Finite simplicial complexes

AUTHORS:

• John H. Palmieri (2009-04)

• D. Benjamin Antieau (2009-06): added is_connected, generated_subcomplex, remove_facet, and is_flag_complex methods; cached the output of the graph() method.

• Travis Scrimshaw (2012-08-17): Made SimplicialComplex have an immutable option, and added __hash__() function which checks to make sure it is immutable. Made SimplicialComplex.remove_face() into a mutator. Deprecated the vertex_set parameter.

• Christian Stump (2011-06): implementation of is_cohen_macaulay

• Travis Scrimshaw (2013-02-16): Allowed SimplicialComplex to make mutable copies.

• Simon King (2014-05-02): Let simplicial complexes be objects of the category of simplicial complexes.

• Jeremy Martin (2016-06-02): added cone_vertices, decone, is_balanced, is_partitionable, intersection methods

This module implements the basic structure of finite simplicial complexes. Given a set \(V\) of “vertices”, a simplicial complex on \(V\) is a collection \(K\) of subsets of \(V\) satisfying the condition that if \(S\) is one of the subsets in \(K\), then so is every subset of \(S\). The subsets \(S\) are called the ‘simplices’ of \(K\).

Note

In Sage, the elements of the vertex set are determined automatically: \(V\) is defined to be the union of the sets in \(K\). So in Sage’s implementation of simplicial complexes, every vertex is included in some face.

A simplicial complex \(K\) can be viewed as a purely combinatorial object, as described above, but it also gives rise to a topological space \(|K|\) (its geometric realization) as follows: first, the points of \(V\) should be in general position in euclidean space. Next, if \(\{v\}\) is in \(K\), then the vertex \(v\) is in \(|K|\). If \(\{v, w\}\) is in \(K\), then the line segment from \(v\) to \(w\) is in \(|K|\). If \(\{u, v, w\}\) is in \(K\), then the triangle with vertices \(u\), \(v\), and \(w\) is in \(|K|\). In general, \(|K|\) is the union of the convex hulls of simplices of \(K\). Frequently, one abuses notation and uses \(K\) to denote both the simplicial complex and the associated topological space.

For any simplicial complex \(K\) and any commutative ring \(R\) there is an associated chain complex, with differential of degree \(-1\). The \(n^{th}\) term is the free \(R\)-module with basis given by the \(n\)-simplices of \(K\). The differential is determined by its value on any simplex: on the \(n\)-simplex with vertices \((v_0, v_1, ..., v_n)\), the differential is the alternating sum with \(i^{th}\) summand \((-1)^i\) multiplied by the \((n-1)\)-simplex obtained by omitting vertex \(v_i\).

In the implementation here, the vertex set must be finite. To define a simplicial complex, specify its facets: the maximal subsets (with respect to inclusion) of the vertex set belonging to \(K\). Each facet can be specified as a list, a tuple, or a set.

Note

This class derives from GenericCellComplex, and so inherits its methods. Some of those methods are not listed here; see the Generic Cell Complex page instead.

EXAMPLES:

sage: SimplicialComplex([[1], [3, 7]])
Simplicial complex with vertex set (1, 3, 7) and facets {(1,), (3, 7)}
sage: SimplicialComplex()   # the empty simplicial complex
Simplicial complex with vertex set () and facets {()}
sage: X = SimplicialComplex([[0,1], [1,2], [2,3], [3,0]])
sage: X
Simplicial complex with vertex set (0, 1, 2, 3) and
 facets {(0, 1), (0, 3), (1, 2), (2, 3)}

Sage can perform a number of operations on simplicial complexes, such as the join and the product, and it can also compute homology:

sage: S = SimplicialComplex([[0,1], [1,2], [0,2]]) # circle
sage: T = S.product(S)  # torus
sage: T
Simplicial complex with 9 vertices and 18 facets
sage: T.homology()   # this computes reduced homology                               # needs sage.modules
{0: 0, 1: Z x Z, 2: Z}
sage: T.euler_characteristic()
0

Sage knows about some basic combinatorial data associated to a simplicial complex:

sage: X = SimplicialComplex([[0,1], [1,2], [2,3], [0,3]])
sage: X.f_vector()
[1, 4, 4]
sage: X.face_poset()
Finite poset containing 8 elements
sage: x0, x1, x2, x3 = X.stanley_reisner_ring().gens()                              # needs sage.libs.singular
sage: x0*x2 == x1*x3 == 0                                                           # needs sage.libs.singular
True
sage: X.is_pure()
True

Mutability (see github issue #12587):

sage: S = SimplicialComplex([[1,4], [2,4]])
sage: S.add_face([1,3])
sage: S.remove_face([1,3]); S
Simplicial complex with vertex set (1, 2, 3, 4) and facets {(3,), (1, 4), (2, 4)}
sage: hash(S)
Traceback (most recent call last):
...
ValueError: this simplicial complex must be immutable; call set_immutable()
sage: S = SimplicialComplex([[1,4], [2,4]])
sage: S.set_immutable()
sage: S.add_face([1,3])
Traceback (most recent call last):
...
ValueError: this simplicial complex is not mutable
sage: S.remove_face([1,3])
Traceback (most recent call last):
...
ValueError: this simplicial complex is not mutable
sage: hash(S) == hash(S)
True

sage: S2 = SimplicialComplex([[1,4], [2,4]], is_mutable=False)
sage: hash(S2) == hash(S)
True

We can also make mutable copies of an immutable simplicial complex (see github issue #14142):

sage: S = SimplicialComplex([[1,4], [2,4]])
sage: S.set_immutable()
sage: T = copy(S)
sage: T.is_mutable()
True
sage: S == T
True

class sage.topology.simplicial_complex.Simplex(X)

Bases: SageObject

Define a simplex.

Topologically, a simplex is the convex hull of a collection of vertices in general position. Combinatorially, it is defined just by specifying a set of vertices. It is represented in Sage by the tuple of the vertices.

Parameters:

X (integer, list, other iterable) – set of vertices

Returns:

simplex with those vertices

X may be a non-negative integer \(n\), in which case the simplicial complex will have \(n+1\) vertices \((0, 1, ..., n)\), or it may be anything which may be converted to a tuple, in which case the vertices will be that tuple. In the second case, each vertex must be hashable, so it should be a number, a string, or a tuple, for instance, but not a list.

Warning

The vertices should be distinct, and no error checking is done to make sure this is the case.

EXAMPLES:

sage: Simplex(4)
(0, 1, 2, 3, 4)
sage: Simplex([3, 4, 1])
(3, 4, 1)
sage: X = Simplex((3, 'a', 'vertex')); X
(3, 'a', 'vertex')
sage: X == loads(dumps(X))
True

Vertices may be tuples but not lists:

sage: Simplex([(1,2), (3,4)])
((1, 2), (3, 4))
sage: Simplex([[1,2], [3,4]])
Traceback (most recent call last):
...
TypeError: unhashable type: 'list'

alexander_whitney(dim)

Subdivide this simplex into a pair of simplices.

If this simplex has vertices \(v_0\), \(v_1\), …, \(v_n\), then subdivide it into simplices \((v_0, v_1, ..., v_{dim})\) and \((v_{dim}, v_{dim + 1}, ..., v_n)\).

INPUT:

• dim – integer between 0 and one more than the dimension of this simplex

OUTPUT:

• a list containing just the triple (1, left, right), where left and right are the two simplices described above.

This method allows one to construct a coproduct from the \(p+q\)-chains to the tensor product of the \(p\)-chains and the \(q\)-chains. The number 1 (a Sage integer) is the coefficient of left tensor right in this coproduct. (The corresponding formula is more complicated for the cubes that make up a cubical complex, and the output format is intended to be consistent for both cubes and simplices.)

Calling this method alexander_whitney is an abuse of notation, since the actual Alexander-Whitney map goes from \(C(X \times Y) \to C(X) \otimes C(Y)\), where \(C(-)\) denotes the chain complex of singular chains, but this subdivision of simplices is at the heart of it.

EXAMPLES:

sage: s = Simplex((0,1,3,4))
sage: s.alexander_whitney(0)
[(1, (0,), (0, 1, 3, 4))]
sage: s.alexander_whitney(2)
[(1, (0, 1, 3), (3, 4))]

dimension()

The dimension of this simplex.

The dimension of a simplex is the number of vertices minus 1.

EXAMPLES:

sage: Simplex(5).dimension() == 5
True
sage: Simplex(5).face(1).dimension()
4

face(n)

The \(n\)-th face of this simplex.

Parameters:

n (integer) – an integer between 0 and the dimension of this simplex

Returns:

the simplex obtained by removing the \(n\)-th vertex from this simplex

EXAMPLES:

sage: S = Simplex(4)
sage: S.face(0)
(1, 2, 3, 4)
sage: S.face(3)
(0, 1, 2, 4)

faces()

The list of faces (of codimension 1) of this simplex.

EXAMPLES:

sage: S = Simplex(4)
sage: S.faces()
[(1, 2, 3, 4), (0, 2, 3, 4), (0, 1, 3, 4), (0, 1, 2, 4), (0, 1, 2, 3)]
sage: len(Simplex(10).faces())
11

is_empty()

Return True iff this simplex is the empty simplex.

EXAMPLES:

sage: [Simplex(n).is_empty() for n in range(-1,4)]
[True, False, False, False, False]

is_face(other)

Return True iff this simplex is a face of other.

EXAMPLES:

sage: Simplex(3).is_face(Simplex(5))
True
sage: Simplex(5).is_face(Simplex(2))
False
sage: Simplex(['a', 'b', 'c']).is_face(Simplex(8))
False

join(right, rename_vertices=True)

The join of this simplex with another one.

The join of two simplices \([v_0, ..., v_k]\) and \([w_0, ..., w_n]\) is the simplex \([v_0, ..., v_k, w_0, ..., w_n]\).

Parameters:

• right – the other simplex (the right-hand factor)

• rename_vertices (boolean; optional, default True) – If this is True, the vertices in the join will be renamed by this formula: vertex “v” in the left-hand factor –> vertex “Lv” in the join, vertex “w” in the right-hand factor –> vertex “Rw” in the join. If this is false, this tries to construct the join without renaming the vertices; this may cause problems if the two factors have any vertices with names in common.

EXAMPLES:

sage: Simplex(2).join(Simplex(3))
('L0', 'L1', 'L2', 'R0', 'R1', 'R2', 'R3')
sage: Simplex(['a', 'b']).join(Simplex(['x', 'y', 'z']))
('La', 'Lb', 'Rx', 'Ry', 'Rz')
sage: Simplex(['a', 'b']).join(Simplex(['x', 'y', 'z']), rename_vertices=False)
('a', 'b', 'x', 'y', 'z')

product(other, rename_vertices=True)

The product of this simplex with another one, as a list of simplices.

Parameters:

• other – the other simplex

• rename_vertices (boolean; optional, default True) – If this is False, then the vertices in the product are the set of ordered pairs \((v,w)\) where \(v\) is a vertex in the left-hand factor (self) and \(w\) is a vertex in the right-hand factor (other). If this is True, then the vertices are renamed as “LvRw” (e.g., the vertex (1,2) would become “L1R2”). This is useful if you want to define the Stanley-Reisner ring of the complex: vertex names like (0,1) are not suitable for that, while vertex names like “L0R1” are.

