Gauge Theory of Topological Entanglements. I: -- General Theory --

A statistical mechanics of mutually entangled long flexible molecules in the presence of topological constraints is developed. The problem is typical example of “disordered systems” in the sense that the distribution of topology is considered to be frozen and is shown, by a mathematical isomorphism, to be equivalent to that of a zero-component superconductor under fluctuating electromagnetic fields. Our theory is a straightforward extension of the de Gennes' theorem on self-avoiding random walks to the case with topological on the paths. Examples are solved for a long polymer entangling with another polymer of various fixed conformations.