Pachner moves on triangulations

A Pachner move on a triangulated n-manifold N works as follows:

- One finds a k-face F of N whose link is combinatorially isomorphic to the link of a k-face in the boundary of triangulated (n+1)-simplex.
- One then replaces the star of F with the complement of the star of a k-face in the boundary of the (n+1)-simplex. This complement is the star of an (n-k)-face.

We call the k-face the attaching face; this is the input to the Pachner move. The corresponding (n-k)-face in the new triangulation is called the belt face (in analogy with surgery).

Since the star of a k-face in the boundary of an (n+1)-simplex has (n+1-k) top-dimensional simplices, the star of the belt (n-k)-face has (k+1) top-dimensional simplices. The Pachner move therefore swaps (n+1-k) top-dimensional simplices for (k+1).

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