AUTHORS:
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 \(K\).
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 \((n1)\)simplex obtained by omitting vertex \(v_i\).
In the implementation here, the vertex set must be finite. To define a simplicial complex, specify its vertex set: this should be a list, tuple, or set, or it can be a nonnegative integer \(n\), in which case the vertex set is \((0, ..., n)\). Also specify the facets: the maximal faces.
Note
The elements of the vertex set are not automatically contained in the simplicial complex: each one is only included if and only if it is a vertex of at least one of the specified facets.
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 {(3, 7), (1,)}
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 {(1, 2), (2, 3), (0, 3), (0, 1)}
sage: X.stanley_reisner_ring()
Quotient of Multivariate Polynomial Ring in x0, x1, x2, x3 over Integer Ring by the ideal (x1*x3, x0*x2)
sage: X.is_pure()
True
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
{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: X.stanley_reisner_ring()
Quotient of Multivariate Polynomial Ring in x0, x1, x2, x3 over Integer Ring by the ideal (x1*x3, x0*x2)
Mutability (see trac ticket #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 {(2, 4), (1, 4), (3,)}
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 trac ticket #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
Bases: sage.structure.sage_object.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 or list, tuple, or other iterable) – set of vertices 

Returns:  simplex with those vertices 
X may be a nonnegative 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'
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
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)
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
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]
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
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: 


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')
The product of this simplex with another one, as a list of simplices.
Parameters: 


Algorithm: see Hatcher, p. 277278 [Hat] (who in turn refers to EilenbergSteenrod, 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 lefthand factor and \(w\) is a vertex in the righthand 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))]
The frozenset attached to this simplex.
EXAMPLES:
sage: Simplex(3).set()
frozenset([0, 1, 2, 3])
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())
<type 'tuple'>
sage: type(Simplex(3))
<class 'sage.homology.simplicial_complex.Simplex'>
Bases: sage.homology.cell_complex.GenericCellComplex
Define a simplicial complex.
Parameters: 


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’.
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.
If sort_facets is True, sort the vertices in each facet. If the vertices in different facets are not ordered compatibly (e.g., if you have facets (1, 3, 5) and (5, 3, 8)), then homology calculations may have unpredictable results.
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 StanleyReisner ring for the simplicial complex.
Warning
Earlier versions of Sage supported a vertex_set argument to specify the vertices. This is now deprecated – see trac ticket #12587 – the set of vertices is determined from the maximal faces.
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, 2), (0, 3), (0,)}
sage: S = SimplicialComplex((('a', 'b'), ['a', 'c'], ('b', 'c')))
sage: S
Simplicial complex with vertex set ('a', 'b', 'c') and facets {('b', 'c'), ('a', 'c'), ('a', 'b')}
Finally, if there is only one argument and it is a simplicial complex, return that complex. If it is an object with a builtin 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()
{0: 0, 1: Z x Z, 2: Z}
TESTS:
Check that we can make mutable copies (see trac ticket #14142):
sage: S = SimplicialComplex([[0,2], [0,3]], is_mutable=False)
sage: S.is_mutable()
False
sage: C = copy(S)
sage: C.is_mutable()
True
sage: SimplicialComplex(S, is_mutable=True).is_mutable()
True
sage: SimplicialComplex(S, is_immutable=False).is_mutable()
True
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 {(1, 2, 3), (0, 1)}
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 {(1, 2, 3), (0, 1)}
Check that the bug reported at trac ticket #14354 has been fixed:
sage: T = SimplicialComplex([range(1,5)]).n_skeleton(1)
sage: T.homology()
{0: 0, 1: Z x Z x Z}
sage: T.add_face([1,2,3])
sage: T.homology()
{0: 0, 1: Z x Z, 2: 0}
Check we’ve fixed the bug reported at trac ticket #14578:
sage: t0 = SimplicialComplex()
sage: t0.add_face(('a', 'b'))
sage: t0.add_face(('c', 'd', 'e'))
sage: t0.add_face(('e', 'f', 'c'))
sage: t0.homology()
{0: Z, 1: 0, 2: 0}
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 {(4,), (2,), (3,), (0,), (1,)}
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 {(1, 3), (0, 2)}
Returns 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 trac ticket #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))
True
sage: P = simplicial_complexes.RealProjectivePlane()
sage: P.automorphism_group().is_isomorphic(AlternatingGroup(5))
True
sage: Z = SimplicialComplex([['1','2'],['2','3','a']])
sage: Z.automorphism_group().is_isomorphic(CyclicPermutationGroup(2))
True
sage: group = Z.automorphism_group()
sage: group.domain()
{'1', '2', '3', 'a'}
The barycentric subdivision of this simplicial complex.
See http://en.wikipedia.org/wiki/Barycentric_subdivision for a definition.
EXAMPLES:
sage: triangle = SimplicialComplex([[0,1], [1,2], [0, 2]])
sage: hexagon = triangle.barycentric_subdivision()
sage: hexagon
Simplicial complex with 6 vertices and 6 facets
sage: hexagon.homology(1) == triangle.homology(1)
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)
sage: S4
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 6 facets
sage: S4.barycentric_subdivision()
Simplicial complex with 62 vertices and 720 facets
Return the category to which this simplicial complex belongs: the category of all simplicial complexes.
EXAMPLES:
sage: SimplicialComplex([[0,1], [1,2,3,4,5]]).category()
Category of simplicial complexes
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()
{0: set([(4,), (2,), (1,)]), 1: set([(1, 2), (1, 4)]), 1: set([()])}
sage: L = SimplicialComplex([[1,2]])
sage: Y.faces(subcomplex=L)
{0: set([(4,)]), 1: set([(1, 4)]), 1: set([])}
The chain complex associated to this simplicial complex.
Parameters: 


Note
If subcomplex is nonempty, then the argument augmented has no effect: the chain complex relative to a nonempty subcomplex is zero in dimension \(1\).
EXAMPLES:
sage: circle = SimplicialComplex([[0,1], [1,2], [0, 2]])
sage: circle.chain_complex()
Chain complex with at most 2 nonzero terms over Integer Ring
sage: circle.chain_complex()._latex_()
'\Bold{Z}^{3} \xrightarrow{d_{1}} \Bold{Z}^{3}'
sage: circle.chain_complex(base_ring=QQ, augmented=True)
Chain complex with at most 3 nonzero terms over Rational Field
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 onepoint simplicial complex.
EXAMPLES:
sage: S = SimplicialComplex([[0], [1]])
sage: S.cone()
Simplicial complex with vertex set ('L0', 'L1', 'R0') and facets {('L0', 'R0'), ('L1', 'R0')}
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: v0 = X.vertices()[0]
sage: v1 = X.vertices()[1]
sage: X.connected_component(Simplex([v0])) == X.connected_component(Simplex([v1]))
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.
The connected sum of this simplicial complex with another one.
Parameters: 


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 nonmanifolds, 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 lefthand factor are renamed by prepending an “L”, and similarly the remaining vertices in the righthand factor are renamed by prepending an “R”.
EXAMPLES:
sage: S1 = simplicial_complexes.Sphere(1)
sage: S1.connected_sum(S1.connected_sum(S1)).homology()
{0: 0, 1: Z}
sage: P = simplicial_complexes.RealProjectivePlane(); P
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 10 facets
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]
Z x C2
Returns 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()
True
The disjoint union of this simplicial complex with another one.
Parameters: 


EXAMPLES:
sage: S1 = simplicial_complexes.Sphere(1)
sage: S2 = simplicial_complexes.Sphere(2)
sage: S1.disjoint_union(S2).homology()
{0: Z, 1: Z, 2: Z}
The set of vertices belonging to some face. Returns the list of vertices.
Warning
This method is deprecated. See trac ticket #12587.
EXAMPLES:
sage: S = SimplicialComplex([[0,1,2,3],[6,7]])
sage: S
Simplicial complex with vertex set (0, 1, 2, 3, 6, 7) and facets {(6, 7), (0, 1, 2, 3)}
sage: S.effective_vertices()
doctest:1: DeprecationWarning: effective_vertices is deprecated. Use vertices instead
See http://trac.sagemath.org/12587 for details.
(0, 1, 2, 3, 6, 7)
An iterator for the faces in this simplicial complex.
INPUTS:
EXAMPLES:
sage: S1 = simplicial_complexes.Sphere(1)
sage: [f for f in S1.face_iterator()]
[(), (2,), (0,), (1,), (1, 2), (0, 2), (0, 1)]
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()
{0: set([(4,), (2,), (1,)]), 1: set([(1, 2), (1, 4)]), 1: set([()])}
sage: L = SimplicialComplex([[1,2]])
sage: Y.faces(subcomplex=L)
{0: set([(4,)]), 1: set([(1, 4)]), 1: set([])}
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: Y.maximal_faces()
{(1, 4), (0, 2)}
facets is a synonym for maximal_faces:
sage: S = SimplicialComplex([[0,1], [0,1,2]])
sage: S.facets()
{(0, 1, 2)}
Return the fundamental group of this simplicial complex.
INPUT:
Algorithm: we compute the edgepath group – see Wikipedia article Fundamental_group. Choose a spanning tree for the 1skeleton, and then the group’s generators are given by the edges in the 1skeleton; there are two types of relations: \(e=1\) if \(e\) is in the spanning tree, and for every 2simplex, 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()
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)
sage: C2
Finitely presented group < e0, e1, e2, e3, e4, e5, e6, e7, e8, e9  e6, e5, e3, e9, e4*e7^1*e6, e9*e7^1*e0, e0*e1^1*e2, e5*e1^1*e8, e4*e3^1*e8, e2 >
sage: C2.simplified()
Finitely presented group < e0  e0^2 >
This is the same answer given if the argument simplify is True (the default):
sage: RP2.fundamental_group()
Finitely presented group < e0  e0^2 >
You must specify a base point to compute the fundamental group of a nonconnected complex:
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: v0 = list(K.vertices())[0]
sage: K.fundamental_group(base_point=v0)
Finitely presented group < e  >
sage: v1 = list(K.vertices())[1]
sage: K.fundamental_group(base_point=v1)
Finitely presented group < e0  e0^2 >
Some other examples:
sage: S1.wedge(S1).fundamental_group()
Finitely presented group < e0, e1  >
sage: simplicial_complexes.Torus().fundamental_group()
Finitely presented group < e0, e3  e0*e3^1*e0^1*e3 >
sage: simplicial_complexes.MooreSpace(5).fundamental_group()
Finitely presented group < e1  e1^5 >
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 \(g_i = h_i  h_{i1}\) for \(i > 0\).
EXAMPLES:
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]
Returns the largest subsimplicial complex of self containing exactly sub_vertex_set as vertices.
Parameters: 


EXAMPLES:
sage: S = simplicial_complexes.Sphere(2)
sage: S
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
sage: S.generated_subcomplex([0,1,2])
Simplicial complex with vertex set (0, 1, 2) and facets {(0, 1, 2)}
The 1skeleton 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()
[(0, 1, None), (0, 2, None), (0, 3, None), (1, 2, None), (1, 3, None), (2, 3, None)]
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 empy simplex), then the \(h\)vector \((h_0, h_1, ..., h_d, h_{d+1})\) is defined by
Alternatively,
EXAMPLES:
The \(f\) and \(h\)vectors of the boundary of an octahedron are computed in Wikipedia’s page on simplicial complexes, http://en.wikipedia.org/wiki/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]
Returns True if self is CohenMacaulay, i.e., if \(\tilde{H}_i(\operatorname{lk}_\Delta(F);\ZZ) = 0\) for all \(F \in \Delta\) and \(i < \operatorname{dim}\operatorname{lk}_\Delta(F)\). Here, \(\Delta\) is self, and \(\operatorname{lk}\) denotes the link operator on self.
INPUT:
For finite simplicial complexes, this is equivalent to the statement that the StanleyReisner ring of self is CohenMacaulay.
EXAMPLES:
Spheres are CohenMacaulay:
sage: S = SimplicialComplex([[1,2],[2,3],[3,1]])
sage: S.is_cohen_macaulay(ncpus=3)
True
The following example is taken from Bruns, Herzog  CohenMacaulay rings, Figure 5.3:
sage: S = SimplicialComplex([[1,2,3],[1,4,5]])
sage: S.is_cohen_macaulay(ncpus=3)
...
False
Returns True if and only if self is connected.
Warning
This may give the wrong answer if the simplicial complex was constructed with maximality_check set to False.
EXAMPLES:
sage: V = SimplicialComplex([[0,1,2],[3]])
sage: V
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 2), (3,)}
sage: V.is_connected()
False
sage: X = SimplicialComplex([[0,1,2]])
sage: X.is_connected()
True
sage: U = simplicial_complexes.ChessboardComplex(3,3)
sage: U.is_connected()
True
sage: W = simplicial_complexes.Sphere(3)
sage: W.is_connected()
True
sage: S = SimplicialComplex([[0,1],[2,3]])
sage: S.is_connected()
False
Returns 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 1skeleton. 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()
sage: 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
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
Checks whether two simplicial complexes are isomorphic
INPUT:
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, certify=True)
(True, {0: 'a', 1: 'b', 2: 'c', 3: 'd', 4: 'e', 5: 'f'})
sage: Z3.is_isomorphic(Z2)
False
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
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 \((d1)\)dimensional simplex is the face of exactly two facets
for every two facets \(S\) and \(T\), there is a sequence of facets
such that for each \(i\), \(f_i\) and \(f_{i1}\) intersect in a \((d1)\)simplex.
By convention, \(S^0\) is the only 0dimensional 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
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
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: 


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 {(1, 3), (1, 2), (0, 2), (0, 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
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: 


