# Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

## Abstract

In this paper we propose to consider two different solutions having a common problem of establishing hyperbolic structure on a 3-manifold. Under the consideration will be a cone 3-manifold with underlying space as a 3-sphere and a singular set nested in it. Furthermore, this paper is divided into two cases: a singular set as the 3_{1} knot with a bridge and a singular set as the 6_{1}^{3} link. The hyperbolic space $H^{3} for the analytical examination in the first case will be a hyperboloid model, in the second using the upper-half space model. To show that the cone-manifold admits a hyperbolic structure, its fundamental set was constructed in the space of each case and the conditions of its existence were provided. In addition, for the first case we will present the analytical formula to calculate the hyperbolic volume of its manifold.

## First Page

213

## Last Page

228

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## Recommended Citation

Grunwald, Lilya and Qutbaev, Aydos
(2022)
"On a fundamental polyhedron of a hyperbolic cone-manifold,"
*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*: Vol. 5:
Iss.
4, Article 1.

DOI: https://doi.org/10.56017/2181-1318.1232