Macaulay2 » Documentation
Packages » SimplicialComplexes :: dim(SimplicialComplex)
next | previous | forward | backward | up | index | toc

dim(SimplicialComplex) -- find the dimension of an abstract simplicial complex

Description

A face $F$ in an abstract simplicial complex $\Delta$ of cardinality $|F| = i + 1$ has dimension $i$. The dimension of $\Delta$ is the maximum of the dimensions of its faces or it is $-\infty$ if $\Delta$ is the void complex (which has no faces).

i1 : S = ZZ[a..e];
 
i2 : Δ = simplicialComplex {b*c*d*e, a*c*d*e, a*b*d*e, a*b*c*e, a*b*c*d}

o2 = simplicialComplex | bcde acde abde abce abcd |

o2 : SimplicialComplex
 
i3 : monomialIdeal Δ

o3 = monomialIdeal(a*b*c*d*e)

o3 : MonomialIdeal of S
 
i4 : dim Δ

o4 = 3
 
i5 : assert (dim Δ === 3 and numgens ideal Δ === 1 and isPure Δ)
 
i6 : R = ZZ/101[a..f];
 
i7 : Γ = simplicialComplex {b*c*d*e, a*c*d*e, a*b*d*e, a*b*c*e, a*b*c*d}

o7 = simplicialComplex | bcde acde abde abce abcd |

o7 : SimplicialComplex
 
i8 : monomialIdeal Γ

o8 = monomialIdeal (a*b*c*d*e, f)

o8 : MonomialIdeal of R
 
i9 : dim Γ

o9 = 3
 
i10 : assert (dim Γ === 3 and numgens ideal Γ === 2 and isPure Γ)

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.

i11 : S' = QQ[a..e];
 
i12 : Δ' = simplicialComplex {e, c*d, b*d, a*b*c}

o12 = simplicialComplex | e cd bd abc |

o12 : SimplicialComplex
 
i13 : monomialIdeal Δ'

o13 = monomialIdeal (a*d, b*c*d, a*e, b*e, c*e, d*e)

o13 : MonomialIdeal of S'
 
i14 : dim Δ'

o14 = 2
 
i15 : assert (dim Δ' === 2 and not isPure Δ')

The irrelevant complex has the empty set as a facet whereas the void complex has no facets. The irrelevant complex has dimension $-1$ while the void complex has dimension $-\infty$.

i16 : irrelevant = simplicialComplex {1_S'};
 
i17 : dim irrelevant

o17 = -1
 
i18 : void = simplicialComplex monomialIdeal 1_S

o18 = simplicialComplex 0

o18 : SimplicialComplex
 
i19 : dim void

o19 = -infinity

o19 : InfiniteNumber
 
i20 : assert (dim irrelevant === -1 and dim void === -infinity)

To avoid repeating a computation, the package caches the result in the CacheTable of the abstract simplicial complex.

i21 : peek Δ.cache

o21 = CacheTable{dim => 3}

See also

Ways to use this method:

The source of this document is in SimplicialComplexes/Documentation.m2:645:0.