Alessia Cattabriga

Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

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 strongly-cyclic 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 .M oreover, 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).

All Strongly-Cyclic Branched Coverings of (1,1)-Knots are Dunwoody Manifolds

It is shown 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 strongly-cyclic branched coverings of (1, 1)-knots. As a consequence, a parametrization of (1, 1)-knots by 4-tuples of integers is obtained. Moreover, using a representation of (1, 1)-knots by the mapping class group of the twice-punctured torus, an algorithm is provided which gives the parametrization of all torus knots in S 3 .

COMPLEXITY, HEEGAARD DIAGRAMS AND GENERALIZED DUNWOODY MANIFOLDS

We deal with Matveev complexity of compact orientable 3- manifolds represented via Heegaard diagrams. This lead us to the definition of modified Heegaard complexity of Heegaard diagrams and of manifolds. We define a class of manifolds which are generalizations of Dunwoody manifolds, including cyclic branched coverings of two-bridge knots and links, torus knots, some pretzel knots, and some theta-graphs. Using modified Heegaard complexity, we obtain upper bounds for their Matveev complexity, which linearly depend on the order of the covering. Moreover, using homology arguments due to Matveev and Pervova we obtain lower bounds.

ON KNOTS AND LINKS IN LENS SPACES

We shortly review some recent results about knots and links in lens spaces. A disk diagram is described together with a Reidemeister-type theorem concerning equivalence. The lift of knots/links in the 3-sphere is discussed, showing examples of different knots and links having equivalent lift. The essentiality respect to the lift of classical invariants on knots/links in lens spaces is discussed.

HILDEN BRAID GROUPS

Let Hg be a genus g handlebody and MCG2n(Tg) be the group of the isotopy classes of orientation preserving homeomorphisms of Tg = ∂Hg, fixing a given set of 2n points. In this paper we study two particular subgroups of MCG2n(Tg) which generalize Hilden groups defined by Hilden in [Generators for two groups related to the braid groups, Pacific J. Math.59 (1975) 475–486]. As well as Hilden groups are related to plat closures of braids, these generalizations are related to Heegaard splittings of manifolds and to bridge decompositions of links. Connections between these subgroups and motion groups of links in closed 3-manifolds are also provided.

THE ALEXANDER POLYNOMIAL OF (1,1)-KNOTS

In this paper we investigate the Alexander polynomial of (1,1)-knots, which are knots lying in a 3-manifold with genus one at most, admitting a particular decomposition. More precisely, we study the connections between the Alexander polynomial and a polynomial associated to a cyclic presentation of the fundamental group of an n-fold strongly-cyclic covering branched over the knot K, which we call the n-cyclic polynomial of K. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S2×S1, a result obtained by Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in S3. As corollaries some properties of the Alexander polynomial of knots in S3 are extended to the case of (1,1)-knots in lens spaces.