On a field theoretical model of polymeric $2s-$plats and some of its consequences

TL;DR

This paper develops a field theoretical model for polymer rings in the form of $2s$-plats, capturing topological interactions and three-body effects, with applications to DNA, confined polymers, and connections to quantum systems.

Contribution

It introduces a novel field theory framework for $2s$-plat polymer links, incorporating three-chain topological interactions and mapping to multi-layer electron gases.

Findings

Three-body topological interactions are significant in entangled polymer systems.

The model enhances understanding of topological effects in confined polymers and DNA.

Connections to quantum systems like high-$T_c$ superconductivity are established.

Abstract

The field theory approach to the statistical mechanics of a system of N polymer rings linked together is generalized to the case of links that have a fixed number $2s$ of maxima and minima. Such kind of links are called plats and appear for instance in the DNA of living organisms. The topological states of the link are distinguished using the Gauss linking number. This is a relatively weak link invariant in the case of a general link, but its efficiency improves when $2s-$plats are considered. It is proved that, if we restrict ourselves to $2s-$plat conformations, the field theoretical model established here is able to take into account also the interactions of topological origin involving three chains simultaneously. It is shown that these three-body interactions have nonvanishing contributions when three or more rings are entangled together, enhancing for instance the attractive forces between monomers. The model can be used to study the statistical mechanics of polymers in confined geometries, for instance when $2s$ extrema of a few polymer rings are attached to membranes. Its partition function is mapped here into that of a multi-layer electron gas. Such quasi-particle systems are studied in connection with several interesting applications, including high-$T_c$ superconductivity and topological quantum computing. At the end an useful connection with the cosh-Gordon equation is shown.