Linking of Lagrangian Tori and Embedding Obstructions in Symplectic 4-Manifolds

We classify weakly exact, rational Lagrangian tori in $T^* $\mathbb{T}$^2- 0_{$\mathbb{T}$^2}$ up to Hamiltonian isotopy. This result is related to the classification theory of closed $1$-forms on ${\mathbb{T}}^n$ and also has applications to symplectic topology. As a first corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in $T^{\ast }{\mathbb{T}}^2-0_{{\mathbb{T}}^2}$ which are homologous to the zero section. As a second corollary, we exhibit pairs of disjoint totally real tori $K_1,K_2\subset T^{\ast }{\mathbb{T}}^2$, each of which is isotopic through totally real tori to the zero section, but such that the union $K_1\cup K_2$ is not even smoothly isotopic to a Lagrangian. In the second part of the paper, we study linking of Lagrangian tori in $({\mathbb{R}}^4,\omega )$ and in rational symplectic $4$-manifolds. We prove that the linking properties of such tori are determined by purely algebro-topological data, which can often be deduced from enumerative disk counts in the monotone case. We also use this result to describe certain Lagrangian embedding obstructions.