Algorithm: see Hatcher, p. 277-278 [Hat2002] (who in turn refers to Eilenberg-Steenrod, p. 68): given S = Simplex(m) and T = Simplex(n), then \(S \times T\) can be triangulated as follows: for each path \(f\) from \((0,0)\) to \((m,n)\) along the integer grid in the plane, going up or right at each lattice point, associate an \((m+n)\)-simplex with vertices \(v_0\), \(v_1\), …, where \(v_k\) is the \(k^{th}\) vertex in the path \(f\).

Note that there are \(m+n\) choose \(n\) such paths. Note also that each vertex in the product is a pair of vertices \((v,w)\) where \(v\) is a vertex in the left-hand factor and \(w\) is a vertex in the right-hand factor.

Note

This produces a list of simplices – not a Simplex, not a SimplicialComplex.

EXAMPLES:

sage: len(Simplex(2).product(Simplex(2)))
6
sage: Simplex(1).product(Simplex(1))
[('L0R0', 'L0R1', 'L1R1'), ('L0R0', 'L1R0', 'L1R1')]
sage: Simplex(1).product(Simplex(1), rename_vertices=False)
[((0, 0), (0, 1), (1, 1)), ((0, 0), (1, 0), (1, 1))]

set()

The frozenset attached to this simplex.

EXAMPLES:

sage: Simplex(3).set()
frozenset({0, 1, 2, 3})

tuple()

The tuple attached to this simplex.

EXAMPLES:

sage: Simplex(3).tuple()
(0, 1, 2, 3)

Although simplices are printed as if they were tuples, they are not the same type:

sage: type(Simplex(3).tuple())
<... 'tuple'>
sage: type(Simplex(3))
<class 'sage.topology.simplicial_complex.Simplex'>

class sage.topology.simplicial_complex.SimplicialComplex(maximal_faces=None, from_characteristic_function=None, maximality_check=True, sort_facets=None, name_check=False, is_mutable=True, is_immutable=False, category=None)

Bases: Parent, GenericCellComplex

Define a simplicial complex.

Parameters:

• maximal_faces – set of maximal faces

• from_characteristic_function – see below

• maximality_check (boolean; optional, default True) – see below

• sort_facets (dict) – see below

• name_check (boolean; optional, default False) – see below

• is_mutable (boolean; optional, default True) – Set to False to make this immutable

• category (category; optional, default finite simplicial complexes) – the category of the simplicial complex

Returns:

a simplicial complex

maximal_faces should be a list or tuple or set (indeed, anything which may be converted to a set) whose elements are lists (or tuples, etc.) of vertices. Maximal faces are also known as ‘facets’. maximal_faces can also be a list containing a single non-negative integer \(n\), in which case this constructs the simplicial complex with a single \(n\)-simplex as the only facet.

Alternatively, the maximal faces can be defined from a monotone boolean function on the subsets of a set \(X\). While defining maximal_faces=None, you can thus set from_characteristic_function=(f,X) where X is the set of points and f a boolean monotone hereditary function that accepts a list of elements from X as input (see subsets_with_hereditary_property() for more information).

If maximality_check is True, check that each maximal face is, in fact, maximal. In this case, when producing the internal representation of the simplicial complex, omit those that are not. It is highly recommended that this be True; various methods for this class may fail if faces which are claimed to be maximal are in fact not.

sort_facets: if not set to None, the default, this should be a dictionary, used for sorting the vertices in each facet. The keys must be the vertices for the simplicial complex, and the values should be distinct sortable objects, for example integers. This should not need to be specified except in very special circumstances; currently the only use in the Sage library is when defining the product of a simplicial complex with itself: in this case, the vertices in the product must be sorted compatibly with the vertices in each factor so that the diagonal map is properly defined.

If name_check is True, check the names of the vertices to see if they can be easily converted to generators of a polynomial ring – use this if you plan to use the Stanley-Reisner ring for the simplicial complex.

EXAMPLES:

sage: SimplicialComplex([[1,2], [1,4]])
Simplicial complex with vertex set (1, 2, 4) and facets {(1, 2), (1, 4)}
sage: SimplicialComplex([[0,2], [0,3], [0]])
Simplicial complex with vertex set (0, 2, 3) and facets {(0, 2), (0, 3)}
sage: SimplicialComplex([[0,2], [0,3], [0]], maximality_check=False)
Simplicial complex with vertex set (0, 2, 3) and facets {(0,), (0, 2), (0, 3)}

Finally, if the first argument is a simplicial complex, return that complex. If it is an object with a built-in conversion to simplicial complexes (via a _simplicial_ method), then the resulting simplicial complex is returned:

sage: S = SimplicialComplex([[0,2], [0,3], [0,6]])
sage: SimplicialComplex(S) == S
True
sage: Tc = cubical_complexes.Torus(); Tc
Cubical complex with 16 vertices and 64 cubes
sage: Ts = SimplicialComplex(Tc); Ts
Simplicial complex with 16 vertices and 32 facets
sage: Ts.homology()                                                             # needs sage.modules
{0: 0, 1: Z x Z, 2: Z}

In the situation where the first argument is a simplicial complex or another object with a built-in conversion, most of the other arguments are ignored. The only exception is is_mutable:

sage: S.is_mutable()
True
sage: SimplicialComplex(S, is_mutable=False).is_mutable()
False

From a characteristic monotone boolean function, e.g. the simplicial complex of all subsets \(S\subseteq \{0,1,2,3,4\}\) such that \(sum(S)\leq 4\):

sage: SimplicialComplex(from_characteristic_function=(lambda x:sum(x)<=4, range(5)))
Simplicial complex with vertex set (0, 1, 2, 3, 4) and facets {(0, 4), (0, 1, 2), (0, 1, 3)}

or e.g. the simplicial complex of all 168 hyperovals of the projective plane of order 4:

sage: l = designs.ProjectiveGeometryDesign(2, 1, GF(4,name='a'))                # needs sage.rings.finite_rings
sage: f = lambda S: not any(len(set(S).intersection(x))>2 for x in l)
sage: SimplicialComplex(from_characteristic_function=(f, l.ground_set()))       # needs sage.rings.finite_rings
Simplicial complex with 21 vertices and 168 facets

Warning

Simplicial complexes are not proper parents as they do not possess element classes. In particular, parents are assumed to be hashable (and hence immutable) by the coercion framework. However this is close enough to being a parent with elements being the faces of self that we currently allow this abuse.

F_triangle(S)

Return the F-triangle of self with respect to one maximal simplex S.

This is the bivariate generating polynomial of all faces, according to the number of elements in S and outside S.

OUTPUT:

an F_triangle

See also

Not to be confused with f_triangle() .

EXAMPLES:

sage: cs = simplicial_complexes.Torus()
sage: cs.F_triangle(cs.facets()[0])                                         # needs sage.combinat
F: x^3 + 9*x^2*y + 3*x*y^2 + y^3 + 6*x^2 + 12*x*y
+ 3*y^2 + 4*x + 3*y + 1

add_face(face)

Add a face to this simplicial complex.

Parameters:

face – a subset of the vertex set

This changes the simplicial complex, adding a new face and all of its subfaces.

EXAMPLES:

sage: X = SimplicialComplex([[0,1], [0,2]])
sage: X.add_face([0,1,2,]); X
Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}
sage: Y = SimplicialComplex(); Y
Simplicial complex with vertex set () and facets {()}
sage: Y.add_face([0,1])
sage: Y.add_face([1,2,3])
sage: Y
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1), (1, 2, 3)}

If you add a face which is already present, there is no effect:

sage: Y.add_face([1,3]); Y
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1), (1, 2, 3)}

alexander_dual(is_mutable=True)

The Alexander dual of this simplicial complex: according to the Macaulay2 documentation, this is the simplicial complex whose faces are the complements of its nonfaces.

Thus find the minimal nonfaces and take their complements to find the facets in the Alexander dual.

Parameters:

is_mutable (boolean; optional, default True) – Determines if the output is mutable

EXAMPLES:

sage: Y = SimplicialComplex([[i] for i in range(5)]); Y
Simplicial complex with vertex set (0, 1, 2, 3, 4) and
 facets {(0,), (1,), (2,), (3,), (4,)}
sage: Y.alexander_dual()
Simplicial complex with vertex set (0, 1, 2, 3, 4) and 10 facets
sage: X = SimplicialComplex([[0,1], [1,2], [2,3], [3,0]])
sage: X.alexander_dual()
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2), (1, 3)}

alexander_whitney(simplex, dim_left)

Subdivide this simplex into a pair of simplices.

If this simplex has vertices \(v_0\), \(v_1\), …, \(v_n\), then subdivide it into simplices \((v_0, v_1, ..., v_{dim})\) and \((v_{dim}, v_{dim + 1}, ..., v_n)\).

See Simplex.alexander_whitney() for more details. This method just calls that one.

INPUT:

• simplex – a simplex in this complex

• dim – integer between 0 and one more than the dimension of this simplex

OUTPUT: a list containing just the triple (1, left, right), where left and right are the two simplices described above.

EXAMPLES:

sage: s = Simplex((0,1,3,4))
sage: X = SimplicialComplex([s])
sage: X.alexander_whitney(s, 0)
[(1, (0,), (0, 1, 3, 4))]
sage: X.alexander_whitney(s, 2)
[(1, (0, 1, 3), (3, 4))]

algebraic_topological_model(base_ring=None)

Algebraic topological model for this simplicial complex with coefficients in base_ring.

The term “algebraic topological model” is defined by Pilarczyk and Réal [PR2015].

INPUT:

• base_ring - coefficient ring (optional, default QQ). Must be a field.

Denote by \(C\) the chain complex associated to this simplicial complex. The algebraic topological model is a chain complex \(M\) with zero differential, with the same homology as \(C\), along with chain maps \(\pi: C \to M\) and \(\iota: M \to C\) satisfying \(\iota \pi = 1_M\) and \(\pi \iota\) chain homotopic to \(1_C\). The chain homotopy \(\phi\) must satisfy

• \(\phi \phi = 0\),

• \(\pi \phi = 0\),

• \(\phi \iota = 0\).

Such a chain homotopy is called a chain contraction.

OUTPUT: a pair consisting of

• chain contraction phi associated to \(C\), \(M\), \(\pi\), and \(\iota\)

• the chain complex \(M\)

Note that from the chain contraction phi, one can recover the chain maps \(\pi\) and \(\iota\) via phi.pi() and phi.iota(). Then one can recover \(C\) and \(M\) from, for example, phi.pi().domain() and phi.pi().codomain(), respectively.

EXAMPLES:

sage: # needs sage.modules
sage: RP2 = simplicial_complexes.RealProjectivePlane()
sage: phi, M = RP2.algebraic_topological_model(GF(2))
sage: M.homology()
{0: Vector space of dimension 1 over Finite Field of size 2,
 1: Vector space of dimension 1 over Finite Field of size 2,
 2: Vector space of dimension 1 over Finite Field of size 2}
sage: T = simplicial_complexes.Torus()
sage: phi, M = T.algebraic_topological_model(QQ)
sage: M.homology()
{0: Vector space of dimension 1 over Rational Field,
 1: Vector space of dimension 2 over Rational Field,
 2: Vector space of dimension 1 over Rational Field}

automorphism_group()

Return the automorphism group of the simplicial complex.

