Link (simplicial complex)


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.
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 ; 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 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 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, 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.