The aim of this paper is to discuss some applications of the relation between Seiberg-Witten theory and two natural norms defined on the first cohomology group of a closed $3$-manifold $N$ --- the Alexander and the Thurston norm. We will start by giving a ``new" proof, applying SW theory, of McMullen's inequality between these two norms, and then use these norms to study two problems related to symplectic $4$-manifolds of the form $S^{1} \times N$. First we will prove that --- as long as $N$ is irreducible --- the unit balls of the Thurston and Alexander norms are related in a way that is similar to the case of fibered $3$-manifolds, supporting the conjecture that $N$ has to be fibered over $S^{1}$. Second, we will provide the first example of a $2$-cohomology class on a symplectic manifold (of the form $S^{1} \times N$) that lies in the positive cone and satisfies Taubes' ``more constraints", but cannot be represented by a symplectic form, disproving a conjecture of Li and Liu (Li-Liu, 2001, Section~{4.1}).