"""
Examples of simplicial complexes
AUTHORS:
- John H. Palmieri (2009-04)
This file constructs some examples of simplicial complexes. There are
two main types: manifolds and examples related to graph theory.
For manifolds, there are functions defining the `n`-sphere for any
`n`, the torus, `n`-dimensional real projective space for any `n`, the
complex projective plane, surfaces of arbitrary genus, and some other
manifolds, 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,
``simplicial_complexes.NotIConnectedGraphs(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::
simplicial_complexes.BarnetteSphere
simplicial_complexes.BrucknerGrunbaumSphere
simplicial_complexes.ChessboardComplex
simplicial_complexes.ComplexProjectivePlane
simplicial_complexes.K3Surface
simplicial_complexes.KleinBottle
simplicial_complexes.MatchingComplex
simplicial_complexes.MooreSpace
simplicial_complexes.NotIConnectedGraphs
simplicial_complexes.PoincareHomologyThreeSphere
simplicial_complexes.PseudoQuaternionicProjectivePlane
simplicial_complexes.RandomComplex
simplicial_complexes.RealProjectivePlane
simplicial_complexes.RealProjectiveSpace
simplicial_complexes.Simplex
simplicial_complexes.Sphere
simplicial_complexes.SumComplex
simplicial_complexes.SurfaceOfGenus
simplicial_complexes.Torus
See the documentation for ``simplicial_complexes`` and for each
particular type of example for full details.
"""
from sage.homology.simplicial_complex import SimplicialComplex, Simplex
from sage.sets.set import Set
from sage.misc.functional import is_even
from sage.combinat.subset import Subsets
import sage.misc.prandom as random
def matching(A, B):
r"""
List of maximal matchings between the sets ``A`` and ``B``.
A matching is a set of pairs `(a,b) \in A \times B` where each `a` and
`b` appears in at most one pair. A maximal matching is one which is
maximal with respect to inclusion of subsets of `A \times 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)])]
"""
answer = []
if len(A) == 0 or len(B) == 0:
return [set([])]
for v in A:
for w in B:
for M in matching(set(A).difference([v]), set(B).difference([w])):
new = M.union([(v,w)])
if new not in answer:
answer.append(new)
return answer
def 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 [Da2007]_.
EXAMPLES::
sage: from sage.homology.examples 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
"""
from sage.groups.perm_gps.permgroup import PermutationGroup
g1 = '(2,7)(4,10)(5,6)(11,12)'
g2 = '(1, 2, 3, 4, 5, 10)(6, 8, 9)(11, 12, 13, 14, 15, 16)'
G = PermutationGroup([g1, g2])
t1 = (1, 2, 4, 5, 11)
t2 = (1, 2, 4, 11, 13)
facets = []
for g in G:
d = g.dict()
for t in [t1, t2]:
new = tuple([d[j] for j in t])
if new not in facets:
facets.append(new)
return facets
def facets_for_K3():
"""
Returns 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.homology.examples import facets_for_K3
sage: A = facets_for_K3() # long time (a few seconds)
sage: SimplicialComplex(A) == simplicial_complexes.K3Surface() # long time
True
"""
from sage.groups.perm_gps.permgroup import PermutationGroup
G = PermutationGroup([[(1,3,8,4,9,16,15,2,14,12,6,7,13,5,10)],
[(1,11,16),(2,10,14),(3,12,13),(4,9,15),(5,7,8)]])
return ([tuple([g(i) for i in (1,2,3,8,12)]) for g in G]
+[tuple([g(i) for i in (1,2,5,8,14)]) for g in G])
SimplicialSurface = SimplicialComplex
class SimplicialComplexExamples():
"""
Some examples of simplicial complexes.
Here are the available examples; you can also type
``simplicial_complexes.`` and hit tab to get a list:
- :meth:`BarnetteSphere`
- :meth:`BrucknerGrunbaumSphere`
- :meth:`ChessboardComplex`
- :meth:`ComplexProjectivePlane`
- :meth:`K3Surface`
- :meth:`KleinBottle`
- :meth:`MatchingComplex`
- :meth:`MooreSpace`
- :meth:`NotIConnectedGraphs`
- :meth:`PoincareHomologyThreeSphere`
- :meth:`PseudoQuaternionicProjectivePlane`
- :meth:`RandomComplex`
- :meth:`RealProjectivePlane`
- :meth:`RealProjectiveSpace`
- :meth:`Simplex`
- :meth:`Sphere`
- :meth:`SumComplex`
- :meth:`SurfaceOfGenus`
- :meth:`Torus`
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}
"""
def Sphere(self,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]]
"""
S = Simplex(n+1)
facets = S.faces()
return SimplicialComplex(facets, is_mutable=False)
def Simplex(self, 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
"""
return SimplicialComplex([Simplex(n)], is_mutable=False)
def Torus(self):
"""
A minimal triangulation of the torus.
EXAMPLES::
sage: simplicial_complexes.Torus().homology(1)
Z x Z
"""
return SimplicialComplex([[0,1,2], [1,2,4], [1,3,4], [1,3,6],
[0,1,5], [1,5,6], [2,3,5], [2,4,5],
[2,3,6], [0,2,6], [0,3,4], [0,3,5],
[4,5,6], [0,4,6]],
is_mutable=False)
def RealProjectivePlane(self):
"""
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
"""
return SimplicialComplex([[0,1,2], [0,2,3], [0,1,5], [0,4,5],
[0,3,4], [1,2,4], [1,3,4], [1,3,5],
[2,3,5], [2,4,5]],
is_mutable=False)
ProjectivePlane = RealProjectivePlane
def KleinBottle(self):
"""
A minimal triangulation of the Klein bottle, as presented for example
in Davide Cervone's thesis [Ce1994]_.
EXAMPLES::
sage: simplicial_complexes.KleinBottle()
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5, 6, 7) and 16 facets
REFERENCES:
.. [Ce1994] D. P. Cervone, "Vertex-minimal simplicial immersions of the Klein
bottle in three-space", Geom. Ded. 50 (1994) 117-141,
http://www.math.union.edu/~dpvc/papers/1993-03.kb/vmkb.pdf.
