the vertex. In this example, the link can be visualized by cutting off the vertex with a plane; formally, intersecting the tetrahedron with a plane near the vertex – the resulting cross-section is the link. Another example is illustrated below. There is a two-dimensional simplicial complex. At the left, a vertex is marked in yellow. At the right, the link of that vertex is marked in green. Properties For any simplicial complex , every link is downward-closed, and therefore it is a simplicial complex too; it is a sub-complex of . Because is simplicial, there is a set isomorphism between and the set : every corresponds to , which is in . Link and star A concept closely-related to the link is the star. Given an abstract simplicial complex and any face ,, its star is a set containing every face such that is a face of . In the special case in which is a 1-dimensional complex (that is: a graph), contains all edges for all vertices that are neighbors of . That is, it is a graph-theoretic star centered at . Given a geometric simplicial complex and any face , its star is a set containing every face such that there is a simplex in having both and as faces: . In other words, it is the closure of the set -- the set of simplices having as a face. So the link is a subset of the star. The star and link are related as follows: For any , . For any , , that is, the star of is the cone of its link at . An example is illustrated below. There is a two-dimensional simplicial complex. At the left, a vertex is marked in yellow. At the right, the star of that vertex is marked in green. See also Vertex figure - a geometric concept similar to the simplicial link. References Geometry