Examples of simplicial complexes¶

AUTHORS:

John H. Palmieri (2009-04)

This file constructs some examples of simplicial complexes. There are two main types: surfaces and examples related to graph theory.

For surfaces (and other manifolds), there are functions defining the 2-sphere, the n-sphere for any n, the torus, the real projective plane, the Klein bottle, and surfaces of arbitrary genus, all as simplicial complexes.

Aside from surfaces, this file also provides some functions for constructing some other simplicial complexes: the simplicial complex of not-i-connected graphs on n vertices, the matching complex on n vertices, and the chessboard complex for an n by i chessboard. These provide examples of large simplicial complexes; for example, not_i_connected_graphs(7,2) has over a million simplices.

All of these examples are accessible by typing “simplicial_complexes.NAME”, where “NAME” is the name of the example; you can get a list by typing “simplicial_complexes.” and hitting the TAB key:

Sphere
Simplex
Torus
RealProjectivePlane
KleinBottle
SurfaceOfGenus
MooreSpace
NotIConnectedGraphs
MatchingComplex
ChessboardComplex
RandomComplex

See the documentation for simplicial_complexes and for each particular type of example for full details.

class sage.homology.examples.SimplicialComplexExamples¶

Some examples of simplicial complexes.

Here are the available examples; you can also type “simplicial_complexes.” and hit tab to get a list:

Sphere
Simplex
Torus
RealProjectivePlane
KleinBottle
SurfaceOfGenus
MooreSpace
NotIConnectedGraphs
MatchingComplex
ChessboardComplex
RandomComplex

EXAMPLES:

sage: S = simplicial_complexes.Sphere(2) # the 2-sphere
sage: S.homology()
{0: 0, 1: 0, 2: Z}
sage: simplicial_complexes.SurfaceOfGenus(3)
Simplicial complex with 15 vertices and 38 facets
sage: M4 = simplicial_complexes.MooreSpace(4)
sage: M4.homology()
{0: 0, 1: C4, 2: 0}
sage: simplicial_complexes.MatchingComplex(6).homology()
{0: 0, 1: Z^16, 2: 0}

ChessboardComplex(n, i)¶

The chessboard complex for an n by i chessboard.

Fix integers $n, i > 0$ and consider sets $V$ of $n$ vertices and $W$ of $i$ vertices. A ‘partial matching’ between $V$ and $W$ is a graph formed by edges $(v,w)$ with $v \in V$ and $w \in W$ so that each vertex is in at most one edge. If $G$ is a partial matching, then so is any graph obtained by deleting edges from $G$ . Thus the set of all partial matchings on $V$ and $W$ , viewed as a set of subsets of the $n+i$ choose 2 possible edges, is closed under taking subsets, and thus forms a simplicial complex called the ‘chessboard complex’. This function produces that simplicial complex. (It is called the chessboard complex because such graphs also correspond to ways of placing rooks on an $n$ by $i$ chessboard so that none of them are attacking each other.)

INPUT:

n, i - positive integers.

See Dumas et al. for information on computing its homology by computer, and see Wachs for an expository article about the theory.

EXAMPLES:

sage: C = simplicial_complexes.ChessboardComplex(5,5)
sage: C.f_vector()
[1, 25, 200, 600, 600, 120]
sage: simplicial_complexes.ChessboardComplex(3,3).homology()
{0: 0, 1: Z x Z x Z x Z, 2: 0}

REFERENCES:

Dumas, Heckenbach, Saunders, Welker, “Computing simplicial homology based on efficient Smith normal form algorithms,” in “Algebra, geometry, and software systems” (2003), 177-206.
Wachs, “Topology of Matching, Chessboard and General Bounded Degree Graph Complexes” (Algebra Universalis Special Issue in Memory of Gian-Carlo Rota, Algebra Universalis, 49 (2003) 345-385)

KleinBottle()¶

A triangulation of the Klein bottle, formed by taking the connected sum of a real projective plane with itself. (This is not a minimal triangulation.)

EXAMPLES:

sage: simplicial_complexes.KleinBottle()
Simplicial complex with 9 vertices and 18 facets

MatchingComplex(n)¶

The matching complex of graphs on n vertices.

Fix an integer $n>0$ and consider a set $V$ of $n$ vertices. A ‘partial matching’ on $V$ is a graph formed by edges so that each vertex is in at most one edge. If $G$ is a partial matching, then so is any graph obtained by deleting edges from $G$ . Thus the set of all partial matchings on $n$ vertices, viewed as a set of subsets of the $n$ choose 2 possible edges, is closed under taking subsets, and thus forms a simplicial complex called the ‘matching complex’. This function produces that simplicial complex.

