The link in a simplicial complex is a generalization of the neighborhood of a vertex in a graph. The link of a vertex encodes information about the local structure of the complex at the vertex. Link of a vertex Given an abstract simplicial complex and a vertex in , its link is a set containing every face such that and is a face of . In the special case in which is a 1-dimensional complex (that is: a graph), contains all vertices such that is an edge in the graph; that is, the neighborhood of in the graph. Given a geometric simplicial complex and , its link is a set containing every face such that and there is a simplex in that has as a vertex and as a face. Equivalently, the join is a face in . As an example, suppose v is the top vertex of the tetrahedron at the left. Then the link of v is the triangle at the base of the tetrahedron. This is because, for each edge of that triangle, the join of v with the edge is a triangle (one of the three triangles at the sides of the tetrahedron); and the join of v with the triangle itself is the entire tetrahedron. An alternative definition is: the link of a vertex is the graph constructed as follows. The vertices of are the edges of incident to . Two such edges are adjacent in iff they are incident to a common 2-cell at . The graph is often given the topology of a ball of small radius centred at ; it is an analog to a sphere centered at a point. Link of a face The definition of a link can be extended from a single vertex to any face. Given an abstract simplicial complex and any face of , its link is a set containing every face such that are disjoint and is a face of : . Given a geometric simplicial complex and any face , its link is a set containing every face such that are disjoint and there is a simplex in that has both and as faces. Examples The link of a vertex of a tetrahedron is a triangle – the three vertices of the link corresponds to the three edges incident to the vertex, and the three edges of the link correspond to the faces incident to