"""
return SimplicialComplex([[2,3,7], [1,2,3], [1,3,5], [1,5,7],
[1,4,7], [2,4,6], [1,2,6], [1,6,0],
[1,4,0], [2,4,0], [3,4,7], [3,4,6],
[3,5,6], [5,6,0], [2,5,0], [2,5,7]],
is_mutable=False)
def SurfaceOfGenus(self, 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
"""
if g == 0:
if not orientable:
raise ValueError, "No non-orientable surface of genus zero."
else:
return simplicial_complexes.Sphere(2)
if orientable:
T = simplicial_complexes.Torus()
else:
T = simplicial_complexes.RealProjectivePlane()
S = T
for i in range(g-1):
S = S.connected_sum(T, is_mutable=False)
return S
def MooreSpace(self, q):
"""
Triangulation of the mod `q` Moore space.
INPUT:
- ``q`` -0 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(2)
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 10 facets
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
"""
if q <= 1:
raise ValueError("The mod q Moore space is only defined if q is at least 2")
if q == 2:
return simplicial_complexes.RealProjectivePlane()
facets = []
for i in range(q):
Ai = "A" + str(i)
Aiplus = "A" + str((i+1)%q)
Bi = "B" + str(i)
facets.append([1, 2, Ai])
facets.append([2, 3, Ai])
facets.append([3, 1, Bi])
facets.append([3, Bi, Ai])
facets.append([1, Bi, Aiplus])
facets.append([Bi, Ai, Aiplus])
for i in range(1, q-1):
Ai = "A" + str(i)
Aiplus = "A" + str((i+1)%q)
facets.append(["A0", Ai, Aiplus])
return SimplicialComplex(facets, is_immutable=True)
def ComplexProjectivePlane(self):
"""
A minimal triangulation of the complex projective plane.
This was constructed by Kühnel and Banchoff [KB1983]_.
REFERENCES:
.. [KB1983] W. Kühnel and T. F. Banchoff, "The 9-vertex complex
projective plane", Math. Intelligencer 5 (1983), no. 3,
11-22.
EXAMPLES::
sage: C = simplicial_complexes.ComplexProjectivePlane()
sage: C.f_vector()
[1, 9, 36, 84, 90, 36]
sage: C.homology(2)
Z
sage: C.homology(4)
Z
"""
return SimplicialComplex(
[[1, 2, 4, 5, 6], [2, 3, 5, 6, 4], [3, 1, 6, 4, 5],
[1, 2, 4, 5, 9], [2, 3, 5, 6, 7], [3, 1, 6, 4, 8],
[2, 3, 6, 4, 9], [3, 1, 4, 5, 7], [1, 2, 5, 6, 8],
[3, 1, 5, 6, 9], [1, 2, 6, 4, 7], [2, 3, 4, 5, 8],
[4, 5, 7, 8, 9], [5, 6, 8, 9, 7], [6, 4, 9, 7, 8],
[4, 5, 7, 8, 3], [5, 6, 8, 9, 1], [6, 4, 9, 7, 2],
[5, 6, 9, 7, 3], [6, 4, 7, 8, 1], [4, 5, 8, 9, 2],
[6, 4, 8, 9, 3], [4, 5, 9, 7, 1], [5, 6, 7, 8, 2],
[7, 8, 1, 2, 3], [8, 9, 2, 3, 1], [9, 7, 3, 1, 2],
[7, 8, 1, 2, 6], [8, 9, 2, 3, 4], [9, 7, 3, 1, 5],
[8, 9, 3, 1, 6], [9, 7, 1, 2, 4], [7, 8, 2, 3, 5],
[9, 7, 2, 3, 6], [7, 8, 3, 1, 4], [8, 9, 1, 2, 5]],
is_mutable=False)
def PseudoQuaternionicProjectivePlane(self):
r"""
Returns a pure simplicial complex of dimension 8 with 490 facets.
.. WARNING::
This is expected to be a triangulation of the projective plane
`HP^2` over the ring of quaternions, but this has not been
proved yet.
This simplicial complex has the same homology as `HP^2`. Its
automorphism group is isomorphic to the alternating group `A_5`
and acts transitively on vertices.
This is defined here using the description in [BrK92]_. This
article deals with three different triangulations. This procedure
returns the only one which has a transitive group of
automorphisms.
EXAMPLES::
sage: HP2 = simplicial_complexes.PseudoQuaternionicProjectivePlane() ; HP2
Simplicial complex with 15 vertices and 490 facets
sage: HP2.f_vector()
[1, 15, 105, 455, 1365, 3003, 4515, 4230, 2205, 490]
Checking its automorphism group::
sage: HP2.automorphism_group().is_isomorphic(AlternatingGroup(5))
True
REFERENCES:
.. [BrK92] Brehm U., Kuhnel W., "15-vertex triangulations of an
8-manifold", Math. Annalen 294 (1992), no. 1, 167-193.
"""
from sage.groups.perm_gps.permgroup import PermutationGroup
P = [(1,2,3,4,5),(6,7,8,9,10),(11,12,13,14,15)]
S = [(1,6,11),(2,15,14),(3,13,8),(4,7,5),(9,12,10)]
start_list = [
(1,2,3,6,8,11,13,14,15),
(1,3,6,8,9,10,11,12,13),
(1,2,6,9,10,11,12,14,15),
(1,2,3,4,7,9,12,14,15),
(1,2,4,7,9,10,12,13,14),
(1,2,6,8,9,10,11,14,15),
(1,2,3,4,5,6,9,11,13),
(1,3,5,6,8,9,10,11,12),
(1,3,5,6,7,8,9,10,11),
(1,2,3,4,5,7,10,12,15),
(1,2,3,7,8,10,12,13,14),
(2,5,6,7,8,9,10,13,14),
(3,4,6,7,11,12,13,14,15),
(3,4,6,7,10,12,13,14,15)]
return SimplicialComplex([ [g(index) for index in tuple]
for tuple in start_list
for g in PermutationGroup([P,S]) ])
def PoincareHomologyThreeSphere(self):
"""
A triangulation of the Poincare homology 3-sphere.
This is a manifold whose integral homology is identical to the
ordinary 3-sphere, but it is not simply connected. In particular,
its fundamental group is the binary icosahedral group, which has
order 120. The triangulation given here has 16 vertices and is
due to Björner and Lutz [BL2000]_.
REFERENCES:
.. [BL2000] Anders Björner and Frank H. Lutz, "Simplicial
manifolds, bistellar flips and a 16-vertex triangulation of
the Poincaré homology 3-sphere", Experiment. Math. 9
(2000), no. 2, 275-289.