This is done by creating a bipartite graph, whose vertices are vertices and facets of the simplicial complex, and computing its automorphism group.

Warning

Since github issue #14319 the domain of the automorphism group is equal to the graph’s vertex set, and the translation argument has become useless.

EXAMPLES:

sage: S = simplicial_complexes.Simplex(3)
sage: S.automorphism_group().is_isomorphic(SymmetricGroup(4))               # needs sage.groups
True

sage: P = simplicial_complexes.RealProjectivePlane()
sage: P.automorphism_group().is_isomorphic(AlternatingGroup(5))             # needs sage.groups
True

sage: Z = SimplicialComplex([['1','2'],['2','3','a']])
sage: Z.automorphism_group().is_isomorphic(CyclicPermutationGroup(2))       # needs sage.groups
True
sage: group = Z.automorphism_group()                                        # needs sage.groups
sage: sorted(group.domain())                                                # needs sage.groups
['1', '2', '3', 'a']

Check that github issue #17032 is fixed:

sage: s = SimplicialComplex([[(0,1),(2,3)]])
sage: s.automorphism_group().cardinality()                                  # needs sage.groups
2

barycentric_subdivision()

The barycentric subdivision of this simplicial complex.

See Wikipedia article Barycentric_subdivision for a definition.

EXAMPLES:

sage: triangle = SimplicialComplex([[0,1], [1,2], [0, 2]])
sage: hexagon = triangle.barycentric_subdivision(); hexagon
Simplicial complex with 6 vertices and 6 facets
sage: hexagon.homology(1) == triangle.homology(1)                           # needs sage.modules
True

Barycentric subdivisions can get quite large, since each \(n\)-dimensional facet in the original complex produces \((n+1)!\) facets in the subdivision:

sage: S4 = simplicial_complexes.Sphere(4); S4
Minimal triangulation of the 4-sphere
sage: S4.barycentric_subdivision()
Simplicial complex with 62 vertices and 720 facets

bigraded_betti_number(a, b, base_ring=Integer Ring, verbose=False)

Return the bigraded Betti number indexed in the form \((-a, 2b)\).

Bigraded Betti number with indices \((-a, 2b)\) is defined as a sum of ranks of \((b-a-1)\)-th (co)homologies of full subcomplexes with exactly \(b\) vertices.

INPUT:

• base_ring – (default: ZZ) the base ring used when computing homology

• verbose – (default: False) if True, print messages during the computation, which indicate in which subcomplexes non-trivial homologies appear

Note

If verbose is True, then caching is avoided.

EXAMPLES:

sage: # needs sage.modules
sage: X = SimplicialComplex([[0,1],[1,2],[2,0],[1,3]])
sage: X.bigraded_betti_number(-1, 4, base_ring=QQ)
2
sage: X.bigraded_betti_number(-1, 8)
0
sage: X.bigraded_betti_number(-2, 5)
0
sage: X.bigraded_betti_number(0, 0)
1
sage: sorted(X.bigraded_betti_numbers().items(), reverse=True)
[((0, 0), 1), ((-1, 6), 1), ((-1, 4), 2), ((-2, 8), 1), ((-2, 6), 1)]
sage: X.bigraded_betti_number(-1, 4, base_ring=QQ)
2
sage: X.bigraded_betti_number(-1, 8)
0
sage: Y = SimplicialComplex([[1,2,3],[1,2,4],[3,5],[4,5]])
sage: Y.bigraded_betti_number(-1, 4, verbose=True)
Non-trivial homology Z in dimension 0 of the full subcomplex
generated by a set of vertices (1, 5)
Non-trivial homology Z in dimension 0 of the full subcomplex
generated by a set of vertices (2, 5)
Non-trivial homology Z in dimension 0 of the full subcomplex
generated by a set of vertices (3, 4)
3

bigraded_betti_numbers(base_ring=Integer Ring, verbose=False)

Return a dictionary of the bigraded Betti numbers of self, with keys \((-a, 2b)\).

INPUT:

• base_ring – (default: ZZ) the base ring used when computing homology

• verbose – (default: False) if True, print messages during the computation, which indicate in which subcomplexes non-trivial homologies appear

Note

If verbose is True, then caching is avoided.

See also

See bigraded_betti_number() for more information.

EXAMPLES:

sage: X = SimplicialComplex([[0,1],[1,2],[1,3],[2,3]])
sage: Y = SimplicialComplex([[1,2,3],[1,2,4],[3,5],[4,5]])
sage: sorted(X.bigraded_betti_numbers(base_ring=QQ).items(), reverse=True)
[((0, 0), 1), ((-1, 6), 1), ((-1, 4), 2), ((-2, 8), 1), ((-2, 6), 1)]
sage: sorted(Y.bigraded_betti_numbers(verbose=True).items(), reverse=True)
(-1, 4): Non-trivial homology Z in dimension 0 of the full
subcomplex generated by a set of vertices (1, 5)
(-1, 4): Non-trivial homology Z in dimension 0 of the full
subcomplex generated by a set of vertices (2, 5)
(-1, 4): Non-trivial homology Z in dimension 0 of the full
subcomplex generated by a set of vertices (3, 4)
(-2, 6): Non-trivial homology Z in dimension 0 of the full
subcomplex generated by a set of vertices (1, 2, 5)
(-2, 8): Non-trivial homology Z in dimension 1 of the full
subcomplex generated by a set of vertices (1, 3, 4, 5)
(-2, 8): Non-trivial homology Z in dimension 1 of the full
subcomplex generated by a set of vertices (2, 3, 4, 5)
(-3, 10): Non-trivial homology Z in dimension 1 of the full
subcomplex generated by a set of vertices (1, 2, 3, 4, 5)
[((0, 0), 1), ((-1, 4), 3), ((-2, 8), 2), ((-2, 6), 1), ((-3, 10), 1)]

If we wish to view them in a form of a table, it is simple enough to create a function as such:

sage: def print_table(bbns):
....:     max_a = max(-p[0] for p in bbns)
....:     max_b = max(p[1] for p in bbns)
....:     bbn_table = [[bbns.get((-a,b), 0) for a in range(max_a+1)]
....:                                       for b in range(max_b+1)]
....:     width = len(str(max(bbns.values()))) + 1
....:     print(' '*width, end=' ')
....:     for i in range(max_a+1):
....:         print(f'{-i:{width}}', end=' ')
....:     print()
....:     for j in range(len(bbn_table)):
....:         print(f'{j:{width}}', end=' ')
....:         for r in bbn_table[j]:
....:             print(f'{r:{width}}', end=' ')
....:         print()
sage: print_table(X.bigraded_betti_numbers())
    0 -1 -2
 0  1  0  0
 1  0  0  0
 2  0  0  0
 3  0  0  0
 4  0  2  0
 5  0  0  0
 6  0  1  1
 7  0  0  0
 8  0  0  1

cells(subcomplex=None)

The faces of this simplicial complex, in the form of a dictionary of sets keyed by dimension. If the optional argument subcomplex is present, then return only the faces which are not in the subcomplex.

Parameters:

subcomplex (optional, default None) – a subcomplex of this simplicial complex. Return faces which are not in this subcomplex.

EXAMPLES:

sage: Y = SimplicialComplex([[1,2], [1,4]])
sage: Y.faces()
{-1: {()}, 0: {(1,), (2,), (4,)}, 1: {(1, 2), (1, 4)}}
sage: L = SimplicialComplex([[1,2]])
sage: Y.faces(subcomplex=L)
{-1: set(), 0: {(4,)}, 1: {(1, 4)}}

chain_complex(subcomplex=None, augmented=False, verbose=False, check=False, dimensions=None, base_ring=Integer Ring, cochain=False)

The chain complex associated to this simplicial complex.

Parameters:

• dimensions – if None, compute the chain complex in all dimensions. If a list or tuple of integers, compute the chain complex in those dimensions, setting the chain groups in all other dimensions to zero.

• base_ring (optional, default ZZ) – commutative ring

• subcomplex (optional, default empty) – a subcomplex of this simplicial complex. Compute the chain complex relative to this subcomplex.

• augmented (boolean; optional, default False) – If True, return the augmented chain complex (that is, include a class in dimension \(-1\) corresponding to the empty cell). This is ignored if dimensions is specified.

• cochain (boolean; optional, default False) – If True, return the cochain complex (that is, the dual of the chain complex).

• verbose (boolean; optional, default False) – If True, print some messages as the chain complex is computed.

• check (boolean; optional, default False) – If True, make sure that the chain complex is actually a chain complex: the differentials are composable and their product is zero.

Note

If subcomplex is nonempty, then the argument augmented has no effect: the chain complex relative to a nonempty subcomplex is zero in dimension \(-1\).

The rows and columns of the boundary matrices are indexed by the lists given by the _n_cells_sorted() method, which by default are sorted.

EXAMPLES:

sage: circle = SimplicialComplex([[0,1], [1,2], [0, 2]])
sage: circle.chain_complex()                                                # needs sage.modules
Chain complex with at most 2 nonzero terms over Integer Ring
sage: circle.chain_complex()._latex_()                                      # needs sage.modules
'\\Bold{Z}^{3} \\xrightarrow{d_{1}} \\Bold{Z}^{3}'
sage: circle.chain_complex(base_ring=QQ, augmented=True)                    # needs sage.modules
Chain complex with at most 3 nonzero terms over Rational Field

cone(is_mutable=True)

The cone on this simplicial complex.

Parameters:

is_mutable (boolean; optional, default True) – Determines if the output is mutable

The cone is the simplicial complex formed by adding a new vertex \(C\) and simplices of the form \([C, v_0, ..., v_k]\) for every simplex \([v_0, ..., v_k]\) in the original simplicial complex. That is, the cone is the join of the original complex with a one-point simplicial complex.

EXAMPLES:

sage: S = SimplicialComplex([[0], [1]])
sage: CS = S.cone()
sage: sorted(CS.vertices())
['L0', 'L1', 'R0']
sage: len(CS.facets())
2
sage: CS.facets() == set([Simplex(['L0', 'R0']), Simplex(['L1', 'R0'])])
True

cone_vertices()

Return the list of cone vertices of self.

A vertex is a cone vertex if and only if it appears in every facet.

EXAMPLES:

sage: SimplicialComplex([[1,2,3]]).cone_vertices()
[1, 2, 3]
sage: SimplicialComplex([[1,2,3], [1,3,4], [1,5,6]]).cone_vertices()
[1]
sage: SimplicialComplex([[1,2,3], [1,3,4], [2,5,6]]).cone_vertices()
[]

connected_component(simplex=None)

Return the connected component of this simplicial complex containing simplex. If simplex is omitted, then return the connected component containing the zeroth vertex in the vertex list. (If the simplicial complex is empty, raise an error.)

EXAMPLES:

sage: S1 = simplicial_complexes.Sphere(1)
sage: S1 == S1.connected_component()
True
sage: X = S1.disjoint_union(S1)
sage: X == X.connected_component()
False
sage: CL0 = X.connected_component(Simplex(['L0']))
sage: CR0 = X.connected_component(Simplex(['R0']))
sage: CL0 == CR0
False

sage: S0 = simplicial_complexes.Sphere(0)
sage: S0.vertices()
(0, 1)
sage: S0.connected_component()
Simplicial complex with vertex set (0,) and facets {(0,)}
sage: S0.connected_component(Simplex((1,)))
Simplicial complex with vertex set (1,) and facets {(1,)}

sage: SimplicialComplex([[]]).connected_component()
Traceback (most recent call last):
...
ValueError: the empty simplicial complex has no connected components

connected_sum(other, is_mutable=True)

The connected sum of this simplicial complex with another one.

Parameters:

• other – another simplicial complex

• is_mutable (boolean; optional, default True) – Determines if the output is mutable

Returns:

the connected sum self # other

Warning

This does not check that self and other are manifolds, only that their facets all have the same dimension. Since a (more or less) random facet is chosen from each complex and then glued together, this method may return random results if applied to non-manifolds, depending on which facet is chosen.

Algorithm: a facet is chosen from each surface, and removed. The vertices of these two facets are relabeled to (0,1,...,dim). Of the remaining vertices, the ones from the left-hand factor are renamed by prepending an “L”, and similarly the remaining vertices in the right-hand factor are renamed by prepending an “R”.

EXAMPLES:

sage: S1 = simplicial_complexes.Sphere(1)
sage: S1.connected_sum(S1.connected_sum(S1)).homology()                     # needs sage.modules
{0: 0, 1: Z}
sage: P = simplicial_complexes.RealProjectivePlane(); P
Minimal triangulation of the real projective plane
sage: P.connected_sum(P)    # the Klein bottle
Simplicial complex with 9 vertices and 18 facets

The notation ‘+’ may be used for connected sum, also:

sage: P + P    # the Klein bottle
Simplicial complex with 9 vertices and 18 facets
sage: (P + P).homology()[1]                                                 # needs sage.modules
Z x C2

decone()

Return the subcomplex of self induced by the non-cone vertices.

EXAMPLES:

sage: SimplicialComplex([[1,2,3]]).decone()
Simplicial complex with vertex set () and facets {()}
sage: SimplicialComplex([[1,2,3], [1,3,4], [1,5,6]]).decone()
Simplicial complex with vertex set (2, 3, 4, 5, 6)
 and facets {(2, 3), (3, 4), (5, 6)}
sage: X = SimplicialComplex([[1,2,3], [1,3,4], [2,5,6]])
sage: X.decone() == X
True

delta_complex(sort_simplices=False)

Return self as a \(\Delta\)-complex.

The \(\Delta\)-complex is essentially identical to the simplicial complex: it has same simplices with the same boundaries.

Parameters:

sort_simplices (boolean; optional, default False) – if True, sort the list of simplices in each dimension

EXAMPLES:

sage: T = simplicial_complexes.Torus()
sage: Td = T.delta_complex()
sage: Td
Delta complex with 7 vertices and 43 simplices
sage: T.homology() == Td.homology()                                         # needs sage.modules
True

disjoint_union(right, rename_vertices=None, is_mutable=True)

The disjoint union of this simplicial complex with another one.

Parameters:

• right – the other simplicial complex (the right-hand factor)

• rename_vertices (boolean; optional, default True) – If this is True, the vertices in the disjoint union will be renamed by the formula: vertex “v” in the left-hand factor –> vertex “Lv” in the disjoint union, vertex “w” in the right-hand factor –> vertex “Rw” in the disjoint union. If this is false, this tries to construct the disjoint union without renaming the vertices; this will cause problems if the two factors have any vertices with names in common.

EXAMPLES:

sage: S1 = simplicial_complexes.Sphere(1)
sage: S2 = simplicial_complexes.Sphere(2)
sage: S1.disjoint_union(S2).homology()                                      # needs sage.modules
{0: Z, 1: Z, 2: Z}

f_triangle()

Compute the \(f\)-triangle of self.

The \(f\)-triangle is given by \(f_{i,j}\) being the number of faces \(F\) of size \(j\) such that \(i = \max_{G \subseteq F} |G|\).

See also

Not to be confused with F_triangle() .

EXAMPLES:

sage: X = SimplicialComplex([[1,2,3], [3,4,5], [1,4], [1,5], [2,4], [2,5]])
sage: X.f_triangle()   # this complex is not pure
[[0],
 [0, 0],
 [0, 0, 4],
 [1, 5, 6, 2]]

A complex is pure if and only if the last row is nonzero:

sage: X = SimplicialComplex([[1,2,3], [3,4,5], [1,4,5]])
sage: X.f_triangle()
[[0], [0, 0], [0, 0, 0], [1, 5, 8, 3]]

face(simplex, i)

The \(i\)-th face of simplex in this simplicial complex

INPUT:

• simplex – a simplex in this simplicial complex

• i – integer

EXAMPLES:

sage: S = SimplicialComplex([[0,1,4], [0,1,2]])
sage: S.face(Simplex((0,2)), 0)
(2,)

sage: S.face(Simplex((0,3)), 0)
Traceback (most recent call last):
...
ValueError: this simplex is not in this simplicial complex

face_iterator(increasing=True)

An iterator for the faces in this simplicial complex.

INPUT:

• increasing – (optional, default True) if True, return faces in increasing order of dimension, thus starting with the empty face. Otherwise it returns faces in decreasing order of dimension.

Note

Among the faces of a fixed dimension, there is no sorting.

EXAMPLES:

sage: S1 = simplicial_complexes.Sphere(1)
sage: sorted(S1.face_iterator())
[(), (0,), (0, 1), (0, 2), (1,), (1, 2), (2,)]

faces(subcomplex=None)

The faces of this simplicial complex, in the form of a dictionary of sets keyed by dimension. If the optional argument subcomplex is present, then return only the faces which are not in the subcomplex.

Parameters:

subcomplex (optional, default None) – a subcomplex of this simplicial complex. Return faces which are not in this subcomplex.

EXAMPLES:

sage: Y = SimplicialComplex([[1,2], [1,4]])
sage: Y.faces()
{-1: {()}, 0: {(1,), (2,), (4,)}, 1: {(1, 2), (1, 4)}}
sage: L = SimplicialComplex([[1,2]])
sage: Y.faces(subcomplex=L)
{-1: set(), 0: {(4,)}, 1: {(1, 4)}}

facets()

The maximal faces (a.k.a. facets) of this simplicial complex.

This just returns the set of facets used in defining the simplicial complex, so if the simplicial complex was defined with no maximality checking, none is done here, either.

EXAMPLES:

sage: Y = SimplicialComplex([[0,2], [1,4]])
sage: sorted(Y.maximal_faces())
[(0, 2), (1, 4)]

facets is a synonym for maximal_faces:

sage: S = SimplicialComplex([[0,1], [0,1,2]])
sage: S.facets()
{(0, 1, 2)}

fixed_complex(G)

Return the fixed simplicial complex \(Fix(G)\) for a subgroup \(G\).

INPUT:

• G – a subgroup of the automorphism group of the simplicial complex or a list of elements of the automorphism group

OUTPUT:

• a simplicial complex \(Fix(G)\)

Vertices in \(Fix(G)\) are the orbits of \(G\) (acting on vertices of self) that form a simplex in self. More generally, simplices in \(Fix(G)\) correspond to simplices in self that are union of such orbits.

A basic example:

sage: S4 = simplicial_complexes.Sphere(4)
sage: S3 = simplicial_complexes.Sphere(3)
sage: fix = S4.fixed_complex([S4.automorphism_group()([(0,1)])]); fix       # needs sage.groups
Simplicial complex with vertex set (0, 2, 3, 4, 5) and 5 facets
sage: fix.is_isomorphic(S3)                                                 # needs sage.groups
True

Another simple example:

sage: T = SimplicialComplex([[1,2,3],[2,3,4]])
sage: G = T.automorphism_group()                                            # needs sage.groups
sage: T.fixed_complex([G([(1,4)])])                                         # needs sage.groups
Simplicial complex with vertex set (2, 3) and facets {(2, 3)}

A more sophisticated example:

sage: RP2 = simplicial_complexes.ProjectivePlane()
sage: CP2 = simplicial_complexes.ComplexProjectivePlane()
sage: G = CP2.automorphism_group()                                          # needs sage.groups
sage: H = G.subgroup([G([(2,3),(5,6),(8,9)])])                              # needs sage.groups
sage: CP2.fixed_complex(H).is_isomorphic(RP2)                               # needs sage.groups
True

flip_graph()

If self is pure, return the flip graph of self, otherwise, return None.

The flip graph of a pure simplicial complex is the (undirected) graph with vertices being the facets, such that two facets are joined by an edge if they meet in a codimension \(1\) face.

The flip graph is used to detect if self is a pseudomanifold.

EXAMPLES:

sage: S0 = simplicial_complexes.Sphere(0)
sage: G = S0.flip_graph()
sage: G.vertices(sort=True); G.edges(sort=True, labels=False)
[(0,), (1,)]
[((0,), (1,))]

sage: G = (S0.wedge(S0)).flip_graph()
sage: G.vertices(sort=True); G.edges(sort=True, labels=False)
[(0,), ('L1',), ('R1',)]
[((0,), ('L1',)), ((0,), ('R1',)), (('L1',), ('R1',))]

sage: S1 = simplicial_complexes.Sphere(1)
sage: S2 = simplicial_complexes.Sphere(2)
sage: G = (S1.wedge(S1)).flip_graph()
sage: len(G.vertices(sort=False))
6
sage: len(G.edges(sort=False))
10

sage: (S1.wedge(S2)).flip_graph() is None
True

sage: G = S2.flip_graph()
sage: G.vertices(sort=True); G.edges(sort=True, labels=False)
[(0, 1, 2), (0, 1, 3), (0, 2, 3), (1, 2, 3)]
[((0, 1, 2), (0, 1, 3)),
 ((0, 1, 2), (0, 2, 3)),
 ((0, 1, 2), (1, 2, 3)),
 ((0, 1, 3), (0, 2, 3)),
 ((0, 1, 3), (1, 2, 3)),
 ((0, 2, 3), (1, 2, 3))]

sage: T = simplicial_complexes.Torus()
sage: G = T.suspension(4).flip_graph()
sage: len(G.vertices(sort=False)); len(G.edges(sort=False, labels=False))
46
161

fundamental_group(base_point=None, simplify=True)

Return the fundamental group of this simplicial complex.

INPUT:

• base_point (optional, default None) – if this complex is not path-connected, then specify a vertex; the fundamental group is computed with that vertex as a base point. If the complex is path-connected, then you may specify a vertex or leave this as its default setting of None. (If this complex is path-connected, then this argument is ignored.)

• simplify (bool, optional True) – if False, then return a presentation of the group in terms of generators and relations. If True, the default, simplify as much as GAP is able to.

Algorithm: we compute the edge-path group – see Wikipedia article Fundamental_group. Choose a spanning tree for the 1-skeleton, and then the group’s generators are given by the edges in the 1-skeleton; there are two types of relations: \(e=1\) if \(e\) is in the spanning tree, and for every 2-simplex, if its edges are \(e_0\), \(e_1\), and \(e_2\), then we impose the relation \(e_0 e_1^{-1} e_2 = 1\).

