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

Usage:

X = skeleton(k,F)

X = skeleton(k,PC)
Inputs:
- k, an integer
- F, an instance of the type Fan
- PC, an instance of the type PolyhedralComplex
Outputs:
- X, an instance of the type Fan, or an instance of the type 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 -1 |, | 0 -1 |, | 1 0 |, | 0 0 |, | 1 0 |, | -1 0 |}
      | 0 -1 |  | 1 -1 |  | 0 0 |  | 1 0 |  | 0 1 |  | -1 0 |
      | 0 -1 |  | 0 -1 |  | 0 1 |  | 0 1 |  | 0 0 |  | -1 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 a method function.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/OldPolyhedra.m2:6615:0.