No abstract

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold stronglycyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus n. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type (k, hk ± 1).

In this paper we show that all 3-manifolds of a family introduced by M. J. Dunwoody are cyclic coverings of lens spaces (eventually S 3 ), branched over genus one 1-bridge knots. As a consequence, we give a positive answer to the Dunwoody conjecture that all the elements of a wide subclass are cyclic coverings of S 3 branched over a knot. Moreover, we show that all branched cyclic coverings of a 2-bridge knot belong to this subclass; this implies that the fundamental group of each branched cyclic covering of a 2-bridge knot admits a geometric cyclic presentation.

We show that every strongly-cyclic branched covering of a (1, 1)-knot is a Dunwoody manifold. This result, together with the converse statement previously obtained by Grasselli and Mulazzani, proves that the class of Dunwoody manifolds coincides with the class of stronglycyclic branched coverings of (1, 1)-knots. As a consequence, we obtain a parametrization of (1, 1)-knots by 4-tuples of integers. Moreover, using a representation of (1, 1)-knots by the mapping class group of the twice punctured torus, we provide an algorithm which gives the parametrization of all torus knots in S 3 .Mathematics Subject Classification 2000: Primary 57M12, 57N10; Secondary 57M25.

In this paper we study some aspects of knots and links in lens spaces. Namely, if we consider lens spaces as quotient of the unit ball B 3 with suitable identification of boundary points, then we can project the links on the equatorial disk of B 3 , obtaining a regular diagram for them. In this contest, we obtain a complete finite set of Reidemeister type moves establishing equivalence, up to ambient isotopy, a Wirtinger type presentation for the fundamental group of the complement of the link and a diagrammatic method giving the first homology group. We also compute Alexander polynomial and twisted Alexander polynomials of this class of links, showing their correlation with Reidemeister torsion.Mathematics Subject Classification 2010: Primary 57M25, 57M27; Secondary 57M05.