Cyclic Branched Coverings Over Some Classes of (1,1)-Knots

Agnese Ilaria Telloni

doi:10.1155/2013/549198

Abstract

We construct a 4-parametric family of combinatorial closed 3-manifolds, obtained by glueing together in pairs the boundary faces of polyhedral 3-balls. Then, we obtain geometric presentations of the fundamental groups of these manifolds and determine the corresponding split extension groups. Finally, we prove that the considered manifolds are cyclic coverings of the 3-sphere branched over well-specified -knots, including torus knots and Montesinos knots.

Highlights

In this paper, we study the topological and covering properties of a class of closed connected orientable 3-manifold M(n, k, r, q), depending on four nonnegative integer parameters n, k, r, q such that n ≥ 2, r ≥ 2, k < n, and q ≥ 0
We obtain finite n-generator and n-relator presentations for the fundamental group G(n, k, r, q) of the manifold M(n, k, r, q) which correspond to spines of the manifold
Some subfamilies of our class of manifolds are known: the manifold M(n, k, 2, 0), with (n, 2 − k) = n, that is, k ≡ 2, is homeomorphic to the closed connected orientable 3-manifold Mn,k considered in [1]; the triangulated 3-cells, from which the manifolds M(n, k, 2, 0) arise, are those used in [2] to construct a family of manifolds with totally geodesic boundary; the manifold M(n, k, r, 0), with (n, 2r − 2 − k) = n, that is, k ≡ 2r − 2, is the unique closed 3-manifolds related with the class of hyperbolic 3-manifolds with totally geodesic boundary, combinatorially constructed in [3]

Summary

Introduction

We study the topological and covering properties of a class of closed connected orientable 3-manifold M(n, k, r, q), depending on four nonnegative integer parameters n, k, r, q such that n ≥ 2, r ≥ 2, k < n, and q ≥ 0. These manifolds are constructed from triangulated 3-cells, whose boundary faces are identified together in pairs.

D11 E1 d10

Split Extension Group

Topological and Covering Properties