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A hollow octahedron is a nice triangulation of the sphere, because once you know the edges, you know everything. The vertices are obviously the ends of the edges, and the faces are any collection of three vertices that are all connected by edges (and the 3-faces are any collection of four vertices that are all connected by edges, but there aren't any of either). If a bunch of vertices are all connected to each other, then their convex hull is contained in the shape (or more combinatorially, they form a face).

A hollow tetrahedron or the bi-cone over a triangle are not very good triangulations of the sphere. In the first case, all 4 vertices are connected to each other, but the convex hull of those connected guys are not in the shape (the tetrahedron wants to be filled in). In the second case, there is a cycle of 3 points, all connected, but the interior of that triangle is not contained in the shape (this is the equator, and filling it in would make the wedge of two spheres instead of a single sphere).

Möbius's 7 vertex triangulation of the torus is also not very nice: it has cycles of length 3 that are spirally candy-stripes around the donut. In fact, the graph is the complete graph on 7 vertices, so we might even think it had 3, 4, 5, and 6 dimensional faces if we just looked at the graph.

I'd like a vertex minimal triangulation of the torus where the underlying graph actually specified the faces (if three vertices are all connected to each other, then they should form a triangle of the triangulation).

In fancier language, I believe I want a simplicial complex such that both it and the clique complex of its 1-skeleton are homotopic to a torus. I'd like the 3-skeleton to be empty.

I don't have too much of an idea of what to search for, so I thought I'd ask the corresponding vocabulary question:

Is there a name for simplicial complexes that are homotopic to the clique complex of their 1-skeleton?

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