Alexander K. Zvonkin

Introduction: What is This Book About

The theory of maps (sometimes also called embedded graphs, or ribbon graphs, or fat graphs, or graphs with rotation), or, otherwise, the topological graph theory, is an old and well established branch of combinatorics. It may justifiably be proud of such classical results as the Euler formula (relating the number of vertices, edges, and faces of a map with the genus of the corresponding surface), and of such notably difficult modern achievements as the Four Color Theorem. But the last two decades witnessed a kind of a volcanic activity in this domain which would have been difficult to predict some 30 years ago.

Belyi functions for Archimedean solids

Abstract The notion of a Belyi function is a main technical tool which relates the combinatorics of maps (i.e., graphs embedded into surfaces) to Galois theory, algebraic number theory, and the theory of Riemann surfaces. In this paper we compute Belyi functions for a class of semi-regular maps which correspond to the so-called Archimedean solids.

Composition of Plane Trees

The connections recently established between combinatorial bicolored plane trees and Shabat polynomials show that the world of plane trees is incredibly rich with different mathematical structures. In this article we use Shabat polynomials to introduce a new operation, that of a composition, for combinatorial bicolored plane trees. The composition may be considered as a generalized symmetry.

Plane trees and Shabat polynomials

Abstract In his unpublished paper [7] Alexandre Grothendieck has indicated that there exist profound relations between the theory of number fields and that of maps on two-dimensional surfaces. This theme was later explored by George Shabat (Moscow) and his students (see [1, 2, 11, 12, 14, 16]). For the simplest class of maps, that of plane trees, this theory leads to a very interesting class of polynomials which generalize Chebyshev polynomials and which we call Shabat polynomials. A catalog of Shabat polynomials for all plane trees up to 8 edges is compiled in [4]. In the present paper we describe the connection between plane trees and Shabat polynomials, give some examples (and counterexamples) and discuss some conjectures.

Towards Topological Classification of Univariate Complex Polynomials

The study of the topological classification of complex polynomials began in the XIX-th century by Luroth (1871), Clebsch (1873) and Hurwitz (1891). However, very few things are known today about the problem. A vast computer experiment allowed us to achieve a classification of all the polynomials up to degree 11 (until 1996, such a classification was known up to degree 6). These new data change entirely the global view of the problem and permit to formulate some plausible conjectures that may eventually lead to a solution of the problem.

Constellations, Coverings, and Maps

We start with a combinatorial—group theoretic notion of constellation. Then we pass on to topology and study ramified coverings of the sphere. Next we introduce various types of “pictures”, or embedded graphs: maps, hypermaps, trees, cacti and so on, and clarify their relation to the ramified coverings and the constellations. Finally, at the end of the chapter we discuss Riemann surfaces; they are constructed as ramified coverings of the complex Riemann sphere.

Introduction to the Matrix Integrals Method

This chapter is devoted to a description of the intriguing connection between map enumeration and matrix integrals. This connection was first established in [143] for the purposes of matrix models of quantum gravity. It was later reinvented by mathematicians [138] in the computation of the Euler characteristic of moduli spaces of complex curves. Our presentation follows [25] and we show, along the lines of [138], how it leads to the enumeration of one-face maps. We also relate, following [87], [299], the universal one-matrix model to the Korteweg—de Vries (KdV) hierarchy of partial differential equations. The results of this chapter will be used in the next one in the description of the Harer—Zagier computation of the Euler characteristic of moduli spaces of curves, as well as in the study of Witten’s conjecture, which is now Kontsevich’s theorem.

Algebraic Structures Associated with Embedded Graphs

This chapter is devoted mainly to the description of algebraic and combinatorial constructions related to Vassiliev’s theory of knot invariants [283], [284], [285]. The main combinatorial object of the theory is a chord diagram, or, which is the same, a one-vertex map. We start with a description of this notion and of the famous 4-term relation for for the chord diagrams. Our presentation follows the main lines of Bar-Natan [18], but we also pay a lot of attention to the recent development in the analysis of intersection graphs of chord diagrams. We do not present the proof of Kontsevich’s theorem for Vassiliev invariants since its machinery is too far from the subject of our book. Good explanations are available in [50], [195].

Meromorphic Functions and Embedded Graphs

The starting point of the research described in this chapter is the question formulated as Problem 1.1.11 in Sec. 1.1. This question consists in enumerating constellations with given passport, and we call it the Hurwitz problem. In this general form, it has no satisfactory answer up to now, and the present chapter describes some partial results reflecting the best of the current knowledge. The known results are, however, rather general.

Geometry of Moduli Spaces of Complex Curves

In this chapter we present an overview of the connection between the geometry of moduli spaces of complex curves with marked points and the topology of embedded graphs. According to Harer [137], the idea of this connection belongs to Mumford. It proved to be extremely fruitful. The most celebrated results here are the calculation of the orbifold Euler characteristic of moduli spaces of smooth curves due to Harer and Zagier [138] (we present also Kontsevich’s calculation based on similar ideas) and Kontsevich’s proof of Witten’s conjecture. We must note that a complete exposition of this proof containing all the details has not yet been published. Our text does not fulfill this mission either. In the next chapter we will show how the geometry of moduli spaces of curves is related to that of Hurwitz spaces, that is, the moduli spaces of meromorphic functions on complex curves. Since a meromorphic function is also associated to an embedded graph, we obtain another facet of the same connection.