bjornerComplex(PolynomialRing) -- make a shellable 2-polyhedron with 6 vertices

Synopsis

Function: bjornerComplex
Usage:

bjornerComplex S
Inputs:
- S, a polynomial ring, that has at least 6 generators
Outputs:
- an abstract simplicial complex,

Description

Attributed to Anders Björner, this method returns a shellable abstract simplicial complex which has non-zero homology.

i1 : S = ZZ/101[a..f];

i2 : Δ = bjornerComplex S

o2 = simplicialComplex | def cef bdf acf abf ade bce abe bcd acd abc |

o2 : SimplicialComplex

i3 : dim Δ

o3 = 2

i4 : fVector Δ

o4 = {1, 6, 15, 11}

o4 : List

i5 : assert(dim Δ === 2 and isPure Δ)

i6 : assert(fVector Δ === {1,6,15,11})

i7 : prune HH chainComplex Δ

o7 = -1 : 0     

      0 : 0     

      1 : 0     

            ZZ 1
      2 : (---)
           101

o7 : GradedModule

This abstract simplicial complex is Cohen-Macaulay and shellable.

A shellable abstract simplicial complex $\Delta$ is extendably shellable if any shelling of a subcomplex can be extended to a shelling of $\Delta$. The Björner complex is not extendably shellable.

Our enumeration of the vertices follows the bjorner example in Masahiro Hachimori's simplicial complex library.

Ways to use this method: