Complex -- The class of all simplicial complexes.

Description

The class of all simplicial complexes on the variables of a PolynomialRing. A complex is represented by its maximal faces.

A complex can be constructed from a reduced monomial Ideal by the method idealToComplex

i1 : K=QQ;

i2 : R=K[x_0..x_4];

i3 : I=ideal(x_0*x_1,x_1*x_2,x_2*x_3,x_3*x_4,x_4*x_0);

o3 : Ideal of R

i4 : D =idealToComplex I

o4 = {x  x  , x  x  , x  x  , x  x  , x  x  }
       2  4    0  3    0  2    1  3    1  4

o4 : complex with 5 facets on the vertices x  x  x  x  x  
                                            0  1  2  3  4

i5 : ring D

o5 = R

o5 : PolynomialRing

i6 : dimension D

o6 = 1

i7 : L=facets D

o7 = {x  x  , x  x  , x  x  , x  x  , x  x  }
       2  4    0  3    0  2    1  3    1  4

o7 : List

i8 : D==complex L

o8 = true

i9 : complexToIdeal D

o9 = ideal (x x , x x , x x , x x , x x )
             3 4   0 4   2 3   1 2   0 1

o9 : Ideal of R

Methods that use an object of class Complex :

Complex == Complex -- Compare two complexes.
complexToIdeal(Complex), see complexToIdeal -- Compute the Stanley-Reisner ideal.
dimension(Complex), see dimension -- The dimension of a simplicial complex or a face of a simplicial complex.
faces(Complex), see faces -- Returns the faces of a complex
faces(Complex,ZZ), see faces -- Returns the faces of a complex
facets(Complex), see facets -- The facets of a simplicial complex.
fvector(Complex), see fvector -- Returns the F-vector of a complex
isFace(Face,Complex), see isFace -- Test whether a face is a face of a given complex.
link(Face,Complex), see link -- The link of a face of a complex.
ring(Complex) -- The underlying polynomial ring of a simplicial complex.
stellarSubdivision(Complex,Face,PolynomialRing), see stellarSubdivision -- Compute the stellar subdivision of a simplicial complex.
substituteComplex(Complex,PolynomialRing), see substituteComplex -- Substitute a complex to a different ring.

For the programmer

The object Complex is a type, with ancestor classes List < VisibleList < BasicList < Thing.

Complex -- The class of all simplicial complexes.

Description

See also

Methods that use an object of class Complex :

For the programmer