EXAMPLES::
sage: S3 = simplicial_complexes.Sphere(3)
sage: Sigma3 = simplicial_complexes.PoincareHomologyThreeSphere()
sage: S3.homology() == Sigma3.homology()
True
sage: Sigma3.fundamental_group().cardinality() # long time
120
"""
return SimplicialComplex(
[[1, 2, 4, 9], [1, 2, 4, 15], [1, 2, 6, 14], [1, 2, 6, 15],
[1, 2, 9, 14], [1, 3, 4, 12], [1, 3, 4, 15], [1, 3, 7, 10],
[1, 3, 7, 12], [1, 3, 10, 15], [1, 4, 9, 12], [1, 5, 6, 13],
[1, 5, 6, 14], [1, 5, 8, 11], [1, 5, 8, 13], [1, 5, 11, 14],
[1, 6, 13, 15], [1, 7, 8, 10], [1, 7, 8, 11], [1, 7, 11, 12],
[1, 8, 10, 13], [1, 9, 11, 12], [1, 9, 11, 14], [1, 10, 13, 15],
[2, 3, 5, 10], [2, 3, 5, 11], [2, 3, 7, 10], [2, 3, 7, 13],
[2, 3, 11, 13], [2, 4, 9, 13], [2, 4, 11, 13], [2, 4, 11, 15],
[2, 5, 8, 11], [2, 5, 8, 12], [2, 5, 10, 12], [2, 6, 10, 12],
[2, 6, 10, 14], [2, 6, 12, 15], [2, 7, 9, 13], [2, 7, 9, 14],
[2, 7, 10, 14], [2, 8, 11, 15], [2, 8, 12, 15], [3, 4, 5, 14],
[3, 4, 5, 15], [3, 4, 12, 14], [3, 5, 10, 15], [3, 5, 11, 14],
[3, 7, 12, 13], [3, 11, 13, 14], [3, 12, 13, 14], [4, 5, 6, 7],
[4, 5, 6, 14], [4, 5, 7, 15], [4, 6, 7, 11], [4, 6, 10, 11],
[4, 6, 10, 14], [4, 7, 11, 15], [4, 8, 9, 12], [4, 8, 9, 13],
[4, 8, 10, 13], [4, 8, 10, 14], [4, 8, 12, 14], [4, 10, 11, 13],
[5, 6, 7, 13], [5, 7, 9, 13], [5, 7, 9, 15], [5, 8, 9, 12],
[5, 8, 9, 13], [5, 9, 10, 12], [5, 9, 10, 15], [6, 7, 11, 12],
[6, 7, 12, 13], [6, 10, 11, 12], [6, 12, 13, 15], [7, 8, 10, 14],
[7, 8, 11, 15], [7, 8, 14, 15], [7, 9, 14, 15], [8, 12, 14, 15],
[9, 10, 11, 12], [9, 10, 11, 16], [9, 10, 15, 16], [9, 11, 14, 16],
[9, 14, 15, 16], [10, 11, 13, 16], [10, 13, 15, 16],
[11, 13, 14, 16], [12, 13, 14, 15], [13, 14, 15, 16]],
is_mutable=False)
def RealProjectiveSpace(self, n):
r"""
A triangulation of `\Bold{R}P^n` for any `n \geq 0`.
INPUT:
- ``n`` -- integer, the dimension of the real projective space
to construct
The first few cases are pretty trivial:
- `\Bold{R}P^0` is a point.
- `\Bold{R}P^1` is a circle, triangulated as the boundary of a
single 2-simplex.
- `\Bold{R}P^2` is the real projective plane, here given its
minimal triangulation with 6 vertices, 15 edges, and 10
triangles.
- `\Bold{R}P^3`: any triangulation has at least 11 vertices by
a result of Walkup [Wa1970]_; this function returns a
triangulation with 11 vertices, as given by Lutz [Lu2005]_.
- `\Bold{R}P^4`: any triangulation has at least 16 vertices by
a result of Walkup; this function returns a triangulation
with 16 vertices as given by Lutz; see also Datta [Da2007]_,
Example 3.12.
- `\Bold{R}P^n`: Lutz has found a triangulation of
`\Bold{R}P^5` with 24 vertices, but it does not seem to have
been published. Kühnel [Ku1987]_ has described a triangulation of
`\Bold{R}P^n`, in general, with `2^{n+1}-1` vertices; see
also Datta, Example 3.21. This triangulation is presumably
not minimal, but it seems to be the best in the published
literature as of this writing. So this function returns it
when `n > 4`.
ALGORITHM: For `n < 4`, these are constructed explicitly by
listing the facets. For `n = 4`, this is constructed by
specifying 16 vertices, two facets, and 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. This is implemented
here by explicitly listing all of the facets; the facets
can be computed by the function :func:`facets_for_RP4`, but
running the function takes a few seconds.
For `n > 4`, the construction is as follows: let `S` denote
the simplicial complex structure on the `n`-sphere given by
the first barycentric subdivision of the boundary of an
`(n+1)`-simplex. This has a simplicial antipodal action: if
`V` denotes the vertices in the boundary of the simplex, then
the vertices in its barycentric subdivision `S` correspond to
nonempty proper subsets `U` of `V`, and the antipodal action
sends any subset `U` to its complement. One can show that
modding out by this action results in a triangulation for
`\Bold{R}P^n`. To find the facets in this triangulation, find
the facets in `S`. These are indentified in pairs to form
`\Bold{R}P^n`, so choose a representative from each pair: for
each facet in `S`, replace any vertex in `S` containing 0 with
its complement.
Of course these complexes increase in size pretty quickly as
`n` increases.
REFERENCES:
.. [Da2007] Basudeb Datta, "Minimal triangulations of manifolds",
J. Indian Inst. Sci. 87 (2007), no. 4, 429-449.
.. [Ku1987] W. Kühnel, "Minimal triangulations of Kummer varieties",
Abh. Math. Sem. Univ. Hamburg 57 (1987), 7-20.
.. [Lu2005] Frank H. Lutz, "Triangulated Manifolds with Few Vertices:
Combinatorial Manifolds", preprint (2005),
arXiv:math/0506372.
.. [Wa1970] David W. Walkup, "The lower bound conjecture for 3- and
4-manifolds", Acta Math. 125 (1970), 75-107.
EXAMPLES::
sage: P3 = simplicial_complexes.RealProjectiveSpace(3)
sage: P3.f_vector()
[1, 11, 51, 80, 40]
sage: P3.homology()
{0: 0, 1: C2, 2: 0, 3: Z}
sage: P4 = simplicial_complexes.RealProjectiveSpace(4)
sage: P4.f_vector()
[1, 16, 120, 330, 375, 150]
sage: P4.homology() # long time
{0: 0, 1: C2, 2: 0, 3: C2, 4: 0}
sage: P5 = simplicial_complexes.RealProjectiveSpace(5) # long time (44s on sage.math, 2012)
sage: P5.f_vector() # long time
[1, 63, 903, 4200, 8400, 7560, 2520]
The following computation can take a long time -- over half an
hour -- with Sage's default computation of homology groups,
but if you have CHomP installed, Sage will use that and the
computation should only take a second or two. (You can
download CHomP from http://chomp.rutgers.edu/, or you can
install it as a Sage package using ``sage -i chomp``). ::
sage: P5.homology() # long time # optional - CHomP
{0: 0, 1: C2, 2: 0, 3: C2, 4: 0, 5: Z}
sage: simplicial_complexes.RealProjectiveSpace(2).dimension()
2
sage: P3.dimension()
3
sage: P4.dimension() # long time
4
sage: P5.dimension() # long time
5
"""
if n == 0:
return self.Simplex(0)
if n == 1:
return self.Sphere(1)
if n == 2:
return self.RealProjectivePlane()
if n == 3:
return SimplicialComplex(
[[1, 2, 3, 7], [1, 4, 7, 9], [2, 3, 4, 8], [2, 5, 8, 10],
[3, 6, 7, 10], [1, 2, 3, 11], [1, 4, 7, 10], [2, 3, 4, 11],
[2, 5, 9, 10], [3, 6, 8, 9], [1, 2, 6, 9], [1, 4, 8, 9],
[2, 3, 7, 8], [2, 6, 9, 10], [3, 6, 9, 10], [1, 2, 6, 11],
[1, 4, 8, 10], [2, 4, 6, 10], [3, 4, 5, 9], [4, 5, 6, 7],
[1, 2, 7, 9], [1, 5, 6, 8], [2, 4, 6, 11], [3, 4, 5, 11],
[4, 5, 6, 11], [1, 3, 5, 10], [1, 5, 6, 11], [2, 4, 8, 10],
[3, 4, 8, 9], [4, 5, 7, 9], [1, 3, 5, 11], [1, 5, 8, 10],
[2, 5, 7, 8], [3, 5, 9, 10], [4, 6, 7, 10], [1, 3, 7, 10],
[1, 6, 8, 9], [2, 5, 7, 9], [3, 6, 7, 8], [5, 6, 7, 8]],
is_mutable=False)
if n == 4:
return SimplicialComplex(
[(1, 3, 8, 12, 13), (2, 7, 8, 13, 16), (4, 8, 9, 12, 14),
(2, 6, 10, 12, 16), (5, 7, 9, 10, 13), (1, 2, 7, 8, 15),
(1, 3, 9, 11, 16), (5, 6, 8, 13, 16), (1, 3, 8, 11, 13),
(3, 4, 10, 13, 15), (4, 6, 9, 12, 15), (2, 4, 6, 11, 13),
(2, 3, 9, 12, 16), (1, 6, 9, 12, 15), (2, 5, 10, 11, 12),
(1, 7, 8, 12, 15), (2, 6, 9, 13, 16), (1, 5, 9, 11, 15),
(4, 9, 10, 13, 14), (2, 7, 8, 15, 16), (2, 3, 9, 12, 14),
(1, 6, 7, 10, 14), (2, 5, 10, 11, 15), (1, 2, 4, 13, 14),
(1, 6, 10, 14, 16), (2, 6, 9, 12, 16), (1, 3, 9, 12, 16),
(4, 5, 7, 11, 16), (5, 9, 10, 11, 15), (3, 5, 8, 12, 14),
(5, 6, 9, 13, 16), (5, 6, 9, 13, 15), (1, 3, 4, 10, 16),
(1, 6, 10, 12, 16), (2, 4, 6, 9, 13), (2, 4, 6, 9, 12),
(1, 2, 4, 11, 13), (7, 9, 10, 13, 14), (1, 7, 8, 12, 13),
(4, 6, 7, 11, 12), (3, 4, 6, 11, 13), (1, 5, 6, 9, 15),
(1, 6, 7, 14, 15), (2, 3, 7, 14, 15), (2, 6, 10, 11, 12),
(5, 7, 9, 10, 11), (1, 2, 4, 5, 14), (3, 5, 10, 13, 15),
(3, 8, 9, 12, 14), (5, 9, 10, 13, 15), (2, 6, 8, 13, 16),
(1, 2, 7, 13, 14), (1, 7, 10, 12, 13), (3, 4, 6, 13, 15),
(4, 9, 10, 13, 15), (2, 3, 10, 12, 16), (1, 2, 5, 14, 15),
(2, 6, 8, 10, 11), (1, 3, 10, 12, 13), (4, 8, 9, 12, 15),
(1, 3, 8, 9, 11), (4, 6, 7, 12, 15), (1, 8, 9, 11, 15),
(4, 5, 8, 14, 16), (1, 2, 8, 11, 13), (3, 6, 8, 11, 13),
(3, 6, 8, 11, 14), (3, 5, 8, 12, 13), (3, 7, 9, 11, 14),
(4, 6, 9, 13, 15), (2, 3, 5, 10, 12), (4, 7, 8, 15, 16),
(1, 2, 7, 14, 15), (3, 7, 9, 11, 16), (3, 6, 7, 14, 15),
(2, 6, 8, 11, 13), (4, 8, 9, 10, 14), (1, 4, 10, 13, 14),
(4, 8, 9, 10, 15), (2, 7, 9, 13, 16), (1, 6, 9, 12, 16),
(2, 3, 7, 9, 14), (4, 8, 10, 15, 16), (1, 5, 9, 11, 16),
(1, 5, 6, 14, 15), (5, 7, 9, 11, 16), (4, 5, 7, 11, 12),
(5, 7, 10, 11, 12), (2, 3, 10, 15, 16), (1, 2, 7, 8, 13),
(1, 6, 7, 10, 12), (1, 3, 10, 12, 16), (7, 9, 10, 11, 14),
(1, 7, 10, 13, 14), (1, 2, 4, 5, 11), (3, 4, 6, 7, 11),
(1, 6, 7, 12, 15), (1, 3, 4, 10, 13), (1, 4, 10, 14, 16),
(2, 4, 6, 11, 12), (5, 6, 8, 14, 16), (3, 5, 6, 8, 13),
(3, 5, 6, 8, 14), (1, 2, 8, 11, 15), (1, 4, 5, 14, 16),
(2, 3, 7, 15, 16), (8, 9, 10, 11, 14), (1, 3, 4, 11, 16),
(6, 8, 10, 14, 16), (8, 9, 10, 11, 15), (1, 3, 4, 11, 13),
(2, 4, 5, 12, 14), (2, 4, 9, 13, 14), (3, 4, 7, 11, 16),
(3, 6, 7, 11, 14), (3, 8, 9, 11, 14), (2, 8, 10, 11, 15),
(1, 3, 8, 9, 12), (4, 5, 7, 8, 16), (4, 5, 8, 12, 14),
(2, 4, 9, 12, 14), (6, 8, 10, 11, 14), (3, 5, 6, 13, 15),
(1, 4, 5, 11, 16), (3, 5, 6, 14, 15), (2, 4, 5, 11, 12),
(4, 5, 7, 8, 12), (1, 8, 9, 12, 15), (5, 7, 8, 13, 16),
(2, 3, 5, 12, 14), (3, 5, 10, 12, 13), (6, 7, 10, 11, 12),
(5, 7, 9, 13, 16), (6, 7, 10, 11, 14), (5, 7, 10, 12, 13),
(1, 2, 5, 11, 15), (1, 5, 6, 9, 16), (5, 7, 8, 12, 13),
(4, 7, 8, 12, 15), (2, 3, 5, 10, 15), (2, 6, 8, 10, 16),
(3, 4, 10, 15, 16), (1, 5, 6, 14, 16), (2, 3, 5, 14, 15),
(2, 3, 7, 9, 16), (2, 7, 9, 13, 14), (3, 4, 6, 7, 15),
(4, 8, 10, 14, 16), (3, 4, 7, 15, 16), (2, 8, 10, 15, 16)],
is_mutable=False)
if n >= 5:
V = set(range(0, n+2))
S = simplicial_complexes.Sphere(n).barycentric_subdivision()
X = S.facets()
facets = set([])
for f in X:
new = []
for v in f:
if 0 in v:
new.append(tuple(V.difference(v)))
else:
new.append(v)
facets.add(tuple(new))
return SimplicialComplex(list(facets), is_mutable=False)
def K3Surface(self):
"""
Returns a minimal triangulation of the K3 surface.
