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.isSimp*, 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 4 5 3 1 4 5 1 5 2 4 2 3 0 4 3 1 0 4 1 2 3 1 0 2 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 |

- CoComplex -- The class of all embedded co-complexes.
- Face -- The class of all faces of complexes or co-complexes.
- HH Complex -- Compute the homology of a complex.

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

- 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.
- isSimp(Complex), see isSimp -- 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.

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