Suhyoung Choi in his article " Geometric structures on low-dimensional manifolds " proposed a concept that I haven't seen anywhere else. He says that if we have a triangulated topological 3-manifold t and its vertex links (linking spheres of each vertex of the triangulation) then the triangulation induces a triangulation on each link sphere. Also he introduced "vertex diagrams" which I can't understand either. Thurston also says that by having the gluing information of a triangulation, the link of vertexes also has a triangulation. Suppose we have two different triangulated linking sphereS and we want to pair their vertexes. Choi says that a vertex on the first sphere paired with another vertex on the second sphere and the paired vertexes have the same number of edges starting from it. Also every edge ending at a vertex is paired with an edge ending at the paired vertex. The pairing corresponds to an edge of the triangulation of M. I just can't see the correspondence.



vertex diagram and triangulation
As far as I can see, what Choi is doing is just the following. Each vertex $i$ of the original triangulation has a sphere as its link. I'm accustomed to thinking of the sphere as consisting of some of the simplices of the triangulation, but for Choi's purposes it seems better to view the link as a much smaller sphere $S\_i$ around $i$. Each edge (of your original triangulation) incident to $i$ pierces the link at a point; each triangle incident to $i$ cuts $s$ in an arc joining two of those points; each tetrahedron incident to $i$ covers a triangular patch in $S\_i$. Those intersections of $S\_i$ with the faces of your triangulation that are incident to $i$ constitute a triangulation of $S\_i$. 


Now each edge of your original triangulation, say an edge $\{i,j\}$, determines in this way a vertex in the triangulation of $S\_i$ and also a vertex in the triangulation of $S\_j$. Those two vertices in two link spheres are to be paired with each other. Note that different vertices on $S\_i$ are not paired with vertices of the same $S\_j$; in fact, the relevant $S\_j$ is different for each of these vertices (because the corresponding edges of the original triangulation connect $i$ to different vertices $j$). 


