  In this work we derive certain topological theories of transverse vector
fields whose amplitudes reproduce topological invariants involving the
interactions among the trajectories of three and four random walks. This result
is applied to the construction of a field theoretical model which describes the
statistical mechanics of an arbitrary number of topologically linked polymers
in the context of the analytical approach of Edwards. With respect to previous
attempts, our approach is very general, as it can treat a system involving an
arbitrary number of polymers and the topological states are not only specified
by the Gauss linking number, but also by higher order topological invariants.
