All strongly-cyclic branched coverings of (1,1)-knots are Dunwoody manifolds
arXiv:math/0309298
Abstract
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.
22 pages, 19 figures. Revised version with minor changes in Proposition 5. Accepted for publication in the Journal of the London Mathematical Society