EXAMPLES:

sage: S1 = simplicial_complexes.Sphere(1)
sage: S1.fundamental_group()                                                # needs sage.groups
Finitely presented group < e |  >

If we pass the argument simplify=False, we get generators and relations in a form which is not usually very helpful. Here is the cyclic group of order 2, for instance:

sage: RP2 = simplicial_complexes.RealProjectiveSpace(2)
sage: C2 = RP2.fundamental_group(simplify=False); C2                        # needs sage.groups
Finitely presented group < e0, e1, e2, e3, e4, e5, e6, e7, e8, e9 | e0, e3,
e4, e7, e9, e5*e2^-1*e0, e7*e2^-1*e1, e8*e3^-1*e1, e8*e6^-1*e4, e9*e6^-1*e5 >
sage: C2.simplified()                                                       # needs sage.groups
Finitely presented group < e1 | e1^2 >

This is the same answer given if the argument simplify is True (the default):

sage: RP2.fundamental_group()                                               # needs sage.groups
Finitely presented group < e1 | e1^2 >

You must specify a base point to compute the fundamental group of a non-connected complex:

sage: # needs sage.groups
sage: K = S1.disjoint_union(RP2)
sage: K.fundamental_group()
Traceback (most recent call last):
...
ValueError: this complex is not connected, so you must specify a base point
sage: K.fundamental_group(base_point='L0')
Finitely presented group < e |  >
sage: K.fundamental_group(base_point='R0').order()
2

Some other examples:

sage: S1.wedge(S1).fundamental_group()                                      # needs sage.groups
Finitely presented group < e0, e1 | >
sage: simplicial_complexes.Torus().fundamental_group()                      # needs sage.groups
Finitely presented group < e1, e4 | e4^-1*e1^-1*e4*e1 >

sage: # needs sage.groups
sage: G = simplicial_complexes.MooreSpace(5).fundamental_group()
sage: G.ngens()
1
sage: x = G.gen(0)
sage: [(x**n).is_one() for n in range(1,6)]
[False, False, False, False, True]

g_vector()

The \(g\)-vector of this simplicial complex.

If the \(h\)-vector of the complex is \((h_0, h_1, ..., h_d, h_{d+1})\) – see h_vector() – then its \(g\)-vector \((g_0, g_1, ..., g_{[(d+1)/2]})\) is defined by \(g_0 = 1\) and \(g_i = h_i - h_{i-1}\) for \(i > 0\).

EXAMPLES:

sage: # needs sage.combinat
sage: S3 = simplicial_complexes.Sphere(3).barycentric_subdivision()
sage: S3.f_vector()
[1, 30, 150, 240, 120]
sage: S3.h_vector()
[1, 26, 66, 26, 1]
sage: S3.g_vector()
[1, 25, 40]

generated_subcomplex(sub_vertex_set, is_mutable=True)

Return the largest sub-simplicial complex of self containing exactly sub_vertex_set as vertices.

Parameters:

• sub_vertex_set – The sub-vertex set.

• is_mutable (boolean; optional, default True) – Determines if the output is mutable

EXAMPLES:

sage: S = simplicial_complexes.Sphere(2)
sage: S
Minimal triangulation of the 2-sphere
sage: S.generated_subcomplex([0,1,2])
Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}

graph()

The 1-skeleton of this simplicial complex, as a graph.

Warning

This may give the wrong answer if the simplicial complex was constructed with maximality_check set to False.

EXAMPLES:

sage: S = SimplicialComplex([[0,1,2,3]])
sage: G = S.graph(); G
Graph on 4 vertices
sage: G.edges(sort=True)
[(0, 1, None), (0, 2, None), (0, 3, None), (1, 2, None), (1, 3, None), (2, 3, None)]

h_triangle()

Compute the \(h\)-triangle of self.

The \(h\)-triangle of a simplicial complex \(\Delta\) is given by

\[h_{i,j} = \sum_{k=0}^j (-1)^{j-k} \binom{i-k}{j-k} f_{i,k},\]

where \(f_{i,k}\) is the \(f\)-triangle of \(\Delta\).

EXAMPLES:

sage: X = SimplicialComplex([[1,2,3], [3,4,5], [1,4], [1,5], [2,4], [2,5]])
sage: X.h_triangle()
[[0],
 [0, 0],
 [0, 0, 4],
 [1, 2, -1, 0]]

h_vector()

The \(h\)-vector of this simplicial complex.

If the complex has dimension \(d\) and \((f_{-1}, f_0, f_1, ..., f_d)\) is its \(f\)-vector (with \(f_{-1} = 1\), representing the empty simplex), then the \(h\)-vector \((h_0, h_1, ..., h_d, h_{d+1})\) is defined by

\[\sum_{i=0}^{d+1} h_i x^{d+1-i} = \sum_{i=0}^{d+1} f_{i-1} (x-1)^{d+1-i}.\]

Alternatively,

\[h_j = \sum_{i=-1}^{j-1} (-1)^{j-i-1} \binom{d-i}{j-i-1} f_i.\]

EXAMPLES:

The \(f\)- and \(h\)-vectors of the boundary of an octahedron are computed in Wikipedia article Simplicial_complex:

sage: square = SimplicialComplex([[0,1], [1,2], [2,3], [0,3]])
sage: S0 = SimplicialComplex([[0], [1]])
sage: octa = square.join(S0) # boundary of an octahedron
sage: octa.f_vector()
[1, 6, 12, 8]
sage: octa.h_vector()
[1, 3, 3, 1]

intersection(other)

Calculate the intersection of two simplicial complexes.

EXAMPLES:

sage: X = SimplicialComplex([[1,2,3], [1,2,4]])
sage: Y = SimplicialComplex([[1,2,3], [1,4,5]])
sage: Z = SimplicialComplex([[1,2,3], [1,4], [2,4]])
sage: sorted(X.intersection(Y).facets())
[(1, 2, 3), (1, 4)]
sage: X.intersection(X) == X
True
sage: X.intersection(Z) == X
False
sage: X.intersection(Z) == Z
True

is_balanced(check_purity=False, certificate=False)

Determine whether self is balanced.

A simplicial complex \(X\) of dimension \(d-1\) is balanced if and only if its vertices can be colored with \(d\) colors such that every face contains at most one vertex of each color. An equivalent condition is that the 1-skeleton of \(X\) is \(d\)-colorable. In some contexts, it is also required that \(X\) be pure (i.e., that all maximal faces of \(X\) have the same dimension).

INPUT:

• check_purity – (default: False) if this is True, require that self be pure as well as balanced

• certificate – (default: False) if this is True and self is balanced, then return a \(d\)-coloring of the 1-skeleton.

EXAMPLES:

A 1-dim simplicial complex is balanced iff it is bipartite:

sage: X = SimplicialComplex([[1,2], [1,4], [3,4], [2,5]])
sage: X.is_balanced()
True
sage: sorted(X.is_balanced(certificate=True))
[[1, 3, 5], [2, 4]]
sage: X = SimplicialComplex([[1,2], [1,4], [3,4], [2,4]])
sage: X.is_balanced()
False

Any barycentric division is balanced:

sage: X = SimplicialComplex([[1,2,3], [1,2,4], [2,3,4]])
sage: X.is_balanced()
False
sage: X.barycentric_subdivision().is_balanced()
True

A non-pure balanced complex:

sage: X = SimplicialComplex([[1,2,3], [3,4]])
sage: X.is_balanced(check_purity=True)
False
sage: sorted(X.is_balanced(certificate=True))
[[1, 4], [2], [3]]

is_cohen_macaulay(base_ring=Rational Field, ncpus=0)

Return True if self is Cohen-Macaulay.

A simplicial complex \(\Delta\) is Cohen-Macaulay over \(R\) iff \(\tilde{H}_i(\mathrm{lk}_\Delta(F);R) = 0\) for all \(F \in \Delta\) and \(i < \dim\mathrm{lk}_\Delta(F)\). Here, \(\Delta\) is self and \(R\) is base_ring, and \(\mathrm{lk}\) denotes the link operator on self.

INPUT:

• base_ring – (default: QQ) the base ring.

• ncpus – (default: 0) number of cpus used for the computation. If this is 0, determine the number of cpus automatically based on the hardware being used.

For finite simplicial complexes, this is equivalent to the statement that the Stanley-Reisner ring of self is Cohen-Macaulay.

EXAMPLES:

Spheres are Cohen-Macaulay:

sage: S = SimplicialComplex([[1,2],[2,3],[3,1]])
sage: S.is_cohen_macaulay(ncpus=3)                                          # needs sage.modules
True

The following example is taken from Bruns, Herzog - Cohen-Macaulay rings, Figure 5.3:

sage: S = SimplicialComplex([[1,2,3],[1,4,5]])
sage: S.is_cohen_macaulay(ncpus=3)                                          # needs sage.modules
False

The choice of base ring can matter. The real projective plane \(\RR P^2\) has \(H_1(\RR P^2) = \ZZ/2\), hence is CM over \(\QQ\) but not over \(\ZZ\).

sage: X = simplicial_complexes.RealProjectivePlane()
sage: X.is_cohen_macaulay()                                                 # needs sage.modules
True
sage: X.is_cohen_macaulay(ZZ)                                               # needs sage.modules
False

is_flag_complex()

Return True if and only if self is a flag complex.

A flag complex is a simplicial complex that is the largest simplicial complex on its 1-skeleton. Thus a flag complex is the clique complex of its graph.

EXAMPLES:

sage: h = Graph({0: [1,2,3,4], 1: [2,3,4], 2: [3]})
sage: x = h.clique_complex(); x
Simplicial complex with vertex set (0, 1, 2, 3, 4)
and facets {(0, 1, 4), (0, 1, 2, 3)}
sage: x.is_flag_complex()
True

sage: X = simplicial_complexes.ChessboardComplex(3,3)
sage: X.is_flag_complex()
True

is_golod()

Return whether self is Golod.

A simplicial complex is Golod if multiplication and all higher Massey operations in the associated Tor-algebra are trivial. This is done by checking the bigraded Betti numbers.

EXAMPLES:

sage: # needs sage.modules
sage: X = SimplicialComplex([[0,1],[1,2],[2,3],[3,0]])
sage: Y = SimplicialComplex([[0,1,2],[0,2],[0,4]])
sage: X.is_golod()
False
sage: Y.is_golod()
True

is_immutable()

Return True if immutable.

EXAMPLES:

sage: S = SimplicialComplex([[1,4], [2,4]])
sage: S.is_immutable()
False
sage: S.set_immutable()
sage: S.is_immutable()
True

is_isomorphic(other, certificate=False)

Check whether two simplicial complexes are isomorphic.

INPUT:

• certificate – if True, then output is (a, b), where a is a boolean and b is either a map or None

This is done by creating two graphs and checking whether they are isomorphic.

EXAMPLES:

sage: Z1 = SimplicialComplex([[0,1],[1,2],[2,3,4],[4,5]])
sage: Z2 = SimplicialComplex([['a','b'],['b','c'],['c','d','e'],['e','f']])
sage: Z3 = SimplicialComplex([[1,2,3]])
sage: Z1.is_isomorphic(Z2)
True
sage: Z1.is_isomorphic(Z2, certificate=True)
(True, {0: 'a', 1: 'b', 2: 'c', 3: 'd', 4: 'e', 5: 'f'})
sage: Z3.is_isomorphic(Z2)
False

We check that github issue #20751 is fixed:

sage: C1 = SimplicialComplex([[1,2,3], [2,4], [3,5], [5,6]])
sage: C2 = SimplicialComplex([['a','b','c'], ['b','d'], ['c','e'], ['e','f']])
sage: C1.is_isomorphic(C2, certificate=True)
(True, {1: 'a', 2: 'b', 3: 'c', 4: 'd', 5: 'e', 6: 'f'})

is_minimally_non_golod()

Return whether self is minimally non-Golod.

If a simplicial complex itself is not Golod, but deleting any vertex gives us a full subcomplex that is Golod, then we say that a simplicial complex is minimally non-Golod.

See also

See is_golod() for more information.

EXAMPLES:

sage: # needs sage.modules
sage: X = SimplicialComplex([[0,1],[1,2],[2,3],[3,0]])
sage: Y = SimplicialComplex([[1,2,3],[1,2,4],[3,5],[4,5]])
sage: X.is_golod()
False
sage: X.is_minimally_non_golod()
True
sage: Y.is_golod()
False
sage: Y.is_minimally_non_golod()
False

is_mutable()

Return True if mutable.

EXAMPLES:

sage: S = SimplicialComplex([[1,4], [2,4]])
sage: S.is_mutable()
True
sage: S.set_immutable()
sage: S.is_mutable()
False
sage: S2 = SimplicialComplex([[1,4], [2,4]], is_mutable=False)
sage: S2.is_mutable()
False
sage: S3 = SimplicialComplex([[1,4], [2,4]], is_mutable=False)
sage: S3.is_mutable()
False

is_partitionable(certificate, solver, integrality_tolerance=False)

Determine whether self is partitionable.

A partitioning of a simplicial complex \(X\) is a decomposition of its face poset into disjoint Boolean intervals \([R,F]\), where \(F\) ranges over all facets of \(X\).

The method sets up an integer program with:

• a variable \(y_i\) for each pair \((R,F)\), where \(F\) is a facet of \(X\) and \(R\) is a subface of \(F\)

• a constraint \(y_i+y_j \leq 1\) for each pair \((R_i,F_i)\), \((R_j,F_j)\) whose Boolean intervals intersect nontrivially (equivalent to \((R_i\subseteq F_j and R_j\subseteq F_i))\)

• objective function equal to the sum of all \(y_i\)

INPUT:

• certificate – (default: False) If True, and self is partitionable, then return a list of pairs \((R,F)\) that form a partitioning.

• solver – (default: None) Specify a Mixed Integer Linear Programming (MILP) solver to be used. If set to None, the default one is used. For more information on MILP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.

• integrality_tolerance – parameter for use with MILP solvers over an inexact base ring; see MixedIntegerLinearProgram.get_values().

EXAMPLES:

Simplices are trivially partitionable:

sage: X = SimplicialComplex([[1,2,3,4]])
sage: X.is_partitionable()                                                  # needs sage.numerical.mip
True
sage: X.is_partitionable(certificate=True)                                  # needs sage.numerical.mip
[((), (1, 2, 3, 4), 4)]

Shellable complexes are partitionable:

sage: # needs sage.numerical.mip
sage: X = SimplicialComplex([[1,3,5], [1,3,6], [1,4,5], [1,4,6],
....:                        [2,3,5], [2,3,6], [2,4,5]])
sage: X.is_partitionable()
True
sage: P = X.is_partitionable(certificate=True)
sage: def n_intervals_containing(f):
....:     return len([RF for RF in P
....:                    if RF[0].is_face(f) and f.is_face(RF[1])])
sage: all(n_intervals_containing(f) == 1
....:     for k in X.faces().keys() for f in X.faces()[k])
True

A non-shellable, non-Cohen-Macaulay, partitionable example, constructed by Björner:

sage: X = SimplicialComplex([[1,2,3], [1,2,4], [1,3,4], [2,3,4], [1,5,6]])
sage: X.is_partitionable()                                                  # needs sage.numerical.mip
True

The bowtie complex is not partitionable:

sage: X = SimplicialComplex([[1,2,3], [1,4,5]])
sage: X.is_partitionable()                                                  # needs sage.numerical.mip
False

is_pseudomanifold()

Return True if self is a pseudomanifold.

A pseudomanifold is a simplicial complex with the following properties:

• it is pure of some dimension \(d\) (all of its facets are \(d\)-dimensional)

• every \((d-1)\)-dimensional simplex is the face of exactly two facets

• for every two facets \(S\) and \(T\), there is a sequence of facets

  \[S = f_0, f_1, ..., f_n = T\]

  such that for each \(i\), \(f_i\) and \(f_{i-1}\) intersect in a \((d-1)\)-simplex.

By convention, \(S^0\) is the only 0-dimensional pseudomanifold.

EXAMPLES:

sage: S0 = simplicial_complexes.Sphere(0)
sage: S0.is_pseudomanifold()
True
sage: (S0.wedge(S0)).is_pseudomanifold()
False
sage: S1 = simplicial_complexes.Sphere(1)
sage: S2 = simplicial_complexes.Sphere(2)
sage: (S1.wedge(S1)).is_pseudomanifold()
False
sage: (S1.wedge(S2)).is_pseudomanifold()
False
sage: S2.is_pseudomanifold()
True
sage: T = simplicial_complexes.Torus()
sage: T.suspension(4).is_pseudomanifold()
True

is_pure()

Return True iff this simplicial complex is pure.

A simplicial complex is pure if and only if all of its maximal faces have the same dimension.

Warning

This may give the wrong answer if the simplicial complex was constructed with maximality_check set to False.

EXAMPLES:

sage: U = SimplicialComplex([[1,2], [1, 3, 4]])
sage: U.is_pure()
False
sage: X = SimplicialComplex([[0,1], [0,2], [1,2]])
sage: X.is_pure()
True

Demonstration of the warning:

sage: S = SimplicialComplex([[0,1], [0]], maximality_check=False)
sage: S.is_pure()
False

is_shellable(certificate=False)

Return if self is shellable.

A simplicial complex is shellable if there exists a shelling order.

Note

1. This method can check all orderings of the facets by brute force, hence can be very slow.

2. This is shellability in the general (nonpure) sense of Bjorner and Wachs [BW1996]. This method does not check purity.

See also

is_shelling_order()

INPUT:

• certificate – (default: False) if True then returns the shelling order (if it exists)

EXAMPLES:

sage: X = SimplicialComplex([[1,2,5], [2,3,5], [3,4,5], [1,4,5]])
sage: X.is_shellable()
True
sage: order = X.is_shellable(True); order
((1, 2, 5), (2, 3, 5), (1, 4, 5), (3, 4, 5))
sage: X.is_shelling_order(order)
True

sage: X = SimplicialComplex([[1,2,3], [3,4,5]])
sage: X.is_shellable()
False

Examples from Figure 1 in [BW1996]:

sage: X = SimplicialComplex([[1,2,3], [3,4], [4,5], [5,6], [4,6]])
sage: X.is_shellable()
True

sage: X = SimplicialComplex([[1,2,3], [3,4], [4,5,6]])
sage: X.is_shellable()
False

REFERENCES:

• Wikipedia article Shelling_(topology)

is_shelling_order(shelling_order, certificate=False)

Return if the order of the facets given by shelling_order is a shelling order for self.

A sequence of facets \((F_i)_{i=1}^N\) of a simplicial complex of dimension \(d\) is a shelling order if for all \(i = 2, 3, 4, \ldots\), the complex

\[X_i = \left( \bigcup_{j=1}^{i-1} F_j \right) \cap F_i\]

is pure and of dimension \(\dim F_i - 1\).

INPUT:

• shelling_order – an ordering of the facets of self

• certificate – (default: False) if True then returns the index of the first facet that violate the condition

See also

is_shellable()

EXAMPLES:

sage: facets = [[1,2,5],[2,3,5],[3,4,5],[1,4,5]]
sage: X = SimplicialComplex(facets)
sage: X.is_shelling_order(facets)
True

sage: b = [[1,2,5], [3,4,5], [2,3,5], [1,4,5]]
sage: X.is_shelling_order(b)
False
sage: X.is_shelling_order(b, True)
(False, 1)

A non-pure example:

sage: facets = [[1,2,3], [3,4], [4,5], [5,6], [4,6]]
sage: X = SimplicialComplex(facets)
sage: X.is_shelling_order(facets)
True

REFERENCES:

• [BW1996]

is_subcomplex(other)

Return True if this is a subcomplex of other.

Parameters:

other – another simplicial complex

EXAMPLES:

sage: S1 = simplicial_complexes.Sphere(1)
sage: S1.is_subcomplex(S1)
True
sage: Empty = SimplicialComplex()
sage: Empty.is_subcomplex(S1)
True
sage: S1.is_subcomplex(Empty)
False

sage: sorted(S1.facets())
[(0, 1), (0, 2), (1, 2)]
sage: T = S1.product(S1)
sage: sorted(T.facets())[0] # typical facet in T
('L0R0', 'L0R1', 'L1R1')
sage: S1.is_subcomplex(T)
False
sage: T._contractible_subcomplex().is_subcomplex(T)
True

join(right, rename_vertices=True, is_mutable=True)

The join of this simplicial complex with another one.

The join of two simplicial complexes \(S\) and \(T\) is the simplicial complex \(S*T\) with simplices of the form \([v_0, ..., v_k, w_0, ..., w_n]\) for all simplices \([v_0, ..., v_k]\) in \(S\) and \([w_0, ..., w_n]\) in \(T\).

Parameters:

• right – the other simplicial complex (the right-hand factor)

• rename_vertices (boolean; optional, default True) – If this is True, the vertices in the join will be renamed by the formula: vertex “v” in the left-hand factor –> vertex “Lv” in the join, vertex “w” in the right-hand factor –> vertex “Rw” in the join. If this is false, this tries to construct the join without renaming the vertices; this will cause problems if the two factors have any vertices with names in common.

• is_mutable (boolean; optional, default True) – Determines if the output is mutable

EXAMPLES:

sage: S = SimplicialComplex([[0], [1]])
sage: T = SimplicialComplex([[2], [3]])
sage: S.join(T)
Simplicial complex with vertex set ('L0', 'L1', 'R2', 'R3') and 4 facets
sage: S.join(T, rename_vertices=False)
Simplicial complex with vertex set (0, 1, 2, 3)
and facets {(0, 2), (0, 3), (1, 2), (1, 3)}

The notation ‘*’ may be used, as well:

sage: S * S
Simplicial complex with vertex set ('L0', 'L1', 'R0', 'R1') and 4 facets
sage: S * S * S * S * S * S * S * S
Simplicial complex with 16 vertices and 256 facets

link(simplex, is_mutable=True)

The link of a simplex in this simplicial complex.

The link of a simplex \(F\) is the simplicial complex formed by all simplices \(G\) which are disjoint from \(F\) but for which \(F \cup G\) is a simplex.

