  Given a symplectomorphism f of a symplectic manifold X, one can form the
`symplectic mapping cylinder' $X_f = (X \times R \times S^1)/Z$ where the Z
action is generated by $(x,s,t)\mapsto (f(x),s+1,t)$. In this paper we compute
the Gromov invariants of the manifolds $X_f$ and of fiber sums of the $X_f$
with other symplectic manifolds. This is done by expressing the Gromov
invariants in terms of the Lefschetz zeta function of f and, in special cases,
in terms of the Alexander polynomials of knots. The result is a large set of
interesting non-Kahler symplectic manifolds with computational ways of
distinguishing them. In particular, this gives a simple symplectic construction
of the `exotic' elliptic surfaces recently discovered by Fintushel and Stern
and of related `exotic' symplectic 6-manifolds.
