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 .  

 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 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