link

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 .