Simplicial Complexes

simplical_complex(generators::Union{Vector{Vector{Int}}, Vector{Set{Int}}})

Construct an abstract simplicial complex from a set of faces. While arbitrary non-negative integers are allowed as vertices, they will be relabeled to consecutive integers starting at 1.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]])
Abstract simplicial complex of dimension 2 on 4 vertices

julia> G = complete_bipartite_graph(2,3)
Undirected graph with 5 nodes and the following edges:
(3, 1)(3, 2)(4, 1)(4, 2)(5, 1)(5, 2)

julia> K = simplicial_complex(G)
Abstract simplicial complex of dimension 1 on 5 vertices

Simplicial complex comprising the empty set only:

julia> empty = simplicial_complex(Vector{Set{Int}}([]))
Abstract simplicial complex of dimension -1 on 0 vertices

The original vertices can be recovered:

julia> L = simplicial_complex([[0,2,17],[2,17,90]]);

julia> facets(L)
2-element Vector{Set{Int64}}:
 Set([2, 3, 1])
 Set([4, 2, 3])

julia> vertexindices(L)
4-element Vector{Int64}:
  0
  2
 17
 90

star_subcomplex(K::SimplicialComplex, sigma::Union{Vector{Int}, Set{Int}})

Return the star of the face sigma in the abstract simplicial complex K.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]]);

julia> star_subcomplex(K,[1])
Abstract simplicial complex of dimension 2 on 3 vertices

link_subcomplex(K::SimplicialComplex, sigma::Union{Vector{Int}, Set{Int}})

Return the link of the face sigma in the abstract simplicial complex K.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]]);

julia> link_subcomplex(K,[2,3])
Abstract simplicial complex of dimension 0 on 2 vertices

torus()

Construct Möbius' (vertex-minimal) 7-vertex triangulation of the torus (surface).

klein_bottle()

Construct a 9-vertex triangulation of the Klein bottle.

real_projective_plane()

Construct the (vertex-minimal) 6-vertex triangulation of the real projective plane.

complex_projective_plane()

Construct the (vertex-minimal) 9-vertex triangulation of the complex projective plane.

n_vertices(K::SimplicialComplex)

Return the number of vertices of the abstract simplicial complex K.

Examples

julia> n_vertices(torus())
7

n_facets(K::SimplicialComplex)

Return the number of facets of the abstract simplicial complex K.

dim(K::SimplicialComplex)

Return the dimension of the abstract simplicial complex K.

f_vector(K::SimplicialComplex)

Return the face vector (number of faces per dimension) of the abstract simplicial complex K.

Examples

julia> f_vector(torus())
3-element Vector{Int64}:
  7
 21
 14

h_vector(K::SimplicialComplex)

Return the h-vector of the abstract simplicial complex K.

Examples

julia> h_vector(torus())
4-element Vector{Int64}:
  1
  4
 10
 -1

euler_characteristic(K::SimplicialComplex)

Return the reduced Euler characteristic of the abstract simplicial complex K.

Examples

julia> euler_characteristic(complex_projective_plane())
2

homology(K::SimplicialComplex, i::Int)

Return i-th integral homology group of K.

Examples

julia> [ homology(real_projective_plane(), i) for i in [0,1,2] ]
3-element Vector{FinGenAbGroup}:
 Z
 Z/2
 Z/1

betti_numbers(K::SimplicialComplex)

Return the reduced rational Betti numbers of the abstract simplicial complex K.

Examples

julia> betti_numbers(klein_bottle())
3-element Vector{Int64}:
 0
 1
 0

cohomology(K::SimplicialComplex, i::Int)

Return i-th integral cohomology group of K.

Examples

julia> K = simplicial_complex([[0,1],[1,2],[0,2]]);

julia> cohomology(K,1)
Z

fundamental_group(K::SimplicialComplex)

Return the fundamental group of the abstract simplicial complex K.

Examples

julia> pi_1 = fundamental_group(torus());

julia> describe(pi_1)
"Z x Z"

is_sphere(K::SimplicialComplex)

Heuristically check if the abstract simplicial complex K is a combinatorial sphere; see [JLLT22]. Note that this is undecidable in general. Returns true if recognized as a sphere. Returns false if not a sphere. Returns nothing if heuristics unsuccessful.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]]);

julia> is_sphere(K)
false

is_ball(K::SimplicialComplex)

Heuristically check if the abstract simplicial complex K is a combinatorial ball; see [JLLT22]. Note that this is undecidable in general. Returns true if recognized as a ball. Returns false if not a ball. Returns nothing if heuristics unsuccessful.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]]);

julia> is_ball(K)
true

is_manifold(K::SimplicialComplex)

Check if the abstract simplicial complex K is a combinatorial manifold, possibly with boundary. Note that this is undecidable in general. Returns true if recognized as a manifold. Returns false if not a manifold. Returns nothing if heuristics unsuccessful.

