Delta * Gamma
The join of two simplicial complexes $\Delta$ and $\Gamma$ is a new simplicial complex whose faces are disjoint unions of a face in $\Delta$ and a face in $\Gamma$.
If $\Gamma$ is the simplicial complex consisting of a single vertex, then the join $\Delta \mathrel{*} \Gamma$ is the cone over $\Delta$. For example, the cone over a bow-tie complex.
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If $\Gamma$ is a $1$-sphere (consisting of two isolated vertices), then the join $\Delta \mathrel{*} \Gamma$ is the suspension of $\Delta$. For example, the octahedron is the suspension of a square.
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The join of a hexagon and a pentagon is a 3-sphere.
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When the variables in the ring of $\Delta$ and the ring of $\Gamma$ are not disjoint, names of vertices in the join may not be intelligible; the same name will be used for two distinct variables.