A study of polymer knots using a simple knot invariant written consisting of multiple contour integrals

TL;DR

This paper investigates the thermodynamic properties of short polymer knots on a lattice using Monte Carlo simulations and introduces a topological invariant based on contour integrals to distinguish knot types efficiently.

Contribution

It combines thermodynamic analysis of polymer knots with a novel topological invariant method for knot identification, reducing computational effort.

Findings

Heat capacity peaks indicate phase transitions in polymer knots.

The topological invariant reliably distinguishes knot types.

The method reduces sampling requirements for knot topology identification.

Abstract

In this work the thermodynamic properties of short polymer knots (up to 120 segments) defined on a simple cubic lattice are studied with the help of the Wang-Landau Monte Carlo algorithm. The sampling process is performed using pivot transformations starting from a given seed conformation. Both cases of short-range attractive and repulsive interactions acting on the monomers are considered. The properties of the specific energy, heat capacity and gyration radius of several knots are discussed. It is found that the heat capacity exhibits a sharp peak. If the interactions are attractive, similar peaks have been observed also in single open chains and have been related to the transition from a frozen crystallite state to an expanded coil state. Some other peculiarities of the behavior of the analyzed observables are presented, like for instance the increasing or decreasing of the knot specific energy at high temperatures with increasing polymer lengths depending if the interactions are attractive or repulsive. Besides the investigation of the thermodynamics of polymer knots, the second goal of this paper is to introduce a method for distinguishing the topology of a knot based on a topological invariant which is in the form of multiple contour integrals and explicitly depends on the physical trajectory of the knot. The chosen invariant is related to the second coefficient of the Conway polynomial. It has been first isolated from the amplitudes of a Chern-Simons field theory with gauge group SU(N). It is shown that this invariant is very reliable in distinguishing the topology of polymer knots. One of the advantages of the proposed approach is that it allows to reduce the number of samples needed by the Wang-Landau algorithm. Some solutions to speed up the calculations exploiting Monte Carlo integration techniques are developed.