On periodic Takahashi manifolds

In this paper we show that periodic Takahashi 3-manifolds are cyclic coverings of the connected sum of two lens spaces (possibly cyclic coverings of the 3-sphere), branched over knots. When the base space is a 3-sphere, we prove that the associated branching set is a two-bridge knot of genus one, and we determine its type. Moreover, a geometric cyclic presentation for the fundamental groups of these manifolds is obtained in several interesting cases, including the ones corresponding to the branched cyclic coverings of the 3-sphere.Comment: 12 pages, 5 figures. To appear in Tsukuba Journal of Mathematic

(1,1)-knots via the mapping class group of the twice punctured torus

We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG(T,2) with a standard element only depending on the ambient space. As a notable examples, we obtain a representation of this type for all torus knots and for all two-bridge knots. Moreover, we give explicit cyclic presentations for the fundamental groups of the cyclic branched coverings of torus knots of type (k,ck+2).Comment: 18 pages, 10 figures. New version with minor changes. Accepted for publication in Advances in Geometr

Lifting Braids

In this paper we study the homeomorphisms of the disk that are liftable with respect to a simple branched covering. Since any such homeomorphism maps the branch set of the covering onto itself and liftability is invariant up to isotopy fixing the branch set, we are dealing in fact with liftable braids. We prove that the group of liftable braids is finitely generated by liftable powers of half-twists around arcs joining branch points. A set of such generators is explicitly determined for the special case of branched coverings from the disk to the disk. As a preliminary result we also obtain the classification of all the simple branched coverings of the disk.Comment: 20 page

Extending homeomorphisms from punctured surfaces to handlebodies

Let $\textup{H}_g$ be a genus $g$ handlebody and $\textup{MCG}_{2n}(\textup{T}_g)$ be the group of the isotopy classes of orientation preserving homeomorphisms of $\textup{T}_g=\partial\textup{H}_g$, fixing a given set of $2n$ points. In this paper we find a finite set of generators for $\mathcal{E}_{2n}^g$, the subgroup of $\textup{MCG}_{2n}(\textup{T}_g)$ consisting of the isotopy classes of homeomorphisms of $\textup{T}_g$ admitting an extension to the handlebody and keeping fixed the union of $n$ disjoint properly embedded trivial arcs. This result generalizes a previous one obtained by the authors for $n=1$. The subgroup $\mathcal{E}_{2n}^g$ turns out to be important for the study of knots and links in closed 3-manifolds via $(g,n)$-decompositions. In fact, the links represented by the isotopy classes belonging to the same left cosets of $\mathcal{E}_{2n}^g$ in $\textup{MCG}_{2n}(\textup{T}_g)$ are equivalent.Comment: We correct the statements of Theorem 9 and 10, by adding missing generators, and improve the statement of Theorem 10, by removing some redundant generator

All strongly-cyclic branched coverings of (1,1)-knots are Dunwoody manifolds

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 strongly-cyclic 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.Comment: 22 pages, 19 figures. Revised version with minor changes in Proposition 5. Accepted for publication in the Journal of the London Mathematical Societ

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.Comment: 20 pages, 15 figure