cellComplex(Ring,PolyhedralComplex) -- creates cell complex from given polyhedral complex

Synopsis

Function: cellComplex
Usage:

cellComplex(Ring,PolyhedralComplex)
Inputs:
- R, a ring, that specifies the base ring
- P, an instance of the type PolyhedralComplex,
Optional inputs:
- Labels => a hash table, default value null, that maps vertices in the polyhedron to labels
Outputs:
- a cell complex, whose cells are the faces of the given polyhedral complex

Description

Given a polyhedral complex, this command returns the cell complex whose cells correspond to the faces of the polyhedral complex. The faces have the default label 1.

i1 : R = QQ[x];

i2 : P1 = convexHull matrix {{2,2,0},{1,-1,0}};

i3 : P2 = convexHull matrix {{2,-2,0},{1,1,0}};

i4 : P3 = convexHull matrix {{-2,-2,0},{1,-1,0}};

i5 : P4 = convexHull matrix {{-2,2,0},{-1,-1,0}};

i6 : F = polyhedralComplex {P1,P2,P3,P4};

i7 : C = cellComplex(R,F);

i8 : facePoset C

o8 = Relation Matrix: | 1 0 0 0 0 0 0 0 1 0 1 1 1 1 1 1 1 |
                      | 0 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 0 |
                      | 0 0 1 0 0 1 0 1 0 0 1 0 0 1 0 1 0 |
                      | 0 0 0 1 0 0 1 0 0 1 0 1 0 0 1 0 1 |
                      | 0 0 0 0 1 0 0 1 0 1 0 0 1 0 0 1 1 |
                      | 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 |
                      | 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 |
                      | 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 |
                      | 0 0 0 0 0 0 0 0 1 0 0 0 0 1 1 0 0 |
                      | 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 |
                      | 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 |
                      | 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 |

o8 : Poset

The labels on the vertices can be controlled via the optional parameter Labels This parameter expects a hash table whose keys are vectors corresponding to the vertices of the polyhedron with desired labels as corresponding values.

Ways to use this method: