A note on T\"uring's 1936

T\"uring's argument that there can be no machine computing the diagonal on the enumeration of the computable sequences is not a demonstration.Comment: 4 pages, for more information see http://paolacattabriga.wordpress.com

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, which we call the n-cyclic polynomial. In this way, we generalize to all (1,1)-knots, with the only exception of those lying in S^2\times S^1, a result obtained by J. Minkus for 2-bridge knots and extended by the author and M. Mulazzani to the case of (1,1)-knots in the 3-sphere. As corollaries some properties of the Alexander polynomial of knots in the 3-sphere are extended to the case of (1,1)-knots in lens spaces.Comment: 11 pages, 1 figure. A corollary has been extended, and a new example added. Accepted for publication on J. Knot Theory Ramification

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

(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

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

Knot quandle decompositions

We show that the fundamental quandle defines a functor from the oriented tangle category to a suitably defined quandle category. Given a tangle decomposition of a link $L$, the fundamental quandle of $L$ may be obtained from the fundamental quandles of tangles. We apply this result to derive a presentation of the fundamental quandle of periodic links, composite knots and satellite knots.Comment: 23 pages, 12 figure