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, an abstract simplicial complex,
Outputs:
- an abstract simplicial complex, 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)

Ways to use this method: