Three-dimensional hyperelliptic manifolds

Let X³ = H³, E³, S³, H² × E¹, S² × E¹, T₁(H²), Nil of Solv be one of the eight 3-dimensional geometrics of Thurston [10] and G be a discrete group of isometrics of X³ acting without fixed points. A manifold M³ = X³/G is said to be hyperelliptic if there is an isometric involution on it such that the factor space M³/<> is diffeomorphic to the 3-sphere S³. In analogy with the theory of Riemann surfaces we call involution.In the present paper the existence of hyperelliptic manifolds in each light of the eight 3-dimensional geometrics will be obtained. All the proofs given there will be written in the language of orbifolds whose basic facts can be found in [9].     

Spherical structures on torus knots and links

We study two infinite families of cone manifolds endowed with a spherical metric. The singular set of the first of them is the torus knot t(2n + 1, 2) and the singular set of the second is the two-component link t(2n, 2). We find the domains of sphericity of these cone manifolds in terms of cone angles and obtain analytic formulas for their volumes.     

Elementary formulas for a hyperbolic tetrahedron

We derive some elementary formulas expressing the relation between the dihedral angles and edge lengths of a tetrahedron in hyperbolic space.     

Asymptotics and Arithmetical Properties of Complexity for Circulant Graphs

Abstract—We study analytical and arithmetical properties of the complexity function for infinite families of circulant C _n (s₁, s_2,…, s _k ) C_2n(s₁, s_2,…, s _k , n). Exact analytical formulas for the complexity functions of these families are derived, and their asymptotics are found. As a consequence, we show that the thermodynamic limit of these families of graphs coincides with the small Mahler measure of the accompanying Laurent polynomials.     

Surgery on Small Volume Hyperbolic 3-Orbifolds

Siberian Mathematical Journal -