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skeleton(ZZ,SimplicialComplex) -- make a new simplicial complex generated by all faces of a bounded dimension

Synopsis

• Function: skeleton
• Usage:
skeleton(n, Delta)
• Inputs:
• k, an integer, that bounds the dimension of the faces
• Delta, ,
• Outputs:
• , that is generated by all the faces in $\Delta$ of dimension less than or equal to $k$

Description

The $k$-skeleton of an abstract simplicial complex is the subcomplex consisting of all of the faces of dimension at most $k$. When the abstract simplicial complex is pure its $k$-skeleton is simply generated by its $k$-dimensional faces.

The boundary of the 4-simplex is a simplicial 3-sphere with 5 vertices, 5 facets, and a minimal nonface that corresponds to the interior of the sphere.

 i1 : S = ZZ[a..e]; i2 : Δ = simplicialComplex monomialIdeal (a*b*c*d*e) o2 = simplicialComplex | bcde acde abde abce abcd | o2 : SimplicialComplex i3 : skeleton (-2, Δ) o3 = simplicialComplex 0 o3 : SimplicialComplex i4 : assert (skeleton (-2, Δ) === simplicialComplex monomialIdeal 1_S) i5 : skeleton (-1, Δ) o5 = simplicialComplex | 1 | o5 : SimplicialComplex i6 : assert (skeleton (-1, Δ) === simplicialComplex {1_S}) i7 : skeleton (0, Δ) o7 = simplicialComplex | e d c b a | o7 : SimplicialComplex i8 : assert (skeleton (0, Δ) === simplicialComplex gens S) i9 : skeleton (1, Δ) o9 = simplicialComplex | de ce be ae cd bd ad bc ac ab | o9 : SimplicialComplex i10 : assert (skeleton (1, Δ) === simplicialComplex apply (subsets (gens S, 2), product)) i11 : skeleton (2, Δ) o11 = simplicialComplex | cde bde ade bce ace abe bcd acd abd abc | o11 : SimplicialComplex i12 : assert (skeleton (2, Δ) === simplicialComplex apply (subsets (gens S, 3), product)) i13 : skeleton (3, Δ) o13 = simplicialComplex | bcde acde abde abce abcd | o13 : SimplicialComplex i14 : assert (skeleton (3, Δ) === Δ) i15 : fVector Δ o15 = {1, 5, 10, 10, 5} o15 : List

The abstract simplicial complex from Example 1.8 of Miller-Sturmfels' Combinatorial Commutative Algebra consists of a triangle (on vertices $a$, $b$, $c$), two edges (connecting $c$ to $d$ and $b$ to $d$), and an isolated vertex (namely $e$). It has six minimal nonfaces. Moreover, its 1-skeleton and 2-skeleton are not pure.

 i16 : R = ZZ/101[a..f]; i17 : Γ = simplicialComplex {e, c*d, b*d, a*b*c} o17 = simplicialComplex | e cd bd abc | o17 : SimplicialComplex i18 : skeleton (-7, Γ) o18 = simplicialComplex 0 o18 : SimplicialComplex i19 : assert (skeleton (-7, Γ) === simplicialComplex monomialIdeal 1_R) i20 : skeleton (-1, Γ) o20 = simplicialComplex | 1 | o20 : SimplicialComplex i21 : assert (skeleton (-1, Γ) === simplicialComplex {1_R}) i22 : skeleton (0, Γ) o22 = simplicialComplex | e d c b a | o22 : SimplicialComplex i23 : assert (skeleton (0, Γ) === simplicialComplex {a, b, c, d, e}) i24 : skeleton (1, Γ) o24 = simplicialComplex | e cd bd bc ac ab | o24 : SimplicialComplex i25 : assert (skeleton (1, Γ) === simplicialComplex {e, c*d, b*d, b*c, a*c, a*b}) i26 : skeleton (2, Γ) o26 = simplicialComplex | e cd bd abc | o26 : SimplicialComplex i27 : assert (skeleton (2, Γ) === Γ)