Knotting probability ͓ P K (N)͔ is defined by the probability of an N-noded random polygon being topologically equivalent to a given knot K. For several nontrivial knots we numerically evaluate the knotting probabilities for Gaussian and rod-bead models. We find that they are well approximated by the following formula: P K (N)ϭC(K)͓Ñ /N(K)͔ m(K) exp͓ϪÑ/N(K)͔ where Ñ ϭNϪN ini (K), and that the fitting parameters C(K), N(K), and N ini (K) are model dependent, while m(K) is not. We suggest that given a knot K, the exponent m(K) should be universal: it is independent of models of random polygon and is determined only by the knot K. ͓S1063-651X͑97͒09004-1͔ PACS number͑s͒: 05.40.ϩj, 61.41.ϩe, 36.20.Ey Recently knotted ring polymers such as knotted DNA molecules are synthesized in various experiments in chemistry and biology ͓1-3͔. In particular, the formation of knotted species on random ring closure of DNA was observed and their fractions were measured ͓4,5͔. In statistical mechanics, the topological constraint that a ring polymer does not change its topology under any thermal fluctuation leads to a great reduction in the available volume of the configuration space ͓6͔. The topological constraint, or the selfentanglement effect, is derived from the fact that any bond between neighboring monomers in the ring polymer is not disconnected when the bonding energy is very large.In the 1960s Delbrück formulated a fundamental question about the self-entanglement of a ring polymer: What fraction of permissible configurations of a chain of given length will contain a knot ͓7,8͔? The fraction of knotted ring polymers has been studied from the following three approaches: numerical experiments using certain knot invariants ͓9-18͔, mathematical discussion of the self-avoiding polygon ͓19-21͔, and biological experiments using DNAs ͓4,5͔.Let us assume that a model of an N-noded random polygon describes a ring polymer with N bonds. Given a knot K, we define knotting probability P K (N) to the model by the fraction of those configurations of the random polygon that have the same knot type K. The main questions in this paper are how the knotting probability P K (N) behaves as a function of step number N for each knot K, and how it depends on models of a random polygon.For the trivial knot (Kϭ0) we call the knotting probability P 0 (N) the unknotting probability. It has been evaluated for several different models of random polygons with different lengths N less than about 2000 ͓9-14͔. The exponential decay of P 0 (N) with respect to N has been discussed for the molecular dynamical model of ring polymers by Michels and Wiegel, and for the rod-bead model by Koniaris and Muthukumar ͓11,14͔.For nontrivial knots, however, the knotting probabilities have been evaluated only for short polygons with NϽ200, where the graphs of P K (N) versus N can be approximated by linear functions of N ͓9͔. When we calculate knot invariants for polygons with large N, there are two technical difficulties: memory-size and computation-time problems ͓22͔.For pol...
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.
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.
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