I'm just learning about simplicial complexes and have been looking at diagrams that conceptualize the closure, star and link of a vertex. In all the cases I saw, the link is the boundary of the star. Is that always true?
Edit:
$\begin{multline} \text{A subcomplex is a simplicial complex L \subseteq K.} \\ Cl(L) = \{ \tau \in K |\tau \leq \sigma \in L\}. \\ St(L) = \{\sigma \in K | \sigma \geq \tau \in L\}. \\ Lk(L) = Cl(St(L)) - St(Cl(L) - \{\emptyset\}). \\~ \end{multline}$