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

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

2003-09-18

arxiv.org/abs/math/0309298v2

Mathematics

Geometric Topology

22 pages, 19 figures. Revised version with minor changes in Proposition 5. Accepted for publication in the Journal of the Lond

Scientific paper

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.

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