Let $\mathcal S$ be a simplicial complex. For a simplex $S \in \mathcal S$, we have the closure, the star and the link
$\operatorname{Cl} S := \{ F \in \mathcal S : F \leq S \}$
$\operatorname{St} S := \{ T \in \mathcal S : S \leq T \}$
$\operatorname{Lk} S := \operatorname{Cl}\operatorname{St} S - \operatorname{St} \operatorname{Cl} S = \{ T \in \mathcal S : S \cap T = \emptyset \}$
We observe that $\operatorname{Lk} S$ is a subcomplex of $\operatorname{Cl}\operatorname{St}S$. I am interested in another subcomplex
$\operatorname{Lk}^\ast S := \{ T \in \mathcal S : S \nleq T \}$
What is known about this complex? Does it have a name?