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# inducedSubcomplex(SimplicialComplex,List) -- make the induced simplicial complex on a subset of vertices

## Synopsis

• Function: inducedSubcomplex
• Usage:
inducedSubcomplex(Delta, V)
• Inputs:
• Delta, ,
• V, a list, of variables in the ring of $\Delta$ representing vertices
• Outputs:
• , the induced subcomplex of $\Delta$ on the given vertices

## Description

Given a simplicial complex $\Delta$ and a subset $V$ of its vertices, the induced subcomplex is the abstract simplicial complexes consisting of all faces in $\Delta$ whose vertices are contained in $V$.

 i1 : S = ZZ[x_0..x_3]; i2 : Δ = simplicialComplex{x_0*x_1*x_2, x_2*x_3, x_1*x_3} o2 = simplicialComplex | x_2x_3 x_1x_3 x_0x_1x_2 | o2 : SimplicialComplex i3 : Γ = inducedSubcomplex(Δ, {x_1, x_2, x_3}) o3 = simplicialComplex | x_2x_3 x_1x_3 x_1x_2 | o3 : SimplicialComplex i4 : vertices Γ o4 = {x , x , x } 1 2 3 o4 : List i5 : assert (isWellDefined Γ and set vertices Γ === set {x_1, x_2, x_3}) i6 : assert all (facets Γ, F -> member(F, faces(#support F - 1, Δ)))

As a special case, we can consider induced subcomplexes of the void and irrelevant complexes.

 i7 : void = simplicialComplex monomialIdeal(1_S); i8 : inducedSubcomplex(void, {}) o8 = simplicialComplex 0 o8 : SimplicialComplex i9 : assert(void === inducedSubcomplex(void, {})) i10 : irrelevant = simplicialComplex {1_S}; i11 : inducedSubcomplex(irrelevant, {}) o11 = simplicialComplex | 1 | o11 : SimplicialComplex i12 : assert(irrelevant === inducedSubcomplex(irrelevant, {}))