Skip to main content
Log in

Catalog of dessins d’enfants with no more than 4 edges

  • Published:
Journal of Mathematical Sciences Aims and scope Submit manuscript

In this work, all the dessins d’enfants with no more than 4 edges are listed and their Belyi pairs are computed. In order to enumerate all dessins, the technique of matrix model computations was used. The total number of dessins is 134; among them 77 are spherical, 53 of genus 1, and 4 of genus 2. The automorphism groups of all the dessins are also found. Dessins are listed by the number of edges. Dessins with the same number of edges are ordered lexicographically by their lists of 0-valencies. The corresponding matrix model for any list of 0-valencies is given and computed. Complex matrix models for dessins with 1–3 edges are used. For the dessins with 4 edges, we use Hermitian matrix model.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Alexandrov, A. Mironov, and A. Morozov, “Partition functions of matrix models: first special functions of string theory,” Int. J. Mod. Phys. A, 19, 4127–4165 (2004); arXiv:hep-th/0310113 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  2. J. Bétréma, D. Péré, and A. Zvonkin, Plane Trees and Their Shabat Polynomials. Catalog, Technical Report LaBRI No. 92-75, Bordeaux (1992).

  3. S. Chada, G. Mahoux, and M. L. Mehta, “A method of integration over matrix variables. II,” J. Phys. A, 14, 589–586 (1981).

    Google Scholar 

  4. P. Di Francesco and C. Itzykson, A Generating Function for Fatgraphs, Preprint (1992).

  5. A. Grothendieck, “Sketch of a programme (Esquisse d’un programme),” in: L. Schneps and P. Lochak, eds., Geometric Galois Actions, Vol. 1, Around Grothendieck’s Esquisse d’un Programme, London Math. Soc. Lect. Note Ser., Vol. 242, Cambridge Univ. Press, Cambridge (1997), pp. 5–48; 243–283.

    Google Scholar 

  6. S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and Their Applications, Encyclopedia Math. Sci., Vol. 141, Springer, Berlin (2004).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to N. M. Adrianov.

Additional information

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 6, pp. 35–112, 2007.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Adrianov, N.M., Amburg, N.Y., Dremov, V.A. et al. Catalog of dessins d’enfants with no more than 4 edges. J Math Sci 158, 22–80 (2009). https://doi.org/10.1007/s10958-009-9373-7

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10958-009-9373-7

Keywords

Navigation