This is a pure simplicial complex of dimension 4 with 16 vertices
and 288 facets. It was constructed by Casella and Kühnel
in [CK2001]_. The construction here uses the labeling from
Spreer and Kühnel [SK2011]_.
REFERENCES:
.. [CK2001] M. Casella and W. Kühnel, "A triangulated K3 surface
with the minimum number of vertices", Topology 40 (2001),
753–772.
.. [SK2011] J. Spreer and W. Kühnel, "Combinatorial properties
of the K3 surface: Simplicial blowups and slicings", Experimental
Mathematics, Volume 20, Issue 2, 2011.
EXAMPLES::
sage: K3=simplicial_complexes.K3Surface() ; K3
Simplicial complex with 16 vertices and 288 facets
sage: K3.f_vector()
[1, 16, 120, 560, 720, 288]
This simplicial complex is implemented just by listing all 288
facets. The list of facets can be computed by the function
:func:`facets_for_K3`, but running the function takes a few
seconds.
"""
return SimplicialComplex(
[(2, 10, 13, 15, 16), (2, 8, 11, 15, 16), (2, 5, 7, 8, 10),
(1, 9, 11, 13, 14), (1, 2, 8, 10, 12), (1, 3, 5, 6, 11),
(1, 5, 6, 9, 12), (1, 2, 6, 13, 16), (1, 4, 10, 13, 14),
(1, 9, 10, 14, 15), (2, 4, 7, 8, 12), (3, 4, 6, 10, 12),
(1, 6, 7, 8, 9), (3, 4, 5, 7, 15), (1, 7, 12, 15, 16),
(4, 5, 7, 13, 16), (5, 8, 11, 12, 15), (2, 4, 7, 12, 14),
(1, 4, 5, 14, 16), (2, 5, 6, 10, 11), (1, 6, 8, 12, 14),
(5, 8, 9, 14, 16), (5, 10, 11, 12, 13), (2, 4, 8, 9, 12),
(7, 9, 12, 15, 16), (1, 2, 6, 9, 15), (1, 5, 14, 15, 16),
(2, 3, 4, 5, 9), (6, 8, 10, 11, 15), (1, 5, 8, 10, 12),
(1, 3, 7, 9, 10), (6, 7, 8, 9, 13), (1, 2, 9, 11, 15),
(2, 8, 11, 14, 16), (2, 4, 5, 13, 16), (1, 4, 8, 13, 15),
(4, 7, 8, 10, 11), (2, 3, 9, 11, 14), (2, 3, 4, 9, 13),
(2, 8, 10, 12, 13), (1, 2, 4, 11, 15), (2, 3, 9, 11, 15),
(3, 5, 10, 13, 15), (3, 4, 5, 9, 11), (6, 10, 13, 15, 16),
(8, 10, 11, 15, 16), (6, 7, 11, 13, 15), (1, 5, 7, 15, 16),
(4, 5, 7, 9, 15), (3, 4, 6, 7, 16), (2, 3, 11, 14, 16),
(3, 4, 9, 11, 13), (1, 2, 5, 14, 15), (2, 3, 9, 13, 14),
(1, 2, 5, 13, 16), (2, 3, 7, 8, 12), (2, 9, 11, 12, 14),
(1, 9, 11, 15, 16), (4, 6, 9, 14, 16), (1, 4, 9, 13, 14),
(1, 2, 3, 12, 16), (8, 11, 12, 14, 15), (2, 4, 11, 12, 14),
(1, 4, 10, 12, 13), (1, 2, 6, 7, 13), (1, 3, 6, 10, 11),
(1, 6, 8, 9, 12), (1, 4, 5, 6, 14), (3, 9, 10, 12, 15),
(5, 8, 11, 12, 16), (5, 9, 10, 14, 15), (3, 9, 12, 15, 16),
(3, 6, 8, 14, 15), (2, 4, 9, 10, 16), (5, 8, 9, 13, 15),
(2, 3, 6, 9, 15), (6, 11, 12, 14, 16), (2, 3, 10, 13, 15),
(2, 8, 9, 10, 13), (3, 4, 8, 11, 13), (3, 4, 5, 7, 13),
(5, 7, 8, 10, 14), (4, 12, 13, 14, 15), (6, 7, 10, 14, 16),
(5, 10, 11, 13, 14), (3, 4, 7, 13, 16), (6, 8, 9, 12, 13),
(1, 3, 4, 10, 14), (2, 4, 6, 11, 12), (1, 7, 9, 10, 14),
(4, 6, 8, 13, 14), (4, 9, 10, 11, 16), (3, 7, 8, 10, 16),
(5, 7, 9, 15, 16), (1, 7, 9, 11, 14), (6, 8, 10, 15, 16),
(5, 8, 9, 10, 14), (7, 8, 10, 14, 16), (2, 6, 7, 9, 11),
(7, 9, 10, 13, 15), (3, 6, 7, 10, 12), (2, 4, 6, 10, 11),
(4, 5, 8, 9, 11), (1, 2, 3, 8, 16), (3, 7, 9, 10, 12),
(1, 2, 6, 8, 14), (3, 5, 6, 13, 15), (1, 5, 6, 12, 14),
(2, 5, 7, 14, 15), (1, 5, 10, 11, 12), (3, 7, 8, 10, 11),
(1, 2, 6, 14, 15), (1, 2, 6, 8, 16), (7, 9, 10, 12, 15),
(3, 4, 6, 8, 14), (3, 7, 13, 14, 16), (2, 5, 7, 8, 14),
(6, 7, 9, 10, 14), (2, 3, 7, 12, 14), (4, 10, 12, 13, 14),
(2, 5, 6, 11, 13), (4, 5, 6, 7, 16), (1, 3, 12, 13, 16),
