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

We classify weakly exact, rational Lagrangian tori in T*T-2 - 0(T2) up to Hamiltonian isotopy. This result is related to the classification theory of closed 1-forms on T-n and also has applications to symplectic topology. As a 1st corollary, we strengthen a result due independently to Eliashberg-Polterovich and to Giroux describing Lagrangian tori in T*T-2 - 0(T2), which are homologous to the zero section. As a 2nd corollary, we exhibit pairs of disjoint totally real tori K-1, K-2 subset of T*T-2, each of which is isotopic through totally real tori to the zero section, but such that the union K-1 boolean OR K-2 is not even smoothly isotopic to a Lagrangian. In the 2nd part of the paper, we study linking of Lagrangian tori in (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.