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[edit]

Let  Δ n + 1 {\displaystyle \Delta _{n+1}} be the  ( n + 1 ) {\displaystyle (n+1)} -simplex.  ∂ Δ n + 1 {\displaystyle \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 (PL) n-manifold  N {\displaystyle N} , and a co-dimension 0 subcomplex  C ⊂ N {\displaystyle C\subset N} together with a simplicial isomorphism  ϕ : C -> C ′ ⊂ ∂ Δ n + 1 {\displaystyle \phi :C\to C'\subset \partial \Delta _{n+1}} , the Pachner move on N associated to C is the triangulated manifold  ( N ∖ C ) ∪ ϕ ( ∂ Δ n + 1 ∖ C ′ ) {\displaystyle (N\setminus C)\cup _{\phi }(\partial \Delta _{n+1}\setminus C')} . By design, this manifold is PL-isomorphic to  N {\displaystyle N} but the isomorphism does not preserve the triangulation.

## See also[edit]

* Flip graph
* Unknotting problem

## References[edit]

* 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.