Predicting Optimal Lengths of Random Knots

In thermally fluctuating long linear polymeric chain in solution, the ends come from time to time into a direct contact or a close vicinity of each other. At such an instance, the chain can be regarded as a closed one and thus will form a knot or rather a virtual knot. Several earlier studies of random knotting demonstrated that simpler knots show their highest occurrence for shorter random walks than more complex knots. However up to now there were no rules that could be used to predict the optimal length of a random walk, i.e. the length for which a given knot reaches its highest occurrence. Using numerical simulations, we show here that a power law accurately describes the relation between the optimal lengths of random walks leading to the formation of different knots and the previously characterized lengths of ideal knots of the corresponding type. keywords: knots, polymers, scaling laws, DNA, random walks. A random walk can frequently lead to the formation of knots and it was proven that as the walk becomes very long the probability of forming nontrivial knots upon closure of such a walk tends to one [1, 2]. Many different simulation approaches were used to study random knotting [3, 4, 5, 6, 7]. Probably the most fundamental one is by simulation of ideal random chains where each segment of the chain is of the same length and has no thickness [4, 8]. In ideal random chains the neighboring segments are not correlated with each other and thus show the average deflection angle of 90◦. Ideal random chain behavior is interesting from physical point of view as it reflects statistical behavior of long