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

 

 

 

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