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Gluings of surfaces with polygonal boundaries

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Abstract

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.

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, Moscow, Russia

    E. T. Akhmedov

  2. Moscow Institute of Physics and Technology, Moscow, Russia

    Sh. Shakirov

Authors
  1. E. T. Akhmedov
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  2. Sh. Shakirov
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Correspondence to E. T. Akhmedov.

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__________

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 43, No. 4, pp. 3–13, 2009

Original Russian Text Copyright © by E. T. Akhmedov and Sh. Shakirov

Supported in part by the Russian Federal Agency for Nuclear Energy. The second author’s research was supported in part by the Program for Support of Leading Scientific Schools (grant no. NSh-8004.2006. 2), the Russian Foundation for Basic Research (grant nos. RFBR-Italy 06-01-92059-CE and 07-02-00642), and the Dynasty Foundation.

In this paper we only consider orientable two-dimensional surfaces.

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Akhmedov, E.T., Shakirov, S. Gluings of surfaces with polygonal boundaries. Funct Anal Its Appl 43, 245–253 (2009). https://doi.org/10.1007/s10688-009-0033-y

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  • DOI: https://doi.org/10.1007/s10688-009-0033-y

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