Parameters:

• simplex – a simplex in this simplicial complex.

• is_mutable (boolean; optional, default True) – Determines if the output is mutable

EXAMPLES:

sage: X = SimplicialComplex([[0,1,2], [1,2,3]])
sage: X.link(Simplex([0]))
Simplicial complex with vertex set (1, 2) and facets {(1, 2)}
sage: X.link([1,2])
Simplicial complex with vertex set (0, 3) and facets {(0,), (3,)}
sage: Y = SimplicialComplex([[0,1,2,3]])
sage: Y.link([1])
Simplicial complex with vertex set (0, 2, 3) and facets {(0, 2, 3)}

maximal_faces()

The maximal faces (a.k.a. facets) of this simplicial complex.

This just returns the set of facets used in defining the simplicial complex, so if the simplicial complex was defined with no maximality checking, none is done here, either.

EXAMPLES:

sage: Y = SimplicialComplex([[0,2], [1,4]])
sage: sorted(Y.maximal_faces())
[(0, 2), (1, 4)]

facets is a synonym for maximal_faces:

sage: S = SimplicialComplex([[0,1], [0,1,2]])
sage: S.facets()
{(0, 1, 2)}

minimal_nonfaces()

Set consisting of the minimal subsets of the vertex set of this simplicial complex which do not form faces.

Algorithm: Proceeds through the faces of the complex increasing the dimension, starting from dimension 0, and add the faces that are not contained in the complex and that are not already contained in a previously seen minimal non-face.

This is used in computing the Stanley-Reisner ring and the Alexander dual.

EXAMPLES:

sage: X = SimplicialComplex([[1,3],[1,2]])
sage: X.minimal_nonfaces()
{(2, 3)}
sage: Y = SimplicialComplex([[0,1], [1,2], [2,3], [3,0]])
sage: sorted(Y.minimal_nonfaces())
[(0, 2), (1, 3)]

moment_angle_complex()

Return the moment-angle complex of self.

A moment-angle complex is a topological space created from this simplicial complex, which holds a lot of information about the simplicial complex itself.

See also

See sage.topology.moment_angle_complex for more information on moment-angle complexes.

EXAMPLES:

sage: X = SimplicialComplex([[0,1,2,3], [1,4], [3,2,4]])
sage: X.moment_angle_complex()
Moment-angle complex of Simplicial complex with vertex set
(0, 1, 2, 3, 4) and facets {(1, 4), (2, 3, 4), (0, 1, 2, 3)}
sage: K = simplicial_complexes.KleinBottle()
sage: K.moment_angle_complex()
Moment-angle complex of Simplicial complex with vertex set
(0, 1, 2, 3, 4, 5, 6, 7) and 16 facets

We can also create it explicitly:

sage: Z = MomentAngleComplex(K); Z
Moment-angle complex of Simplicial complex with vertex set
(0, 1, 2, 3, 4, 5, 6, 7) and 16 facets

n_faces(n, subcomplex=None)

List of cells of dimension n of this cell complex. If the optional argument subcomplex is present, then return the n-dimensional cells which are not in the subcomplex.

Parameters:

• n (non-negative integer) – the dimension

• subcomplex (optional, default None) – a subcomplex of this cell complex. Return the cells which are not in this subcomplex.

Note

The resulting list need not be sorted. If you want a sorted list of \(n\)-cells, use _n_cells_sorted().

EXAMPLES:

sage: delta_complexes.Torus().n_cells(1)
[(0, 0), (0, 0), (0, 0)]
sage: cubical_complexes.Cube(1).n_cells(0)
[[1,1], [0,0]]

n_skeleton(n)

The \(n\)-skeleton of this simplicial complex.

The \(n\)-skeleton of a simplicial complex is obtained by discarding all of the simplices in dimensions larger than \(n\).

Parameters:

n – non-negative integer

EXAMPLES:

sage: X = SimplicialComplex([[0,1], [1,2,3], [0,2,3]])
sage: X.n_skeleton(1)
Simplicial complex with vertex set (0, 1, 2, 3) and
 facets {(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)}
sage: X.set_immutable()
sage: X.n_skeleton(2)
Simplicial complex with vertex set (0, 1, 2, 3) and
 facets {(0, 1), (0, 2, 3), (1, 2, 3)}
sage: X.n_skeleton(4)
Simplicial complex with vertex set (0, 1, 2, 3) and
 facets {(0, 1), (0, 2, 3), (1, 2, 3)}

product(right, rename_vertices=True, is_mutable=True)

The product of this simplicial complex with another one.

Parameters:

• right – the other simplicial complex (the right-hand factor)

• rename_vertices (boolean; optional, default True) – If this is False, then the vertices in the product are the set of ordered pairs \((v,w)\) where \(v\) is a vertex in self and \(w\) is a vertex in right. If this is True, then the vertices are renamed as “LvRw” (e.g., the vertex (1,2) would become “L1R2”). This is useful if you want to define the Stanley-Reisner ring of the complex: vertex names like (0,1) are not suitable for that, while vertex names like “L0R1” are.

• is_mutable (boolean; optional, default True) – Determines if the output is mutable

The vertices in the product will be the set of ordered pairs \((v,w)\) where \(v\) is a vertex in self and \(w\) is a vertex in right.

Warning

If X and Y are simplicial complexes, then X*Y returns their join, not their product.

EXAMPLES:

sage: S = SimplicialComplex([[0,1], [1,2], [0,2]]) # circle
sage: K = SimplicialComplex([[0,1]])   # edge
sage: Cyl = S.product(K)  # cylinder
sage: sorted(Cyl.vertices())
['L0R0', 'L0R1', 'L1R0', 'L1R1', 'L2R0', 'L2R1']
sage: Cyl2 = S.product(K, rename_vertices=False)
sage: sorted(Cyl2.vertices())
[(0, 0), (0, 1), (1, 0), (1, 1), (2, 0), (2, 1)]
sage: T = S.product(S)  # torus
sage: T
Simplicial complex with 9 vertices and 18 facets
sage: T.homology()                                                          # needs sage.modules
{0: 0, 1: Z x Z, 2: Z}

These can get large pretty quickly:

sage: T = simplicial_complexes.Torus(); T
Minimal triangulation of the torus
sage: K = simplicial_complexes.KleinBottle(); K
Minimal triangulation of the Klein bottle
sage: T.product(K)      # long time: 5 or 6 seconds
Simplicial complex with 56 vertices and 1344 facets

remove_face(face, check=False)

Remove a face from this simplicial complex.

Parameters:

• face – a face of the simplicial complex

• check – boolean; optional, default False. If True, raise an error if face is not a face of this simplicial complex

This does not return anything; instead, it changes the simplicial complex.

ALGORITHM:

The facets of the new simplicial complex are the facets of the original complex not containing face, together with those of link(face)*boundary(face).

EXAMPLES:

sage: S = range(1,5)
sage: Z = SimplicialComplex([S]); Z
Simplicial complex with vertex set (1, 2, 3, 4) and facets {(1, 2, 3, 4)}
sage: Z.remove_face([1,2])
sage: Z
Simplicial complex with vertex set (1, 2, 3, 4) and
 facets {(1, 3, 4), (2, 3, 4)}

sage: S = SimplicialComplex([[0,1,2],[2,3]])
sage: S
Simplicial complex with vertex set (0, 1, 2, 3) and
 facets {(2, 3), (0, 1, 2)}
sage: S.remove_face([0,1,2])
sage: S
Simplicial complex with vertex set (0, 1, 2, 3) and
 facets {(0, 1), (0, 2), (1, 2), (2, 3)}

remove_faces(faces, check=False)

Remove a collection of faces from this simplicial complex.

Parameters:

• faces – a list (or any iterable) of faces of the simplicial complex

• check – boolean; optional, default False. If True, raise an error if any element of faces is not a face of this simplicial complex

This does not return anything; instead, it changes the simplicial complex.

ALGORITHM:

Run self.remove_face(f) repeatedly, for f in faces.

EXAMPLES:

sage: S = range(1,5)
sage: Z = SimplicialComplex([S]); Z
Simplicial complex with vertex set (1, 2, 3, 4) and facets {(1, 2, 3, 4)}
sage: Z.remove_faces([[1,2]])
sage: Z
Simplicial complex with vertex set (1, 2, 3, 4) and facets {(1, 3, 4), (2, 3, 4)}

sage: Z = SimplicialComplex([S]); Z
Simplicial complex with vertex set (1, 2, 3, 4) and facets {(1, 2, 3, 4)}
sage: Z.remove_faces([[1,2], [2,3]])
sage: Z
Simplicial complex with vertex set (1, 2, 3, 4) and facets {(2, 4), (1, 3, 4)}

restriction_sets(order)

Return the restriction sets of the facets according to order.

A restriction set of a shelling order is the sequence of smallest new faces that are created during the shelling order.

See also

is_shelling_order()

EXAMPLES:

sage: facets = [[1,2,5], [2,3,5], [3,4,5], [1,4,5]]
sage: X = SimplicialComplex(facets)
sage: X.restriction_sets(facets)
[(), (3,), (4,), (1, 4)]

sage: b = [[1,2,5], [3,4,5], [2,3,5], [1,4,5]]
sage: X.restriction_sets(b)
Traceback (most recent call last):
...
ValueError: not a shelling order

set_immutable()

Make this simplicial complex immutable.

EXAMPLES:

sage: S = SimplicialComplex([[1,4], [2,4]])
sage: S.is_mutable()
True
sage: S.set_immutable()
sage: S.is_mutable()
False

stanley_reisner_ring(base_ring=Integer Ring)

The Stanley-Reisner ring of this simplicial complex.

Parameters:

base_ring (optional, default ZZ) – a commutative ring

Returns:

a quotient of a polynomial algebra with coefficients in base_ring, with one generator for each vertex in the simplicial complex, by the ideal generated by the products of those vertices which do not form faces in it.

Thus the ideal is generated by the products corresponding to the minimal nonfaces of the simplicial complex.

Warning

This may be quite slow!

Also, this may behave badly if the vertices have the ‘wrong’ names. To avoid this, define the simplicial complex at the start with the flag name_check set to True.

More precisely, this is a quotient of a polynomial ring with one generator for each vertex. If the name of a vertex is a non-negative integer, then the corresponding polynomial generator is named 'x' followed by that integer (e.g., 'x2', 'x3', 'x5', …). Otherwise, the polynomial generators are given the same names as the vertices. Thus if the vertex set is (2, 'x2'), there will be problems.

EXAMPLES:

sage: X = SimplicialComplex([[0,1,2], [0,2,3]])
sage: X.stanley_reisner_ring()
Quotient of Multivariate Polynomial Ring in x0, x1, x2, x3 over Integer Ring
 by the ideal (x1*x3)
sage: Y = SimplicialComplex([[0,1,2,3,4]]); Y
Simplicial complex with vertex set (0, 1, 2, 3, 4) and facets {(0, 1, 2, 3, 4)}
sage: Y.add_face([0,1,2,3,4])
sage: Y.stanley_reisner_ring(base_ring=QQ)
Multivariate Polynomial Ring in x0, x1, x2, x3, x4 over Rational Field

star(simplex, is_mutable=True)

Return the star of a simplex in this simplicial complex.

