Surfaces with punctures

n-gon with punctures

 

 

tagged triangulation

• two tagged arcs are compatible if
  (i) they do not cross
  (ii) they are not isotopic except
  

 

 

tagged arc complex is the clique complex where simplices are collections of compatible tagged arcs

 

 

\Theorem (Fomin, Shapiro, Thurston)

(S,M) any marked surface.  Then there exists a cluster algebra associated to it,

cluster variables <=> arcs

clusters <-> ideal triangulations

exchange relations <-> flips