The Jones polynomial in systems with Periodic Boundary Conditions

TL;DR

This paper introduces the Periodic Jones polynomial and Cell Jones polynomial as new topological tools to quantify entanglement in systems with periodic boundary conditions, applicable to polymers and textiles.

Contribution

The paper develops two novel polynomials for assessing topological entanglement in periodic systems, linking finite link invariants to infinite periodic structures.

Findings

The polynomials are sensitive to topology and geometry.

They can measure collective entanglement complexity.

Application to textile and polymer systems shows effectiveness.

Abstract

Entanglement of collections of filaments arises in many contexts, such as in polymer melts, textiles and crystals. Such systems are modeled using periodic boundary conditions (PBC), which create an infinite periodic system whose global entanglement may be impossible to capture and is repetitive. We introduce two new methods to assess topological entanglement in PBC: the Periodic Jones polynomial and the Cell Jones polynomial. These tools capture the grain of entanglement in a periodic system of open or closed chains, by using a finite link as a representative of the global system. These polynomials are topological invariants in some cases, but in general are sensitive to both the topology and the geometry of physical systems. For a general system of 1 closed chain in 1 PBC, we prove that the Periodic Jones polynomial is a recurring factor, up to a remainder, of the Jones polynomial of a conveniently chosen finite cutoff of arbitrary size of the infinite periodic system. We apply the Cell Jones polynomial and the Periodic Jones polynomial to physical PBC systems such as 3D realizations of textile motifs and polymer melts of linear chains obtained from molecular dynamics simulations. Our results demonstrate that the Cell Jones polynomial and the Periodic Jones polynomial can measure collective entanglement complexity in such systems of physical relevance.