abstractSimplicialComplex -- the AbstractSimplicialComplex that is determined by an abstract simplicial complex

Usage:

abstractSimplicialComplex(l)

abstractSimplicialComplex(n)

abstractSimplicialComplex(n,r)
Inputs:
- l, a list,
- n, an integer,
- r, an integer,
Outputs:
- an abstract simplicial complex,

Description

This method returns the AbstractSimplicialComplex that represents a given abstract simplicial complex. The input is either a given collection of generating faces or an integer. These faces need not be facets. Moreover, the elements of the faces need not be written in lexicographic order. When the input is an integer $n$, the output is the corresponding simplex on $[n]$. When the input is a pair of integers, $(n,r)$ the output is the simplicial complex on $[n]$ with complete $r$-skeleton.

i1 : abstractSimplicialComplex({{1,2,3,4}})

o1 = AbstractSimplicialComplex{-1 => {{}}                                           }
                               0 => {{1}, {2}, {3}, {4}}
                               1 => {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}
                               2 => {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}}
                               3 => {{1, 2, 3, 4}}

o1 : AbstractSimplicialComplex

i2 : abstractSimplicialComplex({{4,1,2,3}, {3,2,5},{1,5}})

o2 = AbstractSimplicialComplex{-1 => {{}}                                                                   }
                               0 => {{1}, {2}, {3}, {4}, {5}}
                               1 => {{1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}}
                               2 => {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {2, 3, 5}}
                               3 => {{1, 2, 3, 4}}

o2 : AbstractSimplicialComplex

i3 : abstractSimplicialComplex(4)

o3 = AbstractSimplicialComplex{-1 => {{}}                                           }
                               0 => {{1}, {2}, {3}, {4}}
                               1 => {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}
                               2 => {{1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}}
                               3 => {{1, 2, 3, 4}}

o3 : AbstractSimplicialComplex

The simplicial complex on $[n]$ with complete $r$-skeleton can be constructed as follows.

i4 : abstractSimplicialComplex(4,2)

o4 = AbstractSimplicialComplex{-1 => {{}}                                           }
                               0 => {{1}, {2}, {3}, {4}}
                               1 => {{1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}}

o4 : AbstractSimplicialComplex

Ways to use abstractSimplicialComplex:

abstractSimplicialComplex(List)
abstractSimplicialComplex(ZZ)
abstractSimplicialComplex(ZZ,ZZ)

For the programmer

The object abstractSimplicialComplex is a method function.

The source of this document is in AbstractSimplicialComplexes.m2:903:0.