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# bartnetteSphereComplex(PolynomialRing) -- make a non-polytopal 3-sphere with 8 vertices

## Synopsis

• Function: bartnetteSphereComplex
• Usage:
bartnetteSphereComplex S
• Inputs:
• S, , that has at least 8 generators
• Outputs:

## Description

First described by David Barnette's "Diagrams and Schlegel diagrams" appearing in Combinatorial Structures and their Applications, (Proc. Calgary Internat. Conf. 1969, pp 1-4), Gordon and Breach, New York, 1970, this method returns a pure abstract simplicial complex of dimension 3 with 8 vertices and 19 facets. It is smallest possible non-polytopal simplicial 3-sphere.

 i1 : S = ZZ[a..h]; i2 : Δ = bartnetteSphereComplex S; i3 : matrix {facets Δ} o3 = | defh befh cdfh bcfh adeh abeh acdh abch defg cefg adfg acfg bdeg bceg ------------------------------------------------------------------------ abdg abcg bcef acdf abde | 1 19 o3 : Matrix S <-- S i4 : dim Δ o4 = 3 i5 : fVector Δ o5 = {1, 8, 27, 38, 19} o5 : List i6 : assert(dim Δ === 3 and isPure Δ) i7 : assert(ideal Δ === ideal(b*c*d, a*c*e, c*d*e, a*b*f, b*d*f, a*e*f, c*d*g, a*e*g, b*f*g, b*d*h, c*e*h, a*f*h, g*h)) i8 : assert(fVector Δ === {1,8,27,38,19})

The vertices in the Bartnette sphere will correspond to the first 8 variables of the input polynomial ring.

 i9 : R = QQ[x_0..x_10]; i10 : Γ = bartnetteSphereComplex R; i11 : monomialIdeal Γ o11 = monomialIdeal (x x x , x x x , x x x , x x x , x x x , x x x , x x x , 1 2 3 0 2 4 2 3 4 0 1 5 1 3 5 0 4 5 2 3 6 ----------------------------------------------------------------------- x x x , x x x , x x x , x x x , x x x , x x , x , x , x ) 0 4 6 1 5 6 1 3 7 2 4 7 0 5 7 6 7 8 9 10 o11 : MonomialIdeal of R i12 : assert(dim Γ === 3 and isPure Γ)

Our enumeration of the vertices follows Example 9.5.23 in Jesús A De Loera, Jörg Rambau, and Francisco Santos, Triangulations, structures for algorithms and applications, Algorithms and Computation in Mathematics 25, Springer-Verlag, Berlin, 2010.