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# boundary -- returns the boundary cells along with relative orientations

## Synopsis

• Usage:
boundary(C)
• Inputs:
• Outputs:
• a list, of two-element sequences where the first element is the boundary cell and the second element is an integer representing the orientation of the boundary cell relative to C

## Description

Given a cell C, this command returns a list whose elements are two-element sequences. The first element of each tuple is a boundary cell of C and the second element is the attaching degree of that boundary cell relative to C. This differs from boundaryCells in that it returns the boundary cells and their corresponding attaching degree, whereas boundaryCells returns only the boundary cells.

 i1 : R = QQ[x,y,z]; i2 : vx = newSimplexCell({},x); i3 : vy = newSimplexCell({},y); i4 : vz = newSimplexCell({},z); i5 : exy = newSimplexCell {vx,vy}; i6 : C = cellComplex(R,{exy,vz}); i7 : boundary(exy) o7 = {(Cell of dimension 0 with label x, 1), (Cell of dimension 0 with label ------------------------------------------------------------------------ y, -1)} o7 : List i8 : boundary(vz) o8 = {} o8 : List
 i9 : R = QQ; i10 : P = convexHull matrix {{1,1,-1,-1},{1,-1,1,-1}}; i11 : C = cellComplex(R,P); i12 : f = (cells(2,C))#0; i13 : boundary(f) o13 = {(Cell of dimension 1 with label 1, 1), (Cell of dimension 1 with label ----------------------------------------------------------------------- 1, -1), (Cell of dimension 1 with label 1, -1), (Cell of dimension 1 ----------------------------------------------------------------------- with label 1, 1)} o13 : List