= Link of a simplex
{title2=$\operatorname{lk}_K(\sigma)$}
= Links of a simplex
{synonym}
For a <simplex> $\sigma$ in a <simplicial complex> $K$, its link consists of the <simplexes> disjoint from $\sigma$ whose union with $\sigma$ is a <simplex> of $K$:
$$
\operatorname{lk}_K(\sigma)=\{\rho\in K:\rho\cap\sigma=\varnothing,\ \rho\cup\sigma\in K\}.
$$
This is a <simplicial subcomplex>, because taking a nonempty face preserves both conditions. At a vertex of a <closed combinatorial surface> its link is a <cycle graph>; at a boundary vertex in a triangulated <surface with boundary> it is a <graph path>. Geometrically, the incident triangles form a cyclic fan or a fan with two ends. Deleting one incident triangular face from a cyclic fan changes the vertex link into a path.
Back to article page
