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ALL STRONGLY-CYCLIC BRANCHED COVERINGS OF (1,1)-KNOTS ARE DUNWOODY MANIFOLDS

Published online by Cambridge University Press:  01 October 2004

ALESSIA CATTABRIGA  and
MICHELE MULAZZANI
ALESSIA CATTABRIGA
Affiliation:
Department of Mathematics, University of Bologna, Italycattabri@dm.unibo.it
MICHELE MULAZZANI
Affiliation:
Department of Mathematics and CIRAM, University of Bologna, Italymulazza@dm.unibo.it
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Abstract

It is shown 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, a parametrization of (1,1)-knots by 4-tuples of integers is obtained. Moreover, using a representation of (1,1)-knots by the mapping class group of the twice-punctured torus, an algorithm is provided which gives the parametrization of all torus knots in $\mathbf{S}^3$.

Keywords

57M1257N10 (primary)57M25 (secondary)
Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 70 , Issue 2 , October 2004 , pp. 512 - 528
Copyright
The London Mathematical Society 2004

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Footnotes

This work was performed under the auspices of the GNSAGA, INdAM, Italy and the University of Bologna funds for selected research topics.