A spectral approach to the linking number in the 3-torus

Given a closed Riemannian manifold and a pair of multi-curves in it, we give a formula relating the linking number of the later to the spectral theory of the Laplace operator acting on differential one forms. As an application, we compute the linking number of any two multi-geodesics of the flat torus of dimension 3, generalising a result of P. Dehornoy.