Link (geometry)

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In geometry, the link of a vertex of a 2-dimensional simplicial complex is a graph that encodes information about the local structure of the complex at the vertex.

It is a graph-theoretic analog to a sphere centered at a point.


The tetrahedron is a 2-complex.
The link of a vertex of a tetrahedron is the triangle.

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.


Let be a simplicial complex. The link of a vertex of is the graph constructed as follows. The vertices of correspond to edges of which are incident to . Two such edges are adjacent in if they are incident to a common 2-cells at . In general, for an abstract simplicial complex and a face of , denoted is the set of faces such that G F = and G F X. Because is simplicial, there is a set isomorphism between and such that F .

The graph is often given the topology of a ball of small radius centred at .