Explicit computation of some families of Hurwitz numbers

We compute the number of (weak) equivalence classes of branched covers from a surface of genus g to the sphere, with 3 branching points, degree 2k, and local degrees over the branching points of the form (2,...,2), (2h+1,1,2,...,2), (d_1,...,d_m), for several values of g and h. We obtain explicit formulae of arithmetic nature in terms of the d_i's. Our proofs employ a combinatorial method based on Grothendieck's dessins d'enfant.Comment: To appear in European Journal of Combinatorics (2018); 23 pages, 12 figure

Explicit computation of some families of Hurwitz numbers, II

We continue our computation, using a combinatorial method based on Gronthendieck's dessins d'enfant, of the number of (weak) equivalence classes of surface branched covers matching certain specific branch data. In this note we concentrate on data with the surface of genus g as source surface, the sphere as target surface, 3 branching points, degree 2k, and local degrees over the branching points of the form [2,...,2], [2h+1,3,2,...,2], [d_1,...,d_L]. We compute the corresponding (weak) Hurwitz numbers for several values of g and h, getting explicit arithmetic formulae in terms of the d_i's

Monotone Hurwitz numbers in genus zero

Hurwitz numbers count branched covers of the Riemann sphere with specified ramification data, or equivalently, transitive permutation factorizations in the symmetric group with specified cycle types. Monotone Hurwitz numbers count a restricted subset of the branched covers counted by the Hurwitz numbers, and have arisen in recent work on the the asymptotic expansion of the Harish-Chandra-Itzykson-Zuber integral. In this paper we begin a detailed study of monotone Hurwitz numbers. We prove two results that are reminiscent of those for classical Hurwitz numbers. The first is the monotone join-cut equation, a partial differential equation with initial conditions that characterizes the generating function for monotone Hurwitz numbers in arbitrary genus. The second is our main result, in which we give an explicit formula for monotone Hurwitz numbers in genus zero.Comment: 22 pages, submitted to the Canadian Journal of Mathematic

Computation of highly ramified coverings

An almost Belyi covering is an algebraic covering of the projective line, such that all ramified points except one simple ramified point lie above a set of 3 points of the projective line. In general, there are 1-dimensional families of these coverings with a fixed ramification pattern. (That is, Hurwitz spaces for these coverings are curves.) In this paper, three almost Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We demonstrate how these coverings can be used for computation of several algebraic solutions of the sixth Painleve equation.Comment: 26 page

A GAP package for braid orbit computation, and applications

Let G be a finite group. By Riemann's Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus many problems on algebraic curves require the computation of braid orbits. In this paper we describe an implementation of this computation. We discuss several applications, including the classification of irreducible families of indecomposable rational functions with exceptional monodromy group

On computing Belyi maps

We survey methods to compute three-point branched covers of the projective line, also known as Belyi maps. These methods include a direct approach, involving the solution of a system of polynomial equations, as well as complex analytic methods, modular forms methods, and p-adic methods. Along the way, we pose several questions and provide numerous examples.Comment: 57 pages, 3 figures, extensive bibliography; English and French abstract; revised according to referee's suggestion