Examples

julia> is_manifold(torus())
true

julia> is_manifold(simplicial_complex([[1,2],[2,3]]))
true

julia> is_manifold(simplicial_complex([[1,2],[2,3],[2,4]]))
false

minimal_nonfaces(K::SimplicialComplex)

Return the minimal non-faces of the abstract simplicial complex K.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]]);

julia> minimal_nonfaces(K)
1-element Vector{Set{Int64}}:
 Set([4, 1])

alexander_dual(K::SimplicialComplex)

Return the Alexander dual of the abstract simplicial complex K.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]]);

julia> alexander_dual(K)
Abstract simplicial complex of dimension 1 on 2 vertices

stanley_reisner_ideal(K::SimplicialComplex)

Return the Stanley-Reisner ideal of the abstract simplicial complex K.

Examples

julia> stanley_reisner_ideal(real_projective_plane())
Ideal generated by
  x1*x2*x3
  x1*x2*x4
  x1*x5*x6
  x2*x5*x6
  x1*x3*x6
  x1*x4*x5
  x3*x4*x5
  x3*x4*x6
  x2*x3*x5
  x2*x4*x6

stanley_reisner_ideal(R::MPolyRing, K::SimplicialComplex)

Return the Stanley-Reisner ideal of the abstract simplicial complex K, in the given ring R.

Examples

julia> R, _ = QQ[:a, :b, :c, :d, :e, :f];

julia> stanley_reisner_ideal(R, real_projective_plane())
Ideal generated by
  a*b*c
  a*b*d
  a*e*f
  b*e*f
  a*c*f
  a*d*e
  c*d*e
  c*d*f
  b*c*e
  b*d*f

stanley_reisner_ring(K::SimplicialComplex)

Return the Stanley-Reisner ring of the abstract simplicial complex K.

Examples

julia> K = simplicial_complex([[1,2,3],[2,3,4]]);

julia> stanley_reisner_ring(K)
(Quotient of multivariate polynomial ring by ideal (x1*x4), Map: multivariate polynomial ring -> quotient of multivariate polynomial ring)

stanley_reisner_ring(R::MPolyRing, K::SimplicialComplex)

Return the Stanley-Reisner ring of the abstract simplicial complex K, as a quotient of a given ring R.

Examples

julia>  R, _ = ZZ[:a, :b, :c, :d, :e, :f];

julia> stanley_reisner_ring(R, real_projective_plane())
(Quotient of multivariate polynomial ring by ideal (a*b*c, a*b*d, a*e*f, b*e*f, a*c*f, a*d*e, c*d*e, c*d*f, b*c*e, b*d*f), Map: R -> quotient of multivariate polynomial ring)

is_isomorphic(K1::SimplicialComplex, K2::SimplicialComplex)

Checks if the given simplicial complexes are isomorphic.

Examples

julia> K1 = simplicial_complex([[1,2,3],[2,3,4]]);

julia> K2 = simplicial_complex([[1,2,3],[2,3,4]]);

julia> is_isomorphic(K1, K2)
true

connected_sum(K1::SimplicialComplex, K2::SimplicialComplex, f1::Int=0, f2::Int=0)

Compute the connected sum of two abstract simplicial complexes. Parameters f1 and f2 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 increasing index order).

Examples

julia> K = torus();

julia> surface_genus_2 = connected_sum(K, K)
Abstract simplicial complex of dimension 2 on 11 vertices

julia> homology(surface_genus_2, 1)
Z^4

julia> is_manifold(surface_genus_2)
true

deletion(K::SimplicialComplex, face::Union{<:AbstractSet{Int},<:AbstractVector{Int}})

Remove the given face and all the faces containing it from an abstract simplicial complex K.

Examples

julia> K = simplicial_complex([[1, 2, 3], [2, 3, 4]]);

julia> K_with_deletion = deletion(K, Set([1, 2]));

julia> facets(K_with_deletion)
2-element Vector{Set{Int64}}:
 Set([3, 1])
 Set([4, 2, 3])

automorphism_group(K::SimplicialComplex; action=:on_vertices)

Given a simplicial complex K return its automorphism group as a PermGroup. The group can be returned as a subgroup of the permutation group of the vertices by passing :on_vertices to the action keyword argument or on the facets by passing :on_facets.

Examples

julia> K = simplicial_complex([[1, 2, 3], [2, 3, 4]])
Abstract simplicial complex of dimension 2 on 4 vertices

julia> automorphism_group(K)
Permutation group of degree 4

on_simplicial_complex(K::SimplicialComplex, g::PermGroupElem)

Given a simplicial complex K return the simplicial complex corresponding to a permutation on it's vertices given by g.

Examples

julia> K = simplicial_complex([[1, 2, 3], [2, 3, 4]])
Abstract simplicial complex of dimension 2 on 4 vertices

julia> G = automorphism_group(K)
Permutation group of degree 4

julia> g = collect(G)[2]
(1,4)

julia> facets(on_simplicial_complex(K, g))
2-element Vector{Set{Int64}}:
 Set([2, 3, 1])
 Set([4, 2, 3])