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) and (k,hk-1).Comment: 16 pages, 2 figures. to appear in the Mathematical Proceedings of the
Cambridge Philosophical Societ
