The entropic cost to tie a knot

TL;DR

This paper uses Monte Carlo simulations to analyze the configurational entropy of knotted polygons in a cubic lattice, revealing how knot complexity influences entropy and the asymptotic behavior of the partition function.

Contribution

It provides new insights into the entropic cost of knots, showing that prime knots localize and contribute a logarithmic term, and that the partition function factorizes for composite knots.

Findings

Prime knots localize in small regions of the polygon.

The entropic cost of a knot scales exponentially with its minimal length.

Partition functions for composite knots factorize with combinatorial considerations.

Abstract

We estimate by Monte Carlo simulations the configurational entropy of $N$-steps polygons in the cubic lattice with fixed knot type. By collecting a rich statistics of configurations with very large values of $N$ we are able to analyse the asymptotic behaviour of the partition function of the problem for different knot types. Our results confirm that, in the large $N$ limit, each prime knot is localized in a small region of the polygon, regardless of the possible presence of other knots. Each prime knot component may slide along the unknotted region contributing to the overall configurational entropy with a term proportional to $\ln N$. Furthermore, we discover that the mere existence of a knot requires a well defined entropic cost that scales exponentially with its minimal length. In the case of polygons with composite knots it turns out that the partition function can be simply factorized in terms that depend only on prime components with an additional combinatorial factor that takes into account the statistical property that by interchanging two identical prime knot components in the polygon the corresponding set of overall configuration remains unaltered. Finally, the above results allow to conjecture a sequence of inequalities for the connective constants of polygons whose topology varies within a given family of composite knot types.