $cc = new ChainComplex(new Array<SparseMatrix<Integer>>([[[2,0]]]));
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:
> print $cc->boundary_matrix(1); 2 0
The functions homology, homology_and_cycles and betti_numbers can be used to analyse your complex.
> print homology($cc,0); ({(2 1)} 1) ({} 0)
CycleGroup<Scalar>A group is encoded as a pair of an integer matrix and a vector of faces. The elements of the group can be obtained by symbolic multiplication of both.
Scalar: integer type of matrix elements
coeff()the integer matrix
SparseMatrix<Scalar>
faces()the faces
Filtration<MatrixType>A filtration of chain complexes.
MatrixType
boundary_matrix(Int d, Int t)Returns the d-boundary matrix of the t-th frame of the filtration.
cells()Returns the cells of the filtration, given as array of 3-tuples containing degree, dimension and boundary matrix row number of the cell.
dim()Returns the dimension of the maximal cells in the last frame of the filtration.
n_cells()Returns the number of cells in the last frame of the filtration.
n_frames()Returns the number of frames in of the filtration.
HomologyGroup<Scalar>A finitely generated abelian group is encoded as a sequence ( { (t1 m1) … (tn mn) } f) of non-negative integers, with t1 > t2 > … > tn > 1, plus an extra non-negative integer f. That group is isomorphic to (Z/t1)m1 × … × (Z/tn)mn × Zf, where Z0 is the trivial group.
Scalar: integer type of torsion coefficients
betti_number()the number f
torsion()list of Z-groups
IntersectionForm
Parity and signature of the intersection form of a closed oriented 4k-manifold. See INTERSECTION_FORM.