Editing Triangulation (topology) (section)

=== Pachner Moves ===
[[File:Pachner Move.png|thumb|241x241px|One Pachner-move replaces two tetrahedra by three tetrahedra]]
Pachner moves are a way to manipulate triangulations: Let <math>\mathcal{S} </math> be a simplicial complex. For two simplices <math>K, L,</math> the ''Join'' <math display=inline>K*L = \{ (1-t)k+tl\;|\; k \in K, l \in L, t \in [0,1]\}</math> is the set of points that lie on straights between points in <math>K</math> and in <math>L</math>. Choose <math>S \in \mathcal{S}</math> such that <math>lk(S)= \partial K</math> for any <math>K</math> lying not in  <math>\mathcal{S}</math>. A new complex <math>\mathcal{S'}</math>, can be obtained by replacing <math>S * \partial K</math> by <math>\partial S * K</math>.  This replacement is called a ''Pachner move.'' The theorem of Pachner states that whenever two triangulated manifolds are PL-equivalent, there is a series of Pachner moves transforming both into another.<ref>{{citation|surname1=W B R Lickorish|periodical=Proceedings of the Kirbyfest|title=Simplicial moves on complexes and manifolds|series=Geometry & Topology Monographs |publisher=Mathematical Sciences Publishers|date=1999-11-20|pages=299–320 |doi=10.2140/gtm.1999.2.299
|arxiv=math/9911256 |s2cid=9765634 }}</ref>