# wedge(SimplicialComplex,SimplicialComplex,RingElement,RingElement) -- make the wedge sum of two abstract simplicial complexes

## Synopsis

• Function: wedge
• Usage:
wedge (Delta, Gamma, u, v)
• Inputs:
• Delta, ,
• Gamma, ,
• u, , a vertex of $\Delta$
• v, , a vertex of $\Gamma$
• Optional inputs:
• Variables => a list, default value {}, that provides variables for the polynomial ring in which the wedge sum is represented by its Stanley–Reisner ideal
• Outputs:
• , the wedge sum of $\Delta$ and $\Gamma$ obtained by identifying the vertices $u$ and $v$

## Description

For any two abstract simplicial complexes $\Delta$ and $\Gamma$ with distinguished vertices $u$ and $v$, the wedge sum is the simplicial complex formed by taking the disjoint union of $\Delta$ and $\Gamma$ and then identifying $u$ and $v$.

The bowtie complex is the wedge sum of two 2-simplicies

 i1 : S = QQ[a,b,c]; i2 : Δ = simplexComplex(2, S) o2 = simplicialComplex | abc | o2 : SimplicialComplex i3 : R = QQ[d,e,f]; i4 : Γ = simplexComplex(2, R) o4 = simplicialComplex | def | o4 : SimplicialComplex i5 : ΔvΓ = wedge (Δ, Γ, a, f) o5 = simplicialComplex | ade abc | o5 : SimplicialComplex i6 : vertices ΔvΓ o6 = {a, b, c, d, e} o6 : List i7 : assert (# gens ring ΔvΓ === # gens ring Δ + # gens ring Γ - 1)

When the optional argument $\mathrm{Variables}$ is used, the wedge sum is represented by its Stanley–Reisner ideal in a new polynomial ring having this list as variables. The variables in the ring of $\Delta$ corresponds to the first few variables in this new polynomial ring and the variables in the ring of $\Gamma$ correspond to the next few variables in $R$ (omitting the variable corresponding to $v$). This option is particularly convenient when taking the wedge sum of two abstract simplical complexes already defined in the same ring.

 i8 : ΔvΓ' = wedge (Δ, Γ, a, d, Variables => toList(x_0..x_4)) o8 = simplicialComplex | x_0x_3x_4 x_0x_1x_2 | o8 : SimplicialComplex i9 : vertices ΔvΓ' o9 = {x , x , x , x , x } 0 1 2 3 4 o9 : List i10 : ΔvΓ'' = wedge (Δ, Δ, a, a, Variables => {a,b,c,d,e}) o10 = simplicialComplex | ade abc | o10 : SimplicialComplex i11 : vertices ΔvΓ'' o11 = {a, b, c, d, e} o11 : List i12 : ring ΔvΓ'' o12 = QQ[a..e] o12 : PolynomialRing

## Caveat

When the variables in the ring of $\Delta$ and the ring of $\Gamma$ are not disjoint, names of vertices in the wedge sum may not be intelligible; the same name will apparently be used for two distinct variables.