cellComplex(Ring,SimplicialComplex) -- Creates a cell complex from a given simplicial complex

Synopsis

Function: cellComplex
Usage:

cellComplex(S,D)
Inputs:
- S, a ring,
- D, an abstract simplicial complex, from which the cell complex is created
Optional inputs:
- Labels => a hash table, default value null, that maps simplices represented by monomials to labels
Outputs:
- a cell complex, that matches those of the given simplicial complex

Description

This returns a cellular complex whose faces are those of the given simplicial complex.

i1 : R = QQ[a..f];

i2 : I = monomialIdeal(a*f, b*d, c*e);

o2 : MonomialIdeal of R

i3 : Delta = simplicialComplex I;

i4 : C = cellComplex(QQ,Delta)

o4 = C

o4 : CellComplex

The optional parameter Label can be used to provide labels to the resulting cell complex. By default all cells get a label of 1.

i5 : S = QQ[x,y];

i6 : H = hashTable {a => x^5, b => y*x^4, c=>y^2*x^3, d => y^3*x^2, e => y^4*x, f => x^5};

i7 : C = cellComplex(S,Delta,Labels=>H)

o7 = C

o7 : CellComplex

i8 : applyValues(cells C, l -> apply(l,cellLabel))

                      5   4    3 2   2 3     4   5
o8 = HashTable{0 => {x , x y, x y , x y , x*y , x }                                       }
                      5 3   5 4   5    5 2   5 3   5 4   4 2   4 4   5    3 3   5 2   2 4
               1 => {x y , x y , x y, x y , x y , x y , x y , x y , x y, x y , x y , x y }
                      5 2   5 4   5 3   5 4   5 2   5 4   5 3   5 4
               2 => {x y , x y , x y , x y , x y , x y , x y , x y }

o8 : HashTable

Ways to use this method: