Kyoichi Tsurusaki

A Statistical Study of Random Knotting Using the Vassiliev Invariants

Employing the Vassiliev invariants as tools for determining knot types of polygons in 3 dimensions, we evaluate numerically the knotting probability PK(N) of the Gaussian random polygon being equivalent to a knot type K. For prime knots and composite knots we plot the knotting probability PK(N) against the number N of polygonal nodes. Taking the analogy with the asymptotic scaling behaviors of self-avoiding walks, we propose a formula of fitting curves to the numerical data. The curves fit well the graphs of the knotting probability PK(N) versus N. This agreement suggests to us that the scaling formula for the knotting probability might also work for the random polygons other than the Gaussian random polygon.

Linking probabilities of off-lattice self-avoiding polygons and the effects of excluded volume

We evaluate numerically the probability of linking, i.e. the probability of a given pair of self-avoiding polygons (SAPs) being entangled and forming a nontrivial link type L. In the simulation we generate pairs of SAPs of N spherical segments of radius rd such that they have no overlaps among the segments and each of the SAPs has the trivial knot type. We evaluate the probability of a self-avoiding pair of SAPs forming a given link type L for various link types with fixed distance R between the centers of mass of the two SAPs. We define normalized distance r by where denotes the square root of the mean square radius of gyration of SAP of the trivial knot 01. We introduce formulae expressing the linking probability as a function of normalized distance r, which gives good fitting curves with respect to χ2 values. We also investigate the dependence of linking probabilities on the excluded-volume parameter rd and the number of segments, N. Quite interestingly, the graph of linking probability versus normalized distance r shows no N-dependence at a particular value of the excluded volume parameter, rd = 0.2.

Topology of Closed Random Polygons

We investigate the topology of random walks using derivatives of the Jones polynomial. We enumerate formation probabilities of nontrivial knots 3 1 , 4 1 , 5 1 and 5 2 in closed Gaussian random walks.

A new algorithm for numerical calculation of link invariants

Abstract We introduce an algorithm for numerical calculation of derivatives of the Jones polynomial. This method gives a new tool for determining the topology of knotted closed loops in three dimensions using computers.

Fractions of Particular Knots in Gaussian Random Polygons

Fractions of knotted polygons in Gaussian random polygon are numerically studied. Three dimensional random polygons with N steps are prepared by closed N -step Gaussian random walks. Let P K ( N ) denote the probability that an N -step Gaussian polygon has a knot type K . For prime knots (3 1 , 4 1 , 5 1 , 5 2 ) and composite knots (3 1 #3 1 , 3 1 #4 1 , 3 1 #3 1 #3 1 ), P K ( N )s are evaluated in the range 30≤ N ≤2400. We confirm that a scaling formula gives nice fitting curves for the numerical data plots of P K ( N ) versus N for the different knot types.

Numerical application of knot invariants and universality of random knotting

We study universal properties of random knotting by making an extensive use of isotopy invariants of knots. We define knotting probability (PK(N)) by the probability of an N -noded random polygon being topologically equivalent to a given knot K . The question is the following: for a given model of random polygon how the knotting probability changes with respect to the number N of polygonal nodes? Through numerical simulation we see that the knotting probability can be expressed by a simple function of N . From the result we propose a universal exponent of PK(N), which may be a new numerical invariant of knots.

General Polygonal Length Dependence of the Linking Probability for Ideal Random Polygons

The linking probability that two closed random walks (i.e., random polygons: RPs) are mutually entangled, Plink, is investigated numerically quite accurately. Here, Plink is a function of the distance between two RPs, R, and the number of polygonal segments, N . In a previous paper, we numerically estimated Plink precisely in the wide region of 0 ≤ ξ ≤ 3.0 where ξ denotes the normalized distance, i.e., the ratio of R to the radius of gyration Rg : ξ = R/Rg. We have also shown that the N and R-dependence of Plink can be well approximated by a simple function: Plink(ξ;N) = exp (−κ1ξ1) − C exp (−κ2ξ2), where κ1, ν1, κ2, ν2 and C are fitting parameters which depend on N . In this paper we extract the general N -dependence of Plink. We evaluate numerically the five fitting parameters as functions of N , i.e., κ1(N), ν1(N), κ2(N), ν2(N) and C(N). Considering physical requirements of Plink, we impose constraints on these functions. By taking account of both the numerical data from N = 32 to 512 and the constraints, we propose good approximate functions of N for the above fitting parameters. We find that they are valid from N = 32 to 512. We expect that they should also be effective for approximating the linking probability at least in some region of N > 512, although it is not clear how effective it is for large values of N . This result enables us not only to estimate Plink for an arbitrary N at least roughly, but also to predict the possible asymptotic behavior of Plink at N = ∞.

Crystallization of an Entangled Ring Polymer: Coexistence of Crystal and Amorphous Regions

We discuss a molecular model of crystallization of a quenched ring polymer of a fixed knot type. Crystallized ring polymers contain not only crystal regions but also amorphous regions consisting of highly entangled chain configurations that could not be obtained simply by folding the polymer chains. In the crystallization process, the growth of a lamella region is accompanied by the reduction of a locally entangled region of the polymer chain. The model should effectively describe the separation and formation processes of amorphous and crystal regions in polymer crystallization. Here, the knot type of an initial configuration of the ring polymer is conserved in time evolution. We find that the average fraction of amorphous regions in the ring chain does not depend on the knot type or the chain length.