Link (geometry)

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.

Example

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.

Definition

Let $\scriptstyle X$ be a simplicial complex. The link $\scriptstyle\operatorname{Lk}(v,X)$ of a vertex $\scriptstyle v$ of $\scriptstyle X$ is the graph constructed as follows. The vertices of $\scriptstyle\operatorname{Lk}(v,X)$ correspond to edges of $\scriptstyle X$ which are incident to $\scriptstyle v$. Two such edges are adjacent in $\scriptstyle\operatorname{Lk}(v,X)$ if they are incident to a common 2-cells at $\scriptstyle v$. In general, for an abstract simplicial complex and a face $\scriptstyle F$ of $\scriptstyle X$, denoted $\scriptstyle\operatorname{Lk}(F,X)$ is the set of faces $\scriptstyle G$ such that G $\cap$ F = $\emptyset$ and G $\cup$ F $\in$ X. Because $X$ is simplicial, there is a set isomorphism between $\scriptstyle\operatorname{Lk}(F,X)$ and $X_F = \{G \in X$ such that F $\subset G\}$.

The graph $\scriptstyle\operatorname{Lk}(v,X)$ is often given the topology of a ball of small radius centred at $\scriptstyle v$.