The star of simplex is the simplicial complex formed by all simplices which contain simplex.

INPUT:

• simplex – a simplex in this simplicial complex

• is_mutable – (default: True) boolean; determines if the output is mutable

EXAMPLES:

sage: X = SimplicialComplex([[0,1,2], [1,2,3]])
sage: X.star(Simplex([0]))
Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}
sage: X.star(Simplex([1]))
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 2), (1, 2, 3)}
sage: X.star(Simplex([1,2]))
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 2), (1, 2, 3)}
sage: X.star(Simplex([]))
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 2), (1, 2, 3)}

stellar_subdivision(simplex, inplace=False, is_mutable=True)

Return the stellar subdivision of a simplex in this simplicial complex.

The stellar subdivision of a face is obtained by adding a new vertex to the simplicial complex self joined to the star of the face and then deleting the face simplex to the result.

INPUT:

• simplex – a simplex face of self

• inplace – (default: False) boolean; determines if the operation is done on self or on a copy

• is_mutable – (default: True) boolean; determines if the output is mutable

OUTPUT:

• A simplicial complex obtained by the stellar subdivision of the face simplex

EXAMPLES:

sage: SC = SimplicialComplex([[0,1,2],[1,2,3]])
sage: F1 = Simplex([1,2])
sage: F2 = Simplex([1,3])
sage: F3 = Simplex([1,2,3])
sage: SC.stellar_subdivision(F1)
Simplicial complex with vertex set (0, 1, 2, 3, 4) and
 facets {(0, 1, 4), (0, 2, 4), (1, 3, 4), (2, 3, 4)}
sage: SC.stellar_subdivision(F2)
Simplicial complex with vertex set (0, 1, 2, 3, 4) and
 facets {(0, 1, 2), (1, 2, 4), (2, 3, 4)}
sage: SC.stellar_subdivision(F3)
Simplicial complex with vertex set (0, 1, 2, 3, 4) and
 facets {(0, 1, 2), (1, 2, 4), (1, 3, 4), (2, 3, 4)}
sage: SC.stellar_subdivision(F3, inplace=True);SC
Simplicial complex with vertex set (0, 1, 2, 3, 4) and
 facets {(0, 1, 2), (1, 2, 4), (1, 3, 4), (2, 3, 4)}

The simplex to subdivide should be a face of self:

sage: SC = SimplicialComplex([[0,1,2],[1,2,3]])
sage: F4 = Simplex([3,4])
sage: SC.stellar_subdivision(F4)
Traceback (most recent call last):
...
ValueError: the face to subdivide is not a face of self

One can not modify an immutable simplicial complex:

sage: SC = SimplicialComplex([[0,1,2],[1,2,3]], is_mutable=False)
sage: SC.stellar_subdivision(F1, inplace=True)
Traceback (most recent call last):
...
ValueError: this simplicial complex is not mutable

suspension(n=1, is_mutable=True)

The suspension of this simplicial complex.

Parameters:

• n (optional, default 1) – positive integer – suspend this many times.

• is_mutable (boolean; optional, default True) – Determines if the output is mutable

The suspension is the simplicial complex formed by adding two new vertices \(S_0\) and \(S_1\) and simplices of the form \([S_0, v_0, ..., v_k]\) and \([S_1, v_0, ..., v_k]\) for every simplex \([v_0, ..., v_k]\) in the original simplicial complex. That is, the suspension is the join of the original complex with a two-point simplicial complex.

If the simplicial complex \(M\) happens to be a pseudomanifold (see is_pseudomanifold()), then this instead constructs Datta’s one-point suspension (see [Dat2007], p. 434): choose a vertex \(u\) in \(M\) and choose a new vertex \(w\) to add. Denote the join of simplices by “\(*\)”. The facets in the one-point suspension are of the two forms

• \(u * \alpha\) where \(\alpha\) is a facet of \(M\) not containing \(u\)

• \(w * \beta\) where \(\beta\) is any facet of \(M\).

EXAMPLES:

sage: S0 = SimplicialComplex([[0], [1]])
sage: S0.suspension() == simplicial_complexes.Sphere(1)
True
sage: S3 = S0.suspension(3)  # the 3-sphere
sage: S3.homology()                                                         # needs sage.modules
{0: 0, 1: 0, 2: 0, 3: Z}

For pseudomanifolds, the complex constructed here will be smaller than that obtained by taking the join with the 0-sphere: the join adds two vertices, while this construction only adds one.

sage: T = simplicial_complexes.Torus()
sage: sorted(T.join(S0).vertices())      # 9 vertices
['L0', 'L1', 'L2', 'L3', 'L4', 'L5', 'L6', 'R0', 'R1']
sage: T.suspension().vertices()  # 8 vertices
(0, 1, 2, 3, 4, 5, 6, 7)

vertices()

The vertex set, as a tuple, of this simplicial complex.

EXAMPLES:

sage: S = SimplicialComplex([[i] for i in range(16)] + [[0,1], [1,2]])
sage: S
Simplicial complex with 16 vertices and 15 facets
sage: sorted(S.vertices())
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]

wedge(right, rename_vertices=True, is_mutable=True)

The wedge (one-point union) of this simplicial complex with another one.

Parameters:

• right – the other simplicial complex (the right-hand factor)

• rename_vertices (boolean; optional, default True) – If this is True, the vertices in the wedge will be renamed by the formula: first vertex in each are glued together and called “0”. Otherwise, each vertex “v” in the left-hand factor –> vertex “Lv” in the wedge, vertex “w” in the right-hand factor –> vertex “Rw” in the wedge. If this is False, this tries to construct the wedge without renaming the vertices; this will cause problems if the two factors have any vertices with names in common.

• is_mutable (boolean; optional, default True) – Determines if the output is mutable

Note

This operation is not well-defined if self or other is not path-connected.

EXAMPLES:

sage: S1 = simplicial_complexes.Sphere(1)
sage: S2 = simplicial_complexes.Sphere(2)
sage: S1.wedge(S2).homology()                                               # needs sage.modules
{0: 0, 1: Z, 2: Z}

sage.topology.simplicial_complex.facets_for_K3()

Return the facets for a minimal triangulation of the K3 surface.

This is a pure simplicial complex of dimension 4 with 16 vertices and 288 facets. The facets are obtained by constructing a few facets and a permutation group \(G\), and then computing the \(G\)-orbit of those facets.

See Casella and Kühnel in [CK2001] and Spreer and Kühnel [SK2011]; the construction here uses the labeling from Spreer and Kühnel.

EXAMPLES:

sage: from sage.topology.simplicial_complex import facets_for_K3
sage: A = facets_for_K3()   # long time (a few seconds)
sage: SimplicialComplex(A) == simplicial_complexes.K3Surface()  # long time
True

sage.topology.simplicial_complex.facets_for_RP4()

Return the list of facets for a minimal triangulation of 4-dimensional real projective space.

We use vertices numbered 1 through 16, define two facets, and define a certain subgroup \(G\) of the symmetric group \(S_{16}\). Then the set of all facets is the \(G\)-orbit of the two given facets.

See the description in Example 3.12 in Datta [Dat2007].

EXAMPLES:

sage: from sage.topology.simplicial_complex import facets_for_RP4
sage: A = facets_for_RP4()   # long time (1 or 2 seconds)
sage: SimplicialComplex(A) == simplicial_complexes.RealProjectiveSpace(4) # long time
True

sage.topology.simplicial_complex.lattice_paths(t1, t2, length=None)

Given lists (or tuples or …) t1 and t2, think of them as labelings for vertices: t1 labeling points on the x-axis, t2 labeling points on the y-axis, both increasing. Return the list of rectilinear paths along the grid defined by these points in the plane, starting from (t1[0], t2[0]), ending at (t1[last], t2[last]), and at each grid point, going either right or up. See the examples.

Parameters:

• t1 (list, other iterable) – labeling for vertices

• t2 (list, other iterable) – labeling for vertices

• length (integer or None; optional, default None) – if not None, then an integer, the length of the desired path.

Returns:

list of lists of vertices making up the paths as described above

Return type:

list of lists

This is used when triangulating the product of simplices. The optional argument length is used for \(\Delta\)-complexes, to specify all simplices in a product: in the triangulation of a product of two simplices, there is a \(d\)-simplex for every path of length \(d+1\) in the lattice. The path must start at the bottom left and end at the upper right, and it must use at least one point in each row and in each column, so if length is too small, there will be no paths.

EXAMPLES:

sage: from sage.topology.simplicial_complex import lattice_paths
sage: lattice_paths([0,1,2], [0,1,2])
[[(0, 0), (0, 1), (0, 2), (1, 2), (2, 2)],
 [(0, 0), (0, 1), (1, 1), (1, 2), (2, 2)],
 [(0, 0), (1, 0), (1, 1), (1, 2), (2, 2)],
 [(0, 0), (0, 1), (1, 1), (2, 1), (2, 2)],
 [(0, 0), (1, 0), (1, 1), (2, 1), (2, 2)],
 [(0, 0), (1, 0), (2, 0), (2, 1), (2, 2)]]
sage: lattice_paths(('a', 'b', 'c'), (0, 3, 5))
[[('a', 0), ('a', 3), ('a', 5), ('b', 5), ('c', 5)],
 [('a', 0), ('a', 3), ('b', 3), ('b', 5), ('c', 5)],
 [('a', 0), ('b', 0), ('b', 3), ('b', 5), ('c', 5)],
 [('a', 0), ('a', 3), ('b', 3), ('c', 3), ('c', 5)],
 [('a', 0), ('b', 0), ('b', 3), ('c', 3), ('c', 5)],
 [('a', 0), ('b', 0), ('c', 0), ('c', 3), ('c', 5)]]
sage: lattice_paths(range(3), range(3), length=2)
[]
sage: lattice_paths(range(3), range(3), length=3)
[[(0, 0), (1, 1), (2, 2)]]
sage: lattice_paths(range(3), range(3), length=4)
[[(0, 0), (1, 1), (1, 2), (2, 2)],
 [(0, 0), (0, 1), (1, 2), (2, 2)],
 [(0, 0), (1, 1), (2, 1), (2, 2)],
 [(0, 0), (1, 0), (2, 1), (2, 2)],
 [(0, 0), (0, 1), (1, 1), (2, 2)],
 [(0, 0), (1, 0), (1, 1), (2, 2)]]

sage.topology.simplicial_complex.rename_vertex(n, keep, left=True)

Rename a vertex: the vertices from the list keep get relabeled 0, 1, 2, …, in order. Any other vertex (e.g. 4) gets renamed to by prepending an ‘L’ or an ‘R’ (thus to either ‘L4’ or ‘R4’), depending on whether the argument left is True or False.

Parameters:

• n – a ‘vertex’: either an integer or a string

• keep – a list of three vertices

• left (boolean; optional, default True) – if True, rename for use in left factor

This is used by the connected_sum() method for simplicial complexes.

EXAMPLES:

sage: from sage.topology.simplicial_complex import rename_vertex
sage: rename_vertex(6, [5, 6, 7])
1
sage: rename_vertex(3, [5, 6, 7, 8, 9])
'L3'
sage: rename_vertex(3, [5, 6, 7], left=False)
'R3'