Pachner moves

2-3 Pachner move: a union of 2 tetrahedra gets decomposed into 3 tetrahedra.

In topology, a branch of mathematics, Pachner moves, named after Udo Pachner, are ways of replacing a triangulation of a piecewise linear manifold by a different triangulation of a homeomorphic manifold. Pachner moves are also called bistellar flips. Any two triangulations of a piecewise linear manifold are related by a finite sequence of Pachner moves.

Definition

Let $\Delta _{n+1}$ be the $(n+1)$-simplex. $\partial \Delta _{n+1}$ is a combinatorial n-sphere with its triangulation as the boundary of the n+1-simplex.

Given a triangulated piecewise linear n-manifold $N$, and a co-dimension 0 subcomplex $C\subset N$ together with a simplicial isomorphism $\phi :C\to C'\subset \partial \Delta _{n+1}$, the Pachner move on N associated to C is the triangulated manifold $(N\setminus C)\cup _{\phi }(\partial \Delta _{n+1}\setminus C')$. By design, this manifold is PL-isomorphic to $N$ but the isomorphism does not preserve the triangulation.

References

• Pachner, Udo (1991), "P.L. homeomorphic manifolds are equivalent by elementary shellings", European Journal of Combinatorics, 12 (2): 129–145, doi:10.1016/s0195-6698(13)80080-7.