In Macaulay2, every
simplicial complex is equipped with a polynomial ring, and the resulting matrix of facets is defined over this ring.
i1 : loadPackage "SimplicialComplexes";
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The 3-dimensional sphere has a unique minimal nonface which corresponds to the interior.
i2 : R = ZZ[a..e];
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i3 : sphere = simplicialComplex monomialIdeal(a*b*c*d*e)
o3 = | bcde acde abde abce abcd |
o3 : SimplicialComplex
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i4 : facets sphere
o4 = | bcde acde abde abce abcd |
1 5
o4 : Matrix R <--- R
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The following
faces generate a simplicial complex consisting of a triangle (on vertices
a,b,c), two edges connecting
c to
d and
b to
d, and an isolated vertex
e.
i5 : D = simplicialComplex {e, c*d, b*d, a*b*c, a*b, c}
o5 = | e cd bd abc |
o5 : SimplicialComplex
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i6 : facets D
o6 = | e cd bd abc |
1 4
o6 : Matrix R <--- R
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There are four facets of
D.
Note that no computatation is performed by this routine; all the computation was done while constructing the simplicial complex.
A simplicial complex is displayed by listing its facets, and so this function is frequently unnecessary.