Alexander Mednykh

Enumeration of unrooted maps of a given genus

Let Ng (f) denote the number of rooted maps of genus g having f edges. An exact formula for Ng (f) is known for g = 0 (Tutte, 1963), g = 1 (Arques, 1987), g = 2, 3 (Bender and Canfield, 1991). In the present paper we derive an enumeration formula for the number Θγ (e) of unrooted maps on an orientable surface Sγ of a given genus γ and with a given number of edges e. It has a form of a linear combination Σi,jci,jNgj (fi) of numbers of rooted maps Ngj (fi) for some gj ≤ γ and fi ≤ e. The coefficients ci,j are functions of γ and e. We consider the quotient Sγ/Zl of Sγ by a cyclic group of automorphisms Zl as a two-dimensional orbifold O. The task of determining ci,j requires solving the following two subproblems: (a) to compute the number Epio (Γ, Zl) of order-preserving epimorphisms from the fundamental group Γ of the orbifold O=Sγ/Zl onto Zl; (b) to calculate the number of rooted maps on the orbifold O which lifts along the branched covering Sγ → Sγ/Zl to maps on Sγ with the given number e of edges.The number Epio (Γ, Zl) is expressed in terms of classical number-theoretical functions. The other problem is reduced to the standard enumeration problem of determining the numbers Ng(f) for some g ≤ γ and f ≤ e. It follows that Θγ (e) can be calculated whenever the numbers Ng (f) are known for g ≤ γ and f ≤ e. In the end of the paper the above approach is applied to derive the functions Θγ(e) explicitly for γ ≤ 3. We note that the function Θγ(e) was known only for γ = 0 (Liskovets, 1981). Tables containing the numbers of isomorphism classes of maps with up to 30 edges for genus γ = 1, 2, 3 are presented.

Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces

The aim of this paper is to count subgroups of a given index in the fundamental group of an orientable S 1-bundle over a compact surface. The number of subgroups of index n turns out to be independent of the orientability of the base surface 𝔣, closed or bordered, and is expressed as a linear combination of the numbers of surface subgroups of indices m = n/l l∣n. For a closed base surface of characteristic χ the respective coefficients are equal to l –χm+2 or vanish depending on l,n and the Euler number of the S 1 -bundle.

Volume formula for a ℤ2-symmetric spherical tetrahedron through its edge lengths

The present paper considers volume formulæ, as well as trigonometric identities, that hold for a tetrahedron in 3-dimensional spherical space of constant sectional curvature +1. The tetrahedron possesses a certain symmetry: namely rotation of angle π in the middle points of a certain pair of its skew edges.

Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices

A genus-g map is a 2-cell embedding of a connected graph on a closed, orientable surface of genus g without boundary, that is, a sphere with g handles. Two maps are equivalent if they are related by a homeomorphism between their embedding surfaces that takes the vertices, edges and faces of one map into the vertices, edges and faces, respectively, of the other map, and preserves the orientation of the surfaces. A map is rooted if a dart of the map { half an edge { is distinguished as its root. Two rooted maps are equivalent if they are related by a homeomorphism that has the above properties and that also takes the root of one map into the root of the other. By counting maps, rooted or unrooted, we mean counting equivalence classes of those maps. To count unrooted genus-g maps we rst needed to count rooted maps of

Enumeration of genus-four maps by number of edges

An explicit form of the ordinary generating function for the number of rooted maps on a closed orientable surface of genus four with a given number of edges is given. An analytical formula for the number of unrooted maps of genus four with a given number of edges is obtained through the number of rooted ones. Both results are new.

Enumerating Branched Surface Coverings from Unbranched Ones

The number of non-isomorphic n-fold branched coverings of a given closed surface can be determined by the number of nonisomorphic n-fold unbranched coverings of the surface and the number of nonisomorphic connected n-fold graph coverings of a suitable bouquet of circles. A similar enumeration can also be done for regular branched coverings. Some explicit enumerations are also possible.

Enumeration of unrooted hypermaps

Abstract In this paper we derive an enumeration formula for the number of hypermaps of given genus g and given number of darts n in terms of the numbers of rooted hypermaps of genus γ ⩽ g with m darts, where m | n . Explicit expressions for the number of rooted hypermaps of genus g with n darts were derived by Walsh (1975) for g = 0 , and by Arques (1987) for g = 1 . We apply our general counting formula to derive explicit expressions for the number of unrooted spherical hypermaps and for the number of unrooted toroidal hypermaps with given number of darts. The enumeration results can be expressed in terms of Fuchsian groups.

Volumes of Polytopes in Spaces of Constant Curvature

We overview volume calculations for polyhedra in Euclidean, spherical and hyperbolic spaces. We prove the Sforza formula for the volume of an arbitrary tetrahedron in H 3 and S 3. We also present some results, which provide a solution for the Seidel problem on the volume of non-Euclidean tetrahedron. Finally, we consider a convex hyperbolic quadrilateral inscribed in a circle, horocycle or one branch of equidistant curve. This is a natural hyperbolic analog of the cyclic quadrilateral in the Euclidean plane. We find several versions of the Brahmagupta formula for the area of such quadrilateral. We also present a formula for the area of a hyperbolic trapezoid.

On the enumeration of circular maps with given number of edges

A map is a closed Riemann surface S with an embedded graph G such that S \ G is homeomorphic to a disjoint union of open disks. Tutte began a systematic study of maps in the 1960s, and contemporary authors are actively developing it. We introduce the concept of circular map and establish its equivalence to the concept of map admitting a coloring of the faces in two colors. The main result is a formula for the number of circular maps with given number of edges.

A volume formula for ℤ2-symmetric spherical tetrahedra

We obtain formulas for the volume of a spherical tetrahedron with ℤ2-symmetry realized as rotation about the axis passing through the midpoints of a pair of skew edges. We show the dependence of the volume formula on the edge lengths and dihedral angles of the tetrahedron. Several different formulas result whose scopes are determined by the geometric characteristics of the tetrahedron.