INPUT:

n - positive integer.

See Dumas et al. for information on computing its homology by computer, and see Wachs for an expository article about the theory. For example, the homology of these complexes seems to have only mod 3 torsion, and this has been proved for the bottom non-vanishing homology group for the matching complex $M_n$ .

EXAMPLES:

sage: M = simplicial_complexes.MatchingComplex(7)
sage: H = M.homology()
sage: H
{0: 0, 1: C3, 2: Z^20}
sage: H[2].ngens()
20
sage: simplicial_complexes.MatchingComplex(8).homology(2)  # long time (a few seconds)
Z^132

REFERENCES:

Dumas, Heckenbach, Saunders, Welker, “Computing simplicial homology based on efficient Smith normal form algorithms,” in “Algebra, geometry, and software systems” (2003), 177-206.
Wachs, “Topology of Matching, Chessboard and General Bounded Degree Graph Complexes” (Algebra Universalis Special Issue in Memory of Gian-Carlo Rota, Algebra Universalis, 49 (2003) 345-385)

MooreSpace(q)¶

Triangulation of the mod q Moore space.

INPUT:

q - integer, at least 2

This is a simplicial complex with simplices of dimension 0, 1, and 2, such that its reduced homology is isomorphic to $\ZZ/q\ZZ$ in dimension 1, zero otherwise.

If $q=2$ , this is the real projective plane. If $q>2$ , then construct it as follows: start with a triangle with vertices 1, 2, 3. We take a $3q$ -gon forming a $q$ -fold cover of the triangle, and we form the resulting complex as an identification space of the $3q$ -gon. To triangulate this identification space, put $q$ vertices $A_0$ , ..., $A_{q-1}$ , in the interior, each of which is connected to 1, 2, 3 (two facets each: $[1, 2, A_i]$ , $[2, 3, A_i]$ ). Put $q$ more vertices in the interior: $B_0$ , ..., $B_{q-1}$ , with facets $[3, 1, B_i]$ , $[3, B_i, A_i]$ , $[1, B_i, A_{i+1}]$ , $[B_i, A_i, A_{i+1}]$ . Then triangulate the interior polygon with vertices $A_0$ , $A_1$ , ..., $A_{q-1}$ .

EXAMPLES:

sage: simplicial_complexes.MooreSpace(3).homology()[1]
C3
sage: simplicial_complexes.MooreSpace(4).suspension().homology()[2]
C4
sage: simplicial_complexes.MooreSpace(8)
Simplicial complex with 19 vertices and 54 facets

NotIConnectedGraphs(n, i)¶

The simplicial complex of all graphs on n vertices which are not i-connected.

Fix an integer $n>0$ and consider the set of graphs on $n$ vertices. View each graph as its set of edges, so it is a subset of a set of size $n$ choose 2. A graph is $i$ -connected if, for any $j<i$ , if any $j$ vertices are removed along with the edges emanating from them, then the graph remains connected. Now fix $i$ : it is clear that if $G$ is not $i$ -connected, then the same is true for any graph obtained from $G$ by deleting edges. Thus the set of all graphs which are not $i$ -connected, viewed as a set of subsets of the $n$ choose 2 possible edges, is closed under taking subsets, and thus forms a simplicial complex. This function produces that simplicial complex.

INPUT:

n, i - non-negative integers with $i$ at most $n$

See Dumas et al. for information on computing its homology by computer, and see Babson et al. for theory. For example, Babson et al. show that when $i=2$ , the reduced homology of this complex is nonzero only in dimension $2n-5$ , where it is free abelian of rank $(n-2)!$ .

EXAMPLES:

sage: simplicial_complexes.NotIConnectedGraphs(5,2).f_vector()
[1, 10, 45, 120, 210, 240, 140, 20]
sage: simplicial_complexes.NotIConnectedGraphs(5,2).homology(5).ngens()
6

REFERENCES:

Babson, Bjorner, Linusson, Shareshian, and Welker, “Complexes of not i-connected graphs,” Topology 38 (1999), 271-299
Dumas, Heckenbach, Saunders, Welker, “Computing simplicial homology based on efficient Smith normal form algorithms,” in “Algebra, geometry, and software systems” (2003), 177-206.

ProjectivePlane()¶

A minimal triangulation of the real projective plane.

EXAMPLES:

sage: P = simplicial_complexes.RealProjectivePlane()
sage: Q = simplicial_complexes.ProjectivePlane()
sage: P == Q
True
sage: P.cohomology(1)
0
sage: P.cohomology(2)
C2
sage: P.cohomology(1, base_ring=GF(2))
Vector space of dimension 1 over Finite Field of size 2
sage: P.cohomology(2, base_ring=GF(2))
Vector space of dimension 1 over Finite Field of size 2

RandomComplex(n, d, p=0.5)¶

A random d-dimensional simplicial complex on n vertices.

