Surfaces with punctures

surface with punctures

replace ideal triangulation by tagged triangulation

 

 

notions

• once-punctured monogon
• once-punctured n-gon
• self-folded
• tagged arc
• radius

 

 

tagged triangulation

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

 

 

tagged arc complex

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,

tagged arc complex = cluster complex

tagged arcs,- <=> cluster variables 

tagged flips <-> mutations