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 in the 3-sphere.
A. Cattabriga, M. Mulazzani (2004). All strongly-cyclic branched coverings of (1,1)-knots are Dunwoody manifolds. JOURNAL OF THE LONDON MATHEMATICAL SOCIETY, 70, 512-528 [10.1112/S0024610704005538].
All strongly-cyclic branched coverings of (1,1)-knots are Dunwoody manifolds
CATTABRIGA, ALESSIA;MULAZZANI, MICHELE
2004
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 in the 3-sphere.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.