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.

[edit] 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 a 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 $\subset G\}$ .