(1, 4, 11, 15, 16), (1, 3, 4, 6, 10), (1, 10, 11, 12, 13),
(6, 9, 11, 12, 14), (1, 4, 7, 8, 15), (5, 8, 9, 10, 13),
(1, 2, 5, 7, 15), (1, 7, 12, 13, 16), (3, 11, 13, 14, 16),
(1, 2, 5, 7, 13), (4, 7, 8, 9, 15), (1, 5, 6, 10, 11),
(6, 7, 10, 13, 15), (3, 4, 7, 14, 15), (7, 11, 13, 14, 16),
(3, 4, 10, 12, 14), (3, 6, 8, 10, 16), (2, 7, 8, 14, 16),
(2, 3, 4, 5, 13), (5, 8, 12, 13, 15), (4, 6, 9, 13, 14),
(2, 4, 5, 6, 12), (1, 3, 7, 8, 9), (8, 11, 12, 14, 16),
(1, 7, 12, 13, 15), (8, 12, 13, 14, 15), (2, 8, 9, 12, 13),
(4, 6, 10, 12, 15), (2, 8, 11, 14, 15), (2, 6, 9, 11, 12),
(8, 9, 10, 11, 16), (2, 3, 6, 13, 15), (2, 3, 12, 15, 16),
(1, 3, 5, 9, 12), (2, 5, 6, 9, 12), (2, 10, 12, 13, 14),
(2, 6, 13, 15, 16), (2, 3, 11, 15, 16), (3, 5, 6, 8, 15),
(2, 4, 5, 9, 12), (5, 6, 8, 11, 15), (6, 8, 12, 13, 14),
(1, 2, 3, 8, 12), (1, 4, 7, 8, 11), (3, 5, 7, 14, 15),
(3, 5, 7, 13, 14), (1, 7, 10, 11, 14), (6, 7, 11, 12, 15),
(3, 4, 6, 7, 12), (1, 2, 4, 7, 11), (6, 9, 10, 14, 16),
(4, 10, 12, 15, 16), (5, 6, 7, 12, 16), (3, 9, 11, 13, 14),
(5, 9, 14, 15, 16), (4, 5, 6, 7, 12), (1, 3, 9, 10, 15),
(4, 7, 8, 9, 12), (5, 9, 10, 13, 15), (1, 3, 8, 13, 16),
(2, 9, 12, 13, 14), (6, 7, 10, 12, 15), (2, 6, 8, 14, 15),
(3, 5, 6, 8, 11), (3, 4, 7, 12, 14), (1, 3, 10, 14, 15),
(7, 11, 12, 13, 16), (3, 11, 12, 13, 16), (3, 4, 5, 8, 15),
(2, 4, 7, 8, 10), (2, 4, 7, 14, 15), (1, 2, 10, 12, 16),
(1, 6, 8, 13, 16), (1, 7, 8, 13, 15), (3, 9, 11, 15, 16),
(4, 6, 10, 11, 15), (2, 4, 11, 14, 15), (1, 3, 8, 9, 12),
(1, 3, 6, 14, 15), (2, 4, 5, 6, 10), (1, 4, 9, 14, 16),
(5, 7, 9, 12, 16), (1, 3, 7, 10, 11), (7, 8, 9, 13, 15),
(3, 5, 10, 14, 15), (1, 4, 10, 12, 16), (3, 4, 5, 8, 11),
(1, 2, 6, 7, 9), (1, 3, 11, 12, 13), (1, 5, 7, 13, 16),
(5, 7, 10, 11, 14), (2, 10, 12, 15, 16), (3, 6, 7, 10, 16),
(1, 2, 5, 8, 10), (4, 10, 11, 15, 16), (5, 8, 10, 12, 13),
(3, 6, 8, 10, 11), (4, 5, 7, 9, 12), (6, 7, 11, 12, 16),
(3, 5, 9, 11, 16), (8, 9, 10, 14, 16), (3, 4, 6, 8, 16),
(1, 10, 11, 13, 14), (2, 9, 10, 13, 16), (1, 2, 5, 8, 14),
(2, 4, 5, 10, 16), (1, 2, 7, 9, 11), (1, 3, 5, 6, 9),
(5, 7, 11, 13, 14), (3, 5, 10, 13, 14), (2, 4, 8, 9, 10),
(4, 11, 12, 14, 15), (2, 3, 7, 14, 16), (3, 4, 8, 13, 16),
(6, 7, 9, 11, 14), (5, 6, 11, 13, 15), (4, 5, 6, 14, 16),
(3, 4, 8, 14, 15), (4, 5, 8, 9, 15), (1, 4, 8, 11, 13),
(5, 6, 12, 14, 16), (2, 3, 10, 12, 14), (1, 2, 5, 10, 16),
(2, 5, 7, 10, 11), (2, 6, 7, 11, 13), (1, 4, 5, 10, 16),
(2, 6, 8, 15, 16), (2, 3, 10, 12, 15), (7, 11, 12, 13, 15),
(1, 3, 8, 11, 13), (4, 8, 9, 10, 11), (1, 9, 14, 15, 16),
(1, 3, 6, 9, 15), (6, 9, 12, 13, 14), (2, 3, 10, 13, 14),
(2, 5, 7, 11, 13), (2, 3, 5, 6, 13), (4, 6, 8, 13, 16),
(6, 7, 9, 10, 13), (5, 8, 12, 14, 16), (4, 6, 9, 13, 16),
(5, 8, 9, 11, 16), (2, 3, 5, 6, 9), (1, 3, 5, 11, 12),
(3, 7, 8, 9, 12), (4, 6, 11, 12, 15), (3, 5, 9, 12, 16),
(5, 11, 12, 13, 15), (1, 3, 4, 6, 14), (3, 5, 11, 12, 16),
(1, 5, 8, 12, 14), (4, 8, 13, 14, 15), (1, 3, 7, 8, 11),
(6, 9, 10, 13, 16), (2, 4, 9, 13, 16), (1, 6, 7, 8, 13),
(1, 4, 12, 13, 15), (2, 4, 7, 10, 11), (1, 4, 9, 11, 13),
(6, 7, 11, 14, 16), (1, 4, 9, 11, 16), (1, 4, 12, 15, 16),
(1, 2, 4, 7, 15), (2, 3, 7, 8, 16), (1, 4, 5, 6, 10)],
is_mutable=False)
def BarnetteSphere(self):
r"""
Returns Barnette's triangulation of the 3-sphere.
