# skeleton -- computes the k-skeleton of a Fan or PolyhedralComplex

## Description

For a Fan F and an integer k between 0 and the dimension of F, skeleton computes the k-skeleton of the Fan F, i.e. the Fan F1 generated by all cones of dimension k in F.

For example, we can look at the 2-skeleton of the fan of projective 3-space:

 i1 : P = convexHull matrix{{1,0,0,0},{0,1,0,0},{0,0,1,0}} o1 = {ambient dimension => 3 } dimension of lineality space => 0 dimension of polyhedron => 3 number of facets => 4 number of rays => 0 number of vertices => 4 o1 : Polyhedron i2 : F = normalFan P o2 = {ambient dimension => 3 } number of generating cones => 4 number of rays => 4 top dimension of the cones => 3 o2 : Fan i3 : F1 = skeleton(2,F) o3 = {ambient dimension => 3 } number of generating cones => 6 number of rays => 4 top dimension of the cones => 2 o3 : Fan i4 : apply(maxCones F1,rays) o4 = {| -1 0 |, | 1 -1 |, | 0 -1 |, | 1 0 |, | 1 0 |, | 0 0 |} | -1 0 | | 0 -1 | | 1 -1 | | 0 1 | | 0 0 | | 1 0 | | -1 1 | | 0 -1 | | 0 -1 | | 0 0 | | 0 1 | | 0 1 | o4 : List

For a PolyhedralComplex PC and an integer k between 0 and the dimension of PC, skeleton computes the k-skeleton of the PolyhedralComplex PC, i.e. the PolyhedralComplex PC1 generated by all polyhedra of dimension k in PC.

 i5 : PC = polyhedralComplex hypercube 3 o5 = {ambient dimension => 3 } number of generating polyhedra => 1 top dimension of the polyhedra => 3 o5 : PolyhedralComplex i6 : PC1 = skeleton(2,PC) o6 = {ambient dimension => 3 } number of generating polyhedra => 6 top dimension of the polyhedra => 2 o6 : PolyhedralComplex i7 : apply(maxPolyhedra PC1,vertices) o7 = {| -1 1 -1 1 |, | -1 1 -1 1 |, | -1 -1 -1 -1 |, | -1 1 -1 1 |, | 1 | -1 -1 1 1 | | -1 -1 1 1 | | -1 1 -1 1 | | 1 1 1 1 | | -1 | 1 1 1 1 | | -1 -1 -1 -1 | | -1 -1 1 1 | | -1 -1 1 1 | | -1 ------------------------------------------------------------------------ 1 1 1 |, | -1 1 -1 1 |} 1 -1 1 | | -1 -1 -1 -1 | -1 1 1 | | -1 -1 1 1 | o7 : List

## Ways to use skeleton :

• "skeleton(ZZ,Fan)"
• "skeleton(ZZ,PolyhedralComplex)"

## For the programmer

The object skeleton is .