Michele Mulazzani

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).

CYCLIC PRESENTATIONS OF GROUPS AND CYCLIC BRANCHED COVERINGS OF (1;1)-KNOTS

In this paper we study the connections between cyclic presentations of groups and cyclic branched coverings of (1;1)- knots. In particular, we prove that every n-fold strongly-cyclic branched covering of a (1;1)-knot admits a cyclic presentation for the fundamental group encoded by a Heegaard diagram of genus n.

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 .

Torus Knots and Dunwoody Manifolds

We obtain an explicit representation as Dunwoody manifolds of all cyclic branched coverings of torus knots of type (p,mp±1), with p > 1 and m > 0.

Generalized Takahashi manifolds

We introduce a family of closed 3-dimensional manifolds, which are a generalization of certain manifolds studied by M. Takahashi. The manifolds are represented by Dehn surgery with rational coefficients on the 3-sphere, along an n-periodic 2n-component link. A presentation of their fundamental group is obtained, and covering properties of these manifolds are studied. In particular, this family of manifolds includes the whole class of cyclic branched coverings of two-bridge knots. As a consequence we obtain a simple explicit surgery presentation for this important class of manifolds.

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.

4-colored Graphs and Knot/Link Complements

A representation for compact 3-manifolds with non-empty non-spherical boundary via 4-colored graphs (i.e. 4-regular graphs endowed with a proper edge-coloration with four colors) has been recently introduced by two of the authors, and an initial classification of such manifolds has been obtained up to 8 vertices of the representing graphs. Computer experiments show that the number of graphs/manifolds grows very quickly as the number of vertices increases. As a consequence, we have focused on the case of orientable 3-manifolds with toric boundary, which contains the important case of complements of knots and links in the 3-sphere. In this paper we obtain the complete catalogation/classification of these 3-manifolds up to 12 vertices of the associated graphs, showing the diagrams of the involved knots and links. For the particular case of complements of knots, the research has been extended up to 16 vertices.

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.

ALL LINS-MANDEL SPACES ARE BRANCHED CYCLIC COVERINGS OF S3

In this paper we show that all Lins-Mandel spaces S (b, l, t, c) are branched cyclic coverings of the 3-sphere. When the space is a 3-manifold, the branching set of the covering is a two-bridge knot or link of type (l, t) and otherwise is a graph with two vertices joined by three edges (a θ-graph). In the latter case the singular set of the space is always composed by two points with homeomorphic links. The first homology groups of the Lins-Mandel manifolds are computed when t=1 and when the branching set is a knot of genus one. Furthermore the family of spaces has been extended in order to contain all branched cyclic coverings of two-bridge knots or links.

COMPACT 3-MANIFOLDS VIA 4-COLORED GRAPHS

We introduce a representation of compact 3-manifolds without spherical boundary components via (regular) 4-colored graphs, which turns out to be very convenient for computer aided study and tabulation. Our construction is a direct generalization of the one given in the 1980s by S. Lins for closed 3-manifolds, which is in turn dual to the earlier construction introduced by Pezzana’s school in Modena. In this context we establish some results concerning fundamental groups, connected sums, moves between graphs representing the same manifold, Heegaard genus and complexity, as well as an enumeration and classification of compact 3-manifolds representable by graphs with few vertices (