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# vertices(SimplicialComplex) -- get the list of the vertices for an abstract simplicial complex

## Synopsis

• Function: vertices
• Usage:
vertices Delta
• Inputs:
• Delta, ,
• Outputs:
• a list, of variables in a polynomial ring corresponding to the vertices of $\Delta$

## Description

In this package, an abstract simplicial complex is represented by its Stanley–Reisner ideal in a polynomial ring, so the vertices are identified with a subset of the variables. This method returns the list of variables in this polynomial ring that corresponds to the vertices.

 i1 : S = QQ[a..e]; i2 : vertices simplexComplex(4, S) o2 = {a, b, c, d, e} o2 : List i3 : Δ = simplicialComplex monomialIdeal(a*b, b*c, c*d, d*e) o3 = simplicialComplex | be bd ad ace | o3 : SimplicialComplex i4 : vertices Δ o4 = {a, b, c, d, e} o4 : List i5 : faces(0, Δ) o5 = {a, b, c, d, e} o5 : List i6 : assert(vertices Δ === gens S)

The vertices may correspond to a proper subset of the variables in the ambient polynomial ring.

 i7 : vertices simplexComplex(2, S) o7 = {a, b, c} o7 : List i8 : Γ = simplicialComplex monomialIdeal(a, a*b, b*c, c*d) o8 = simplicialComplex | ce bde | o8 : SimplicialComplex i9 : vertices Γ o9 = {b, c, d, e} o9 : List i10 : faces(0, Γ) o10 = {b, c, d, e} o10 : List i11 : assert(vertices Γ === {b, c, d, e})

There are two "trivial" simplicial complexes having no vertices: the irrelevant complex has the empty set as a facet whereas the void complex has no facets.

 i12 : irrelevant = simplicialComplex monomialIdeal gens S o12 = simplicialComplex | 1 | o12 : SimplicialComplex i13 : vertices irrelevant o13 = {} o13 : List i14 : assert(vertices irrelevant === {}) i15 : void = simplicialComplex monomialIdeal 1_S o15 = simplicialComplex 0 o15 : SimplicialComplex i16 : vertices void o16 = {} o16 : List i17 : assert(vertices void === {})