Discovering Topological Invariants in Inhomogeneous Polymeric Systems

In this talk, I will present our work related to understanding topological invariants in melts of linear and ring polymers. In particular, issue of Gauge invariance in the field theory of polymers will be discussed and it will be shown that Gauge fixing can be used to discover topological invariants. A specific example using the Coulomb gauge will be used to demonstrate that the helicity is one of the topological invariants for both, linear and ring polymers. Furthermore, a numerical recipe to generate knotted vector fields will be presented for studying topological configurations near equilibrium using the self-consistent field theory of polymers.