rudinBallComplex(PolynomialRing) -- make a nonshellable 3-ball with 14 vertices and 41 facets

Synopsis

Function: rudinBallComplex
Usage:

rudinBallComplex S
Inputs:
- S, a polynomial ring, that has at least 14 generators
Outputs:
- an abstract simplicial complex,

Description

As described in Mary Ellen Rudin's "An unshellable triangulation of a tetrahedron", Bulletin of the American Mathematical Society 64 (1958) 90–91, this method returns triangulation of a 3-ball with 14 vertices and 41 facets that is not shellable. This abstract simplicial complex has a convex realization.

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

i2 : Δ = rudinBallComplex S;

i3 : matrix {facets Δ}

o3 = | klmn hlmn ikmn fjmn djmn fimn dhmn jkln bjln bhln cjkn eikn egkn cgkn
     ------------------------------------------------------------------------
     cfjn bdjn efin bdhn gklm eilm cilm ehlm cglm aikm agkm efim acim dehm
     ------------------------------------------------------------------------
     acgm fjkl fhkl dhkl dgkl bfjl bfhl dehl cdgl cfjk aeik aegk cdgk |

             1      41
o3 : Matrix S  <-- S

i4 : dim Δ

o4 = 3

i5 : fVector Δ

o5 = {1, 14, 66, 94, 41}

o5 : List

i6 : assert(dim Δ === 3 and isPure Δ)

i7 : assert(fVector Δ === {1,14,66,94,41})

This abstract simplicial complex is Cohen-Macaulay but not shellable.

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

Ways to use this method: