I don't really understand this definition (from here):
Ok, so we're considering a face $v$ (as $K$ is a simplicial complex, vertices are faces). I get that the vertex set of $\text{lk}(v)$ is the set of vertices which are adjacent to $v$. What I don't understand is why the edges between them are the link's simplices. How is the union between one of those edges and $v$ in $\Sigma$?
Also, not sure I understand the idea behind the star. Wouldn't the union of all the triangles also be in $\Sigma$? If so, then wouldn't the union of the insides of simplices containing $v$ be much bigger?