We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus n. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type (k, hk ± 1).

Cattabriga A., Mulazzani M. (2003). Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 135(1), 137-146 [10.1017/S0305004103006686].

Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups

Cattabriga A.;Mulazzani M.
2003

Abstract

We study the connections among the mapping class group of the twice punctured torus, the cyclic branched coverings of (1, 1)-knots and the cyclic presentations of groups. We give the necessary and sufficient conditions for the existence and uniqueness of the n-fold strongly-cyclic branched coverings of (1, 1)-knots, through the elements of the mapping class group. We prove that every n-fold strongly-cyclic branched covering of a (1, 1)-knot admits a cyclic presentation for the fundamental group, arising from a Heegaard splitting of genus n. Moreover, we give an algorithm to produce the cyclic presentation and illustrate it in the case of cyclic branched coverings of torus knots of type (k, hk ± 1).
2003
Cattabriga A., Mulazzani M. (2003). Strongly-cyclic branched coverings of (1, 1)-knots and cyclic presentations of groups. MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY, 135(1), 137-146 [10.1017/S0305004103006686].
Cattabriga A.; Mulazzani M.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11585/880627
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