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

## Synopsis

• Function: cellComplex
• Usage:
cellComplex(Ring,PolyhedralComplex)
• Inputs:
• Optional inputs:
• Labels => , default value null, that maps vertices in the polyhedron to labels
• Outputs:
• , 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.