Symplectic Structures with Non-Isomorphic Primitive Cohomology on open 4-Manifolds

We analyze two different fibrations of a link complement M constructed by McMullen-Taubes, and studied further by Vidussi. These examples lead to inequivalent symplectic forms on a 4-manifold X = S x M, which can be distinguished by the dimension of the primitive cohomologies of differential forms. We provide a general algorithm for computing the monodromies of the fibrations explicitly, which are needed to determine the primitive cohomologies. We also investigate a similar phenomenon coming from fibrations of a class of graph links, whose primitive cohomology provides information about the fibration structure.