INPUT:

n - number of vertices
d - dimension of the complex
p - floating point number between 0 and 1 (optional, default 0.5)

A random $d$ -dimensional simplicial complex on $n$ vertices, as defined for example by Meshulam and Wallach, is constructed as follows: take $n$ vertices and include all of the simplices of dimension strictly less than $d$ , and then for each possible simplex of dimension $d$ , include it with probability $p$ .

EXAMPLES:

sage: simplicial_complexes.RandomComplex(6, 2)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6) and 15 facets

If $d$ is too large (if $d > n+1$ , so that there are no $d$ -dimensional simplices), then return the simplicial complex with a single $(n+1)$ -dimensional simplex:

sage: simplicial_complexes.RandomComplex(6,12)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and facets {(0, 1, 2, 3, 4, 5, 6, 7)}

REFERENCES:

Meshulam and Wallach, “Homological connectivity of random $k$ -dimensional complexes”, preprint, math.CO/0609773.

RealProjectivePlane()¶

A minimal triangulation of the real projective plane.

EXAMPLES:

sage: P = simplicial_complexes.RealProjectivePlane()
sage: Q = simplicial_complexes.ProjectivePlane()
sage: P == Q
True
sage: P.cohomology(1)
0
sage: P.cohomology(2)
C2
sage: P.cohomology(1, base_ring=GF(2))
Vector space of dimension 1 over Finite Field of size 2
sage: P.cohomology(2, base_ring=GF(2))
Vector space of dimension 1 over Finite Field of size 2

Simplex(n)¶

An $n$ -dimensional simplex, as a simplicial complex.

INPUT:

n - a non-negative integer

OUTPUT: the simplicial complex consisting of the $n$ -simplex on vertices $(0, 1, ..., n)$ and all of its faces.

EXAMPLES:

sage: simplicial_complexes.Simplex(3)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 1, 2, 3)}
sage: simplicial_complexes.Simplex(5).euler_characteristic()
1

Sphere(n)¶

A minimal triangulation of the n-dimensional sphere.

INPUT:

n - positive integer

EXAMPLES:

sage: simplicial_complexes.Sphere(2)
Simplicial complex with vertex set (0, 1, 2, 3) and facets {(0, 2, 3), (0, 1, 2), (1, 2, 3), (0, 1, 3)}
sage: simplicial_complexes.Sphere(5).homology()
{0: 0, 1: 0, 2: 0, 3: 0, 4: 0, 5: Z}
sage: [simplicial_complexes.Sphere(n).euler_characteristic() for n in range(6)]
[2, 0, 2, 0, 2, 0]
sage: [simplicial_complexes.Sphere(n).f_vector() for n in range(6)]
[[1, 2],
 [1, 3, 3],
 [1, 4, 6, 4],
 [1, 5, 10, 10, 5],
 [1, 6, 15, 20, 15, 6],
 [1, 7, 21, 35, 35, 21, 7]]

SurfaceOfGenus(g, orientable=True)¶

A surface of genus g.

INPUT:

g - a non-negative integer. The desired genus
orientable - boolean (optional, default True). If True, return an orientable surface, and if False, return a non-orientable surface.

In the orientable case, return a sphere if $g$ is zero, and otherwise return a $g$ -fold connected sum of a torus with itself.

In the non-orientable case, raise an error if $g$ is zero. If $g$ is positive, return a $g$ -fold connected sum of a real projective plane with itself.

EXAMPLES:

sage: simplicial_complexes.SurfaceOfGenus(2)
Simplicial complex with 11 vertices and 26 facets
sage: simplicial_complexes.SurfaceOfGenus(1, orientable=False)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 10 facets

Torus()¶

A minimal triangulation of the torus.

EXAMPLES:

sage: simplicial_complexes.Torus().homology(1)
Z x Z

sage.homology.examples.matching(A, B)¶

List of maximal matchings between the sets A and B: a matching is a set of pairs (a,b) in A x B where each a, b appears in at most one pair. A maximal matching is one which is maximal with respect to inclusion of subsets of A x B.

INPUT:

A, B - list, tuple, or indeed anything which can be converted to a set.

EXAMPLES:

sage: from sage.homology.examples import matching
sage: matching([1,2], [3,4])
[set([(1, 3), (2, 4)]), set([(2, 3), (1, 4)])]
sage: matching([0,2], [0])
[set([(0, 0)]), set([(2, 0)])]

Examples of simplicial complexes¶

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