Modeling polymer systems in the presence of non-trivial topological relations: a combined analytical-numerical approach*

The news that will be delivered in this talk is that field theories are back to the realm of polymer physics and may be effectively used in order to understand the statistical behavior of knotted and concatenated polymer rings in a melt or in solution. This is a good news because for a long time analytical models of polymer rings subjected to topological constraints have been considered as mathematically intractable.

Concretely, an algorithm is presented to construct field theory models describing the fluctuations of circular polymers in melts or solutions whose topological properties are controlled using numerical knot invariants. A scheme for perturbative calculations is provided.

This set-up allows to predict how topological effects scale down with increasing polymer length or to estimate how the topological complexity of a system containing a few rings increases with increasing polymer lengths and rigidity.

Finally, field theoretical models of polymers in the presence of topological relations establish new correspondences with other systems in which the topological relations between quasi one-dimensional objects become relevant, such as for instance the lines of the solar magnetic fields and quasiparticles.