The entropic cost to tie a knot

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.

References in corpus (8)

• Critical Exponents of the N-vector model
• Accurate estimate of the critical exponent $\nu$ for self-avoiding walks via a fast implementation of the pivot algorithm
• What is the length of a knot in a polymer?
• Knotting of random ring polymers in confined spaces
• On the size of knots in ring polymers
• Ranking knots of random, globular polymer rings
• Interplay between writhe and knotting for swollen and compact polymers
• Elastic Lattice Polymers

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• Chirality modifies the interaction between knots
• On the factorization of composite polymer knots into separated prime components
• Knot probabilities in equilateral random polygons
• The free energy of compressed lattice knots
• Random knotting in very long off-lattice self-avoiding polygons
• Thermodynamic and topological properties of copolymer rings with a segregation/mixing transition