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 {(3,), (0,)}
sage: Y = SimplicialComplex([[0,1,2,3]])
sage: Y.link([1])
Simplicial complex with vertex set (0, 2, 3) and facets {(0, 2, 3)}
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: Y.maximal_faces()
{(1, 4), (0, 2)}
facets is a synonym for maximal_faces:
sage: S = SimplicialComplex([[0,1], [0,1,2]])
sage: S.facets()
{(0, 1, 2)}
Set consisting of the minimal subsets of the vertex set of this simplicial complex which do not form faces.
Algorithm: first take the complement (within the vertex set) of each facet, obtaining a set \((f_1, f_2, ...)\) of simplices. Now form the set of all simplices of the form \((v_1, v_2, ...)\) where vertex \(v_i\) is in face \(f_i\). This set will contain the minimal nonfaces and may contain some nonminimal nonfaces also, so loop through the set to find the minimal ones. (The last two steps are taken care of by the _transpose_simplices routine.)
This is used in computing the StanleyReisner 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: Y.minimal_nonfaces()
{(1, 3), (0, 2)}
The set of simplices of dimension n of this simplicial complex. If the optional argument subcomplex is present, then return the ndimensional faces which are not in the subcomplex.
Parameters: 


EXAMPLES:
sage: S = Set(range(1,5))
sage: Z = SimplicialComplex(S.subsets())
sage: Z
Simplicial complex with vertex set (1, 2, 3, 4) and facets {(1, 2, 3, 4)}
sage: Z.n_faces(2)
set([(1, 3, 4), (1, 2, 3), (2, 3, 4), (1, 2, 4)])
sage: K = SimplicialComplex([[1,2,3], [2,3,4]])
sage: Z.n_faces(2, subcomplex=K)
set([(1, 3, 4), (1, 2, 4)])
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 – nonnegative 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 {(2, 3), (0, 2), (1, 3), (1, 2), (0, 3), (0, 1)}
sage: X.set_immutable()
sage: X.n_skeleton(2)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (1, 2, 3), (0, 1)}
The product of this simplicial complex with another one.
Parameters: 


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: S.product(K).vertices() # cylinder
('L0R0', 'L0R1', 'L1R0', 'L1R1', 'L2R0', 'L2R1')
sage: S.product(K, rename_vertices=False).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()
{0: 0, 1: Z x Z, 2: Z}
These can get large pretty quickly:
sage: T = simplicial_complexes.Torus(); T
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6) and 14 facets
sage: K = simplicial_complexes.KleinBottle(); K
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 16 facets
sage: T.product(K) # long time: 5 or 6 seconds
Simplicial complex with 56 vertices and 1344 facets
Remove a face from this simplicial complex and return the resulting simplicial complex.
Parameters:  face – a face of the simplicial complex 

This 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 {(0, 1, 2), (2, 3)}
sage: S.remove_face([0,1,2])
sage: S
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(1, 2), (2, 3), (0, 2), (0, 1)}
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
The StanleyReisner 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 nonnegative 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], [1,2], [2,3], [0,3]])
sage: X.stanley_reisner_ring()
Quotient of Multivariate Polynomial Ring in x0, x1, x2, x3 over Integer Ring by the ideal (x1*x3, x0*x2)
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
The suspension of this simplicial complex.
Parameters: 


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 twopoint simplicial complex.
If the simplicial complex \(M\) happens to be a pseudomanifold (see is_pseudomanifold()), then this instead constructs Datta’s onepoint suspension (see p. 434 in the cited article): choose a vertex \(u\) in \(M\) and choose a new vertex \(w\) to add. Denote the join of simplices by “\(*\)”. The facets in the onepoint suspension are of the two forms
REFERENCES:
EXAMPLES:
sage: S0 = SimplicialComplex([[0], [1]])
sage: S0.suspension() == simplicial_complexes.Sphere(1)
True
sage: S3 = S0.suspension(3) # the 3sphere
sage: S3.homology()
{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 0sphere: the join adds two vertices, while this construction only adds one.
sage: T = simplicial_complexes.Torus()
sage: 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)
The vertex set 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: S.vertices()
(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)
Note that this actually returns a simplex:
sage: type(S.vertices())
<class 'sage.homology.simplicial_complex.Simplex'>
The wedge (onepoint union) of this simplicial complex with another one.
Parameters: 


Note
This operation is not welldefined if self or other is not pathconnected.
EXAMPLES:
sage: S1 = simplicial_complexes.Sphere(1)
sage: S2 = simplicial_complexes.Sphere(2)
sage: S1.wedge(S2).homology()
{0: 0, 1: Z, 2: Z}
Given lists (or tuples or ...) t1 and t2, think of them as labelings for vertices: t1 labeling points on the xaxis, t2 labeling points on the yaxis, 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: 


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.homology.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)]]
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: 


This is used by the connected_sum() method for simplicial complexes.
EXAMPLES:
sage: from sage.homology.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'