Vertex link of a Salvetti complex

The unique vertex of $S_{\Gamma }$ has two link vertices $v^{+},v^{-}$ for each $v\in V(\Gamma )$. Signed vertices $v_1^{{ϵ}_1},\dots ,v_k^{{ϵ}_k}$ span a simplex exactly when the distinct underlying vertices $v_1,\dots ,v_k$ form a clique of $\Gamma$.