  We analyze four-dimensional symplectic manifolds of type $X=S^1 \times M^3$
where $M^3$ is an open $3$-manifold admitting inequivalent fibrations leading
to inequivalent symplectic structures on $X$. For the case where $M^3 \subset
S^3$ is the complement of a $4$-component link constructed by McMullen-Taubes,
we provide a general algorithm for computing the monodromy of the fibrations
explicitly. We use this algorithm to show that certain inequivalent symplectic
structures are distinguished by the dimensions of the primitive cohomologies of
differential forms on $X$. We also calculate the primitive cohomologies on $X$
for a class of open $3$-manifolds that are complements of a family of fibered
graph links in $S^3$. In this case, we show that there exist pairs of
symplectic forms on $X$, arising from either equivalent or inequivalent pairs
of fibrations on the link complement, that have different dimensions of the
primitive cohomologies.
