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

Synopsis

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 = P

o1 : Polyhedron

i2 : F = normalFan P

o2 = F

o2 : Fan

i3 : F1 = skeleton(2,F)

o3 = F1

o3 : Fan

i4 : raysF = rays F

o4 = | 1 0 -1 0 |
     | 0 1 -1 0 |
     | 0 0 -1 1 |

              3       4
o4 : Matrix ZZ  <-- ZZ

i5 : apply(maxCones F1, mc -> raysF_mc)

o5 = {| 1 0 |, | 1 -1 |, | 1 0 |, | 0 -1 |, | 0 0 |, | -1 0 |}
      | 0 1 |  | 0 -1 |  | 0 0 |  | 1 -1 |  | 1 0 |  | -1 0 |
      | 0 0 |  | 0 -1 |  | 0 1 |  | 0 -1 |  | 0 1 |  | -1 1 |

o5 : 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.

i6 : PC = polyhedralComplex hypercube 3

o6 = PC

o6 : PolyhedralComplex

i7 : PC1 = skeleton(2,PC)

o7 = PC1

o7 : PolyhedralComplex

i8 : vertPC1 = vertices PC1

o8 = | -1 1  -1 1  -1 1  -1 1 |
     | -1 -1 1  1  -1 -1 1  1 |
     | -1 -1 -1 -1 1  1  1  1 |

              3       8
o8 : Matrix QQ  <-- QQ

i9 : apply(maxPolyhedra PC1, mp -> vertPC1_(mp#0))

o9 = {| -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 |

o9 : List

Ways to use skeleton :

skeleton(ZZ,Fan)
skeleton(ZZ,PolyhedralComplex)

For the programmer

The object skeleton is a method function.