Statistical mechanics with topological constraints: II

It is shown that the full specification of an assembly of long flexible molecules, needed for a statistical-mechanical study, requires an infinite set of topological invariants, and the first two of these are derived in detail. It is argued that these invariants provide a better description of the topology of the system than a more intuitively obvious one, for example, to state the condition that a molecule contains a single knot is very complicated requiring an infinite number of invariants, just as the specification of a function at a point requires an infinite number of Fourier coefficients. It is shown that the probability of molecules taking up configurations with given values for the invariants is a problem in quantum field theory, and that for example the first invariant leads to a formalism isomorphic with the electrodynamics of scalar bosons, and the governing differential equations for one and two molecules are derived. The transition from a real polymer to its representation by a continuous curve leads to divergences, but these can be absorbed by renormalizing the step length and entropy per monomer; within these two changes the topological properties are independent of monomer structure.