Gluings of surfaces with polygonal boundaries

  • E. T. Akhmedov
  • Sh. Shakirov


By pairwise gluing edges of a polygon, one obtains two-dimensional surfaces with handles and holes. We compute the number N g,L (n 1, ..., n L ) of distinct ways to obtain a surface of given genus g whose boundary consists of L polygonal components with given numbers n 1, ..., n L of edges. Using combinatorial relations between graphs on real two-dimensional surfaces, we derive recursion relations between the N g,L . We show that the Harer-Zagier numbers arise as a special case of N g,L and derive a new closed-form expression for them.

Key words

graph on surface number of graphs generating function 


  1. [1]
    J. Harer and D. Zagier, “The Euler characteristic of the moduli space of curves,” Invent Math., 85:3 (1986), 457–485.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S. Lando, Lectures on generating functions [in Russian], MCCME, Moscow, 2002.Google Scholar
  3. [3]
    S. Lando and A. Zvonkin, Graphs on Surfaces and Their Applications, Encyclopedia of Mathematical Sciences, vol. 141, Springer-Verlag, Berlin, 2004.Google Scholar
  4. [4]
    J. Stasheff, “Homotopy associativity of H-spaces. I,” Trans. Amer. Math. Soc., 108 (1963), 275–292.CrossRefMathSciNetGoogle Scholar
  5. [5]
    I. Gelfand, M. Kapranov, and A. Zelevinsky, Discriminants, Resultants and Multidimensional Determinants, Birkhauser, Boston, 1994.MATHCrossRefGoogle Scholar
  6. [6]
    M. Kontsevich, “Feynman diagrams and low-dimensional topology,” in: First European Congress of Mathematics, Vol. II (Paris, 1992), Progr. Math., vol. 120, Birkhauser, Basel, 1994, 97–121.Google Scholar
  7. [7]
    M. Kontsevich, “Intersection theory on the moduli space of curves,” Funkts. Anal. Prilozhen., 25:2 (1991), 50–57; English transl.: Functional Anal. Appl., 25:2 (1991), 123–129.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Moscow Institute of Physics and TechnologyMoscowRussia

Personalised recommendations