Link of a simplex (${\mathrm{lk}}_K(\sigma )$)

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 $K$: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.