Complex -- The class of all embedded complexes.

Description

The class of all embedded complexes, not necessarily simplicial or compact or equidimensional. These are complexes with coordinates assigned to their vertices.

Creating complexes:

The following functions return complexes:

simplex -- Simplex in the variables of a polynomial ring

boundaryCyclicPolytope -- The boundary complex of a cyclic polytope with standard projective space vertices

fullCyclicPolytope -- The full cyclic polytope with moment curve vertices

convHull -- The convex hull

hull -- The positive hull

boundaryOfPolytope -- The boundary of a polytope

newEmptyComplex -- Generates an empty complex.

idealToComplex -- The complex associated to a reduced monomial ideal

dualize -- The dual of a co-complex.

complement -- The complement of a co-complex.

complex -- Make a complex from a list of faces

complexFromFacets -- Make a complex from a list of facets

embeddingComplex -- The complex containing a subcomplex

For examples see the documentation of these functions.

The data stored in a complex C:

C.simplexRing, the polynomial ring of vertices of C.

C.grading, is C.simplexRing.grading, a matrix with the coordinates of the vertices of C in its rows.

C.facets, a list with the facets of C sorted into lists by dimension.

C.edim, the embedding dimension of C, i.e., rank source C.grading.

C.dim, the dimension of the complex.

C.isSimplicial, a Boolean indicating whether C is simplicial.

C.isEquidimensional, a Boolean indicating whether C is equidimensional.

If not just the facets but the faces of C a known (e.g., after computed with fc) then the following data is present:

C.fc, a ScriptedFunctor with the faces of C sorted and indexed by dimension.

C.fvector, a List with the F-vector of C.

The following may be present (if known due to creation of C or due to calling some function):

C.dualComplex, the dual co-complex of C in the sense of dual faces of a polytope. See dualize.

C.isPolytope, a Boolean indicating whether C is a polytope.

C.polytopalFacets, a List with the boundary faces of the polytope C.

C.complementComplex, the complement co-complex of C (if C is a subcomplex of a simplex). See complement.

i1 : R=QQ[x_0..x_5]

o1 = R

o1 : PolynomialRing

i2 : C=boundaryCyclicPolytope(3,R)

o2 = 2: x x x  x x x  x x x  x x x  x x x  x x x  x x x  x x x  
         0 1 2  0 2 3  0 3 4  0 1 5  1 2 5  2 3 5  0 4 5  3 4 5

o2 : complex of dim 2 embedded in dim 5 (printing facets)
     equidimensional, simplicial, F-vector {1, 6, 12, 8, 0, 0, 0}, Euler = 1

i3 : C.simplexRing

o3 = R

o3 : PolynomialRing

i4 : C.grading

o4 = | -1 -1 -1 -1 -1 |
     | 1  0  0  0  0  |
     | 0  1  0  0  0  |
     | 0  0  1  0  0  |
     | 0  0  0  1  0  |
     | 0  0  0  0  1  |

              6        5
o4 : Matrix ZZ  <--- ZZ

i5 : C.fc_2

o5 = {x x x , x x x , x x x , x x x , x x x , x x x , x x x , x x x }
       0 1 2   0 2 3   0 3 4   0 1 5   1 2 5   2 3 5   0 4 5   3 4 5

o5 : List

i6 : C.facets

o6 = {{}, {}, {}, {x x x , x x x , x x x , x x x , x x x , x x x , x x x ,
                    0 1 2   0 2 3   0 3 4   0 1 5   1 2 5   2 3 5   0 4 5 
     ------------------------------------------------------------------------
     x x x }, {}, {}, {}}
      3 4 5

o6 : List

i7 : dualize C

o7 = 2: v v v  v v v  v v v  v v v  v v v  v v v  v v v  v v v  
         0 1 2  0 1 4  0 3 4  1 2 3  1 2 5  1 4 5  2 3 4  3 4 5

o7 : co-complex of dim 2 embedded in dim 5 (printing facets)
     equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1

i8 : complement C

o8 = 2: x x x  x x x  x x x  x x x  x x x  x x x  x x x  x x x  
         3 4 5  1 4 5  2 1 5  2 3 4  0 3 4  1 0 4  2 1 3  2 1 0

o8 : co-complex of dim 2 embedded in dim 5 (printing facets)
     equidimensional, simplicial, F-vector {0, 0, 0, 8, 12, 6, 1}, Euler = 1

i9 : R=QQ[x_0..x_5]

o9 = R

o9 : PolynomialRing

i10 : C=simplex R

o10 = 5: x x x x x x  
          0 1 2 3 4 5

o10 : complex of dim 5 embedded in dim 5 (printing facets)
      equidimensional, simplicial, F-vector {1, 6, 15, 20, 15, 6, 1}, Euler = 0

i11 : C.isPolytope

o11 = true

i12 : C.polytopalFacets

o12 = {x x x x x , x x x x x , x x x x x , x x x x x , x x x x x ,
        0 1 2 3 4   0 1 2 3 5   0 1 2 4 5   0 1 3 4 5   0 2 3 4 5 
      -----------------------------------------------------------------------
      x x x x x }
       1 2 3 4 5

o12 : List

Types of embedded complex :

CoComplex -- The class of all embedded co-complexes.

