Lagrangian tori in homotopy elliptic surfaces

Let $E(1)_K$ denote the symplectic four-manifold, homotopy equivalent to the rational elliptic surface, corresponding to a fibred knot $K$ in $S^3$ constructed by R. Fintushel and R. J. Stern in 1998. We construct a family of nullhomologous Lagrangian tori in $E(1)_K$ and prove that infinitely many of these tori have complements with mutually non-isomorphic fundamental groups if the Alexander polynomial of $K$ has some irreducible factor which does not divide $t^n-1$ for any positive integer $n$. We also show how these tori can be non-isotopically embedded as nullhomologous Lagrangian submanifolds in other symplectic $4$-manifolds.