This is a pure simplicial complex of dimension 3 with 8
vertices and 19 facets, which is a non-polytopal triangulation
of the 3-sphere. It was constructed by Barnette in
[B1970]_. The construction here uses the labeling from De
Loera, Rambau and Santos [DLRS2010]_. Another reference is chapter
III.4 of Ewald [E1996]_.
EXAMPLES::
sage: BS = simplicial_complexes.BarnetteSphere() ; BS
Simplicial complex with vertex set (1, 2, 3, 4, 5, 6, 7, 8) and 19 facets
sage: BS.f_vector()
[1, 8, 27, 38, 19]
TESTS:
Checks that this is indeed the same Barnette Sphere as the one
given on page 87 of [E1996]_.::
sage: BS2 = SimplicialComplex([[1,2,3,4],[3,4,5,6],[1,2,5,6],
... [1,2,4,7],[1,3,4,7],[3,4,6,7],
... [3,5,6,7],[1,2,5,7],[2,5,6,7],
... [2,4,6,7],[1,2,3,8],[2,3,4,8],
... [3,4,5,8],[4,5,6,8],[1,2,6,8],
... [1,5,6,8],[1,3,5,8],[2,4,6,8],
... [1,3,5,7]])
sage: BS.is_isomorphic(BS2)
True
REFERENCES:
.. [B1970] Barnette, "Diagrams and Schlegel diagrams", in
Combinatorial Structures and Their Applications, Proc. Calgary
Internat. Conference 1969, New York, 1970, Gordon and Breach.
.. [DLRS2010] De Loera, Rambau and Santos, "Triangulations:
Structures for Algorithms and Applications", Algorithms and
Computation in Mathematics, Volume 25, Springer, 2011.
.. [E1996] Ewald, "Combinatorial Convexity and Algebraic Geometry",
vol. 168 of Graduate Texts in Mathematics, Springer, 1996
"""
return SimplicialComplex([
(1,2,4,5),(2,3,5,6),(1,3,4,6),(1,2,3,7),(4,5,6,7),(1,2,4,7),
(2,4,5,7),(2,3,5,7),(3,5,6,7),(3,1,6,7),(1,6,4,7),(1,2,3,8),
(4,5,6,8),(1,2,5,8),(1,4,5,8),(2,3,6,8),(2,5,6,8),(3,1,4,8),
(3,6,4,8)])
def BrucknerGrunbaumSphere(self):
r"""
Returns Bruckner and Grunbaum's triangulation of the 3-sphere.
This is a pure simplicial complex of dimension 3 with 8
vertices and 20 facets, which is a non-polytopal triangulation
of the 3-sphere. It appeared first in [Br1910]_ and was studied in
[GrS1967]_.
It is defined here as the link of any vertex in the unique minimal
triangulation of the complex projective plane, see chapter 4 of
[Ku1995]_.
EXAMPLES::
sage: BGS = simplicial_complexes.BrucknerGrunbaumSphere() ; BGS
Simplicial complex with vertex set (1, 2, 3, 4, 5, 6, 7, 8) and 20 facets
sage: BGS.f_vector()
[1, 8, 28, 40, 20]
REFERENCES:
.. [Br1910] Bruckner, "Uber die Ableitung der allgemeinen
Polytope und die nach Isomorphismus verschiedenen Typen der
allgemeinen Achtzelle (Oktatope)", Verhand. Konik. Akad. Wetenschap,
Erste Sectie, 10 (1910)
.. [GrS1967] Grunbaum and Sreedharan, "An enumeration of simplicial
4-polytopes with 8 vertices", J. Comb. Th. 2, 437-465 (1967)
.. [Ku1995] Kuhnel, "Tight Polyhedral Submanifolds and Tight Triangulations"
Lecture Notes in Mathematics Volume 1612, 1995
"""
return simplicial_complexes.ComplexProjectivePlane().link([9])
def NotIConnectedGraphs(self, 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. [DHSW2003]_ for information on computing its homology
by computer, and see Babson et al. [BBLSW1999]_ 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:
.. [BBLSW1999] Babson, Bjorner, Linusson, Shareshian, and Welker,
"Complexes of not i-connected graphs," Topology 38 (1999),
271-299
.. [DHSW2003] Dumas, Heckenbach, Saunders, Welker, "Computing simplicial
homology based on efficient Smith normal form algorithms,"
in "Algebra, geometry, and software systems" (2003),
177-206.
