(2016-10-19)
The simplicial complex Consider the (abstract) simplicial $T$ defined as follows: vertices are $$ X=\{ 0,1,2,3,4,5,6 \} = \{ j \mod 7 \}. $$ Now, for each $x\in X$ let $\Phi_ x$ denote the following subsets of $2^X$ : $$ \begin{aligned} H_ x &= \{ \{j,j+1\} : j =0 \ldots 6 \} = \{ \{j,j-1\} : j =0 \ldots 6 \} \\ V_ x &= \{ \{ j,j+x \} : j =0 \ldots 6 \} = \{ \{ j,j-x \} : j =0 \ldots 6 \} \\ D_ x &= \{ \{ j, j + x+1 \} : j = 0 \ldots 6 \} = \{ \{ j, j - x-1 \} : j = 0 \ldots 6 \} \\ \Phi_ x & = H_ x \cup V_ x \cup D_ x \end{aligned} $$ where all the integers $j$ are meant $\mod 7$.

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