  In this paper we investigate the Alexander polynomial of (1,1)-knots, which
are knots lying in a 3-manifold with genus one at most, admitting a particular
decomposition. More precisely, we study the connections between the Alexander
polynomial and a polynomial associated to a cyclic presentation of the
fundamental group of an n-fold strongly-cyclic covering branched over the knot,
which we call the n-cyclic polynomial. In this way, we generalize to all
(1,1)-knots, with the only exception of those lying in S^2\times S^1, a result
obtained by J. Minkus for 2-bridge knots and extended by the author and M.
Mulazzani to the case of (1,1)-knots in the 3-sphere. As corollaries some
properties of the Alexander polynomial of knots in the 3-sphere are extended to
the case of (1,1)-knots in lens spaces.