"""
G_list = range(1,n+1)
G_vertices = Set(G_list)
E_list = []
for w in G_list:
for v in range(1,w):
E_list.append((v,w))
E = Set(E_list)
facets = []
i_minus_one_sets = list(G_vertices.subsets(size=i-1))
for A in i_minus_one_sets:
G_minus_A = G_vertices.difference(A)
for B in G_minus_A.subsets():
if len(B) > 0 and len(B) < len(G_minus_A):
C = G_minus_A.difference(B)
facet = E
for v in B:
for w in C:
bad_edge = (min(v,w), max(v,w))
facet = facet.difference(Set([bad_edge]))
facets.append(facet)
return SimplicialComplex(facets, is_mutable=False)
def MatchingComplex(self, 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. [DHSW2003]_ for information on computing its homology
by computer, and see Wachs [Wa2003]_ 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 (6s on sage.math, 2012)
Z^132
REFERENCES:
.. [Wa2003] 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)
"""
G_vertices = Set(range(1,n+1))
facets = []
if is_even(n):
half = int(n/2)
half_n_sets = list(G_vertices.subsets(size=half))
else:
half = int((n-1)/2)
half_n_sets = list(G_vertices.subsets(size=half))
for X in half_n_sets:
Xcomp = G_vertices.difference(X)
if is_even(n):
if 1 in X:
A = X
B = Xcomp
else:
A = Xcomp
B = X
for M in matching(A, B):
facet = []
for pair in M:
facet.append(tuple(sorted(pair)))
facets.append(facet)
else:
for w in Xcomp:
if 1 in X or (w == 1 and 2 in X):
A = X
B = Xcomp.difference([w])
else:
B = X
A = Xcomp.difference([w])
for M in matching(A, B):
facet = []
for pair in M:
facet.append(tuple(sorted(pair)))
facets.append(facet)
return SimplicialComplex(facets, is_mutable=False)
def ChessboardComplex(self, n, i):
r"""
The chessboard complex for an `n \times 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. [DHSW2003]_ for information on computing its homology
by computer, and see Wachs [Wa2003]_ 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}
"""
A = range(n)
B = range(i)
E_dict = {}
index = 0
for v in A:
for w in B:
E_dict[(v,w)] = index
index += 1
facets = []
for M in matching(A, B):
facet = []
for pair in M:
facet.append(E_dict[pair])
facets.append(facet)
return SimplicialComplex(facets, is_mutable=False)
def RandomComplex(self, 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 [MW2009]_, 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: X = simplicial_complexes.RandomComplex(6, 2); X
Simplicial complex with vertex set (0, 1, 2, 3, 4, 5) and 10 facets
sage: len(list(X.vertices()))
6
If `d` is too large (if `d+1 > n`, 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) and facets {(0, 1, 2, 3, 4, 5)}
REFERENCES:
.. [MW2009] Meshulam and Wallach, "Homological connectivity of random
`k`-dimensional complexes", preprint, math.CO/0609773.
"""
if d+1 > n:
return simplicial_complexes.Simplex(n-1)
else:
vertices = range(n)
facets = Subsets(vertices, d).list()
maybe = Subsets(vertices, d+1)
facets.extend([f for f in maybe if random.random() <= p])
return SimplicialComplex(facets, is_mutable=False)
def SumComplex(self, n, A):
r"""
The sum complexes of Linial, Meshulam, and Rosenthal [LMR2010]_.
If `k+1` is the cardinality of `A`, then this returns a
`k`-dimensional simplicial complex `X_A` with vertices
`\ZZ/(n)`, and facets given by all `k+1`-tuples `(x_0, x_1,
..., x_k)` such that the sum `\sum x_i` is in `A`. See the
paper by Linial, Meshulam, and Rosenthal [LMR2010]_, in which
they prove various results about these complexes; for example,
if `n` is prime, then `X_A` is rationally acyclic, and if in
addition `A` forms an arithmetic progression in `\ZZ/(n)`,
then `X_A` is `\ZZ`-acyclic. Throughout their paper, they
assume that `n` and `k` are relatively prime, but the
construction makes sense in general.
In addition to the results from the cited paper, these
complexes can have large torsion, given the number of
vertices; for example, if `n=10`, and `A=\{0,1,2,3,6\}`, then
`H_3(X_A)` is cyclic of order 2728, and there is a
4-dimensional complex on 13 vertices with `H_3` having a
cyclic summand of order
.. MATH::
706565607945 = 3 \cdot 5 \cdot 53 \cdot 79 \cdot 131
\cdot 157 \cdot 547.
See the examples.
INPUT:
- ``n`` -- a positive integer
- ``A`` -- a subset of `\ZZ/(n)`
REFERENCES:
.. [LMR2010] N. Linial, R. Meshulam and M. Rosenthal, "Sum
complexes -- a new family of hypertrees", Discrete &
Computational Geometry, 2010, Volume 44, Number 3, Pages
622-636
EXAMPLES::
sage: S = simplicial_complexes.SumComplex(10, [0,1,2,3,6]); S
Simplicial complex with 10 vertices and 126 facets
sage: S.homology()
{0: 0, 1: 0, 2: 0, 3: C2728, 4: 0}
sage: factor(2728)
2^3 * 11 * 31
sage: S = simplicial_complexes.SumComplex(11, [0, 1, 3]); S
Simplicial complex with 11 vertices and 45 facets
sage: S.homology(1)
C23
sage: S = simplicial_complexes.SumComplex(11, [0,1,2,3,4,7]); S
Simplicial complex with 11 vertices and 252 facets
sage: S.homology() # long time
{0: 0, 1: 0, 2: 0, 3: 0, 4: C645679, 5: 0}
sage: factor(645679)
23 * 67 * 419
sage: S = simplicial_complexes.SumComplex(13, [0, 1, 3]); S
Simplicial complex with 13 vertices and 66 facets
sage: S.homology(1)
C159
sage: factor(159)
3 * 53
sage: S = simplicial_complexes.SumComplex(13, [0,1,2,5]); S
Simplicial complex with 13 vertices and 220 facets
sage: S.homology() # long time
{0: 0, 1: 0, 2: C146989209, 3: 0}
sage: factor(1648910295)
3^2 * 5 * 53 * 521 * 1327
sage: S = simplicial_complexes.SumComplex(13, [0,1,2,3,5]); S
Simplicial complex with 13 vertices and 495 facets
sage: S.homology() # long time
{0: 0, 1: 0, 2: 0, 3: C3 x C237 x C706565607945, 4: 0}
sage: factor(706565607945)
3 * 5 * 53 * 79 * 131 * 157 * 547
sage: S = simplicial_complexes.SumComplex(17, [0, 1, 4]); S
Simplicial complex with 17 vertices and 120 facets
sage: S.homology(1)
C140183
sage: factor(140183)
103 * 1361
sage: S = simplicial_complexes.SumComplex(19, [0, 1, 4]); S
Simplicial complex with 19 vertices and 153 facets
sage: S.homology(1)
C5670599
sage: factor(5670599)
11 * 191 * 2699
sage: S = simplicial_complexes.SumComplex(31, [0, 1, 4]); S
Simplicial complex with 31 vertices and 435 facets
sage: S.homology(1) # long time
C5 x C5 x C5 x C5 x C26951480558170926865
sage: factor(26951480558170926865)
5 * 311 * 683 * 1117 * 11657 * 1948909
"""
from sage.rings.all import Integers
Zn = Integers(n)
A = frozenset([Zn(x) for x in A])
facets = []
for f in Set(Zn).subsets(len(A)):
if sum(f) in A:
facets.append(tuple(f))
return SimplicialComplex(facets, is_mutable=False)
simplicial_complexes = SimplicialComplexExamples()
