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

Representations of (1,1)-knots

We present two different representations of (1,1)-knots and study some connections between them. The first representation is algebraic: every (1,1)-knot is represented by an element of the pure mapping class group of the twice punctured torus. The second representation is parametric: every (1,1)-knot can be represented by a 4-tuple of integer parameters. The strict connection of this representation with the class of Dunwoody manifolds is illustrated. The above representations are explicitly obtained in some interesting cases, including two-bridge knots and torus knots.Comment: 16 pages, 7 figures. To appear in Fundamenta Mathematicae, special volume Proceedings of Knots in Poland, vol. I

A Markov theorem for generalized plat decomposition

We prove a Markov theorem for tame links in a connected closed orientable 3-manifold $M$ with respect to a plat-like representation. More precisely, given a genus $g$ Heegaard surface $\Sigma_g$ for $M$ we represent each link in $M$ as the plat closure of a braid in the surface braid group $B_{g,2n}=\pi_1(C_{2n}(\Sigma_g))$ and analyze how to translate the equivalence of links in $M$ under ambient isotopy into an algebraic equivalence in $B_{g,2n}$. First, we study the equivalence problem in $\Sigma_g\times [0,1]$, and then, to obtain the equivalence in $M$, we investigate how isotopies corresponding to "sliding" along meridian discs change the braid representative. At the end we provide explicit constructions for Heegaard genus 1 manifolds, i.e. lens spaces and $S^2\times S^1$.Comment: Acknowledgements added. Accepted for publication on Ann. Sc. Norm. Super. Pisa Cl. Sc