Methods that use an embedded complex :

addFaceDataToComplex(Complex,List), see addFaceDataToComplex -- Adds to a complex face data.
addFaceDataToComplex(Complex,List,List), see addFaceDataToComplex -- Adds to a complex face data.
addFacetDataToComplex(Complex,List), see addFacetDataToComplex -- Adds to a complex facet data.
boundaryOfPolytope(Complex), see boundaryOfPolytope -- The boundary of a polytope.
closedStar(Face,Complex), see closedStar -- The closed star of a face of a complex.
complement(Complex) -- Compute the complement CoComplex.
Complex == Complex -- Compare two complexes.
complexToIdeal(Complex), see complexToIdeal -- The monomial ideal associated to a complex.
coordinates(Face,Complex), see coordinates -- The coordinates of a face.
deform(Complex), see deform -- Compute the deformations associated to a Stanley-Reisner complex.
deformationsFace(Face,Complex), see deformationsFace -- Compute the deformations associated to a face.
deformationsFace(Face,Complex,Ideal), see deformationsFace -- Compute the deformations associated to a face.
dim(Complex) -- Compute the dimension of a complex or co-complex.
dim(Face,Complex) -- Compute the dimension of a face.
dualGrading(Complex), see dualGrading -- The dual vertices of a polytope.
dualize(Complex), see dualize -- The dual of a face or complex.
edim(Complex), see edim -- The embedding dimension of a complex or co-complex.
embeddingComplex(Complex), see embeddingComplex -- The embedding complex of a complex or co-complex.
eulerCharacteristic(Complex), see eulerCharacteristic -- The Euler characteristic of a complex.
face(List,Complex), see face -- Generate a face.
face(List,Complex,ZZ,ZZ), see face -- Generate a face.
facets(Complex), see facets -- The maximal faces of a complex.
fc(Complex), see fc -- The faces of a complex.
fc(Complex,ZZ), see fc -- The faces of a complex.
fvector(Complex), see fvector -- The F-vector of a complex.
grading(Complex) -- The grading of a complex.
HH Complex -- Compute the homology of a complex.
idealToCoComplex(Ideal,Complex), see idealToCoComplex -- The co-complex associated to a reduced monomial ideal.
idealToCoComplex(MonomialIdeal,Complex), see idealToCoComplex -- The co-complex associated to a reduced monomial ideal.
idealToComplex(Ideal,Complex), see idealToComplex -- The complex associated to a reduced monomial ideal.
idealToComplex(MonomialIdeal,Complex), see idealToComplex -- The complex associated to a reduced monomial ideal.
isEquidimensional(Complex), see isEquidimensional -- Check whether a complex or co-complex is equidimensional.
isPolytope(Complex), see isPolytope -- Check whether a complex is a polytope.
isSimplicial(Complex), see isSimplicial -- Check whether a complex or co-complex is simplicial.
link(Face,Complex), see link -- The link of a face of a complex.
loadDeformations(Complex,String), see loadDeformations -- Read the deformation data of a complex from a file.
minimalNonFaces(Complex), see minimalNonFaces -- The minimal non-faces of a complex.
net(Complex) -- Printing complexes.
polytopalFacets(Complex), see polytopalFacets -- The facets of a polytope.
PT1(Complex), see PT1 -- Compute the deformation polytope associated to a Stanley-Reisner complex.
saveDeformations(Complex,String), see saveDeformations -- Store the deformation data of a complex in a file.
simplexRing(Complex) -- The underlying polynomial ring of a complex.
trivialDeformations(Complex), see trivialDeformations -- Compute the trivial deformations.
tropDef(Complex,Complex), see tropDef -- The co-complex of tropical faces of the deformation polytope.
variables(Complex), see variables -- The variables of a complex or co-complex.
vert(Complex), see vert -- The vertices of a face or complex.
verticesDualPolytope(Complex), see verticesDualPolytope -- The dual vertices of a polytope.

For the programmer

The object Complex is a type, with ancestor classes MutableHashTable < HashTable < Thing.

Complex -- The class of all embedded complexes.

Description

See also

Types of embedded complex :

Methods that use an embedded complex :

For the programmer