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.

Definition

Let $scriptstyle\; X$ be a simplicial complex. The link $scriptstyleoperatorname\{Lk\}(v,X)$ of a vertex $scriptstyle\; v$ of $scriptstyle\; X$ is the graph constructed as follows. The vertices of $scriptstyleoperatorname\{Lk\}(v,X)$ correspond to edges of $scriptstyle\; X$ which are incident to $scriptstyle\; v$. Two such edges are adjacent in $scriptstyleoperatorname\{Lk\}(v,X)$ if they are incident to a common 2-cells at $scriptstyle\; v$. In general, for a abstract simplicial complex and a face $scriptstyle\; F$ of $scriptstyle\; X$, denoted $scriptstyleoperatorname\{Lk\}(F,X)$ 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 $scriptstyleoperatorname\{Lk\}(F,X)$ and $X\_F\; =\; \{G\; in\; X$ such that F $subset\; G\}$.

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

Examples

To follow.