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We define the knotting probability of a knot K by the probability for a random polygon or self-avoiding polygon (SAP) of N segments having the knot type K. We show fundamental and generic properties of the knotting probability particularly its dependence on the excluded volume. We investigate them for the SAP consisting of hard cylindrical segments of unit length and radius rex. For various prime and composite knots, we numerically show that a compact formula describes the knotting probabilities for the cylindrical SAP as a function of segment number N and radius rex. It connects the small-N to the large-N behavior and even to lattice knots in the case of large values of radius. As the excluded volume increases, the maximum of the knotting probability decreases for prime knots except for the trefoil knot. If it is large, the trefoil knot and its descendants are dominant among the nontrivial knots in the SAP. From the factorization property of the knotting probability, we derive a sum rule among the estimates of a fitting parameter for all prime knots, which suggests the local knot picture and the dominance of the trefoil knot in the case of large excluded volumes. Here we remark that the cylindrical SAP gives a model of circular DNA which is negatively charged and semiflexible, where radius rex corresponds to the screening length. [ABSTRACT FROM AUTHOR]

For various polymers with different structures in chemical connectivity expressed by graphs, we numerically evaluate the mean-square radius of gyration and the hydrodynamic radius systematically through simulation. We call polymers with nontrivial structures in chemical connectivity and those of nontrivial topology of spatial graphs as embeddings in three dimensions topological polymers. We evaluate the two quantities both for ideal and real chain models and show that the ratios of the quantities among different structures in chemical connectivity do not depend on the existence of excluded volume if the topological polymers have only up to trivalent vertices, as far as the polymers investigated. We also evaluate the ratio of the gyration radius to the hydrodynamic radius, which we expect to be universal from the viewpoint of renormalization group. Furthermore, we show that the short-distance intrachain correlation is much enhanced for real topological polymers (the Kremer-Grest model) expressed with complex graphs. [ABSTRACT FROM AUTHOR]

For a double-ring polymer in solution we evaluate the mean-square radius of gyration and the diffusion coefficient through simulation of off-lattice self-avoiding double polygons consisting of cylindrical segments with radius rex of unit length. Here, a self-avoiding double polygon consists of twin self-avoiding polygons which are connected by a cylindrical segment. We show numerically that several statistical and dynamical properties of double-ring polymers in solution depend on the linking number of the constituent twin ring polymers. The ratio of the mean-square radius of gyration of self-avoiding double polygons with zero linking number to that of no topological constraint is larger than 1, in particular, when the radius of cylindrical segments rex is small. However, the ratio is almost constant with respect to the number of vertices, N, and does not depend on N. The large-N behavior of topological swelling is thus quite different from the case of knotted random polygons. [ABSTRACT FROM AUTHOR]

Abstract We use a simple physical mapping to adapt the known asymptotic expressions for the knotting probabilities of self-avoiding polygons to the case of semiflexible rings of beads. We thus obtain analytical expressions that approximate the abundance of the simplest knots as a function of the length and bending rigidity of the rings. We validate the predictions against previously published data from stochastic simulations of rings of beads showing that they reproduce the intriguing non-monotonic dependence of knotting probability on bending rigidity. The mapping thus provides a useful theoretical tool not only for a physically-transparent interpretation of previous results, but especially to predict the knotting probabilities for previously unexplored combinations of chain lengths and bending rigidities. In particular, our mapping suggests that for rings longer than 20,000 beads, the rigidity-dependent knotting probability profile switches from unimodal to bimodal. [ABSTRACT FROM AUTHOR]

Abstract We revisit the numerical quaternionic study on the mean-square radius of gyration and the hydrodynamic radius for topological or graph-shaped polymers. We show that it is consistent with other approaches although we apply a nontrivial modification of the quaternionic method [51] for generating random polygons. In the modified method we generate random walks that connect two given points by rescaling the bond length and assign them some weight. We evaluate by it the mean-square radius of gyration and the hydrodynamic radius for several topological polymers. We correct the plots of Ref. [46] for the hydrodynamic radius versus the segment number and for the ratio of the gyration to the hydrodynamic radius versus the segment number. The estimated ratios are close to the values derived from an analytic assumption of the pair distribution function. The gyration radius of the multi-theta chain evaluated by the modified method agrees with exact Gaussian results [48]. We derive the moments of the bond vectors' coordinates distribution in random polygons generated by the quaternionic method. [ABSTRACT FROM AUTHOR]

Highlights • For the first time the two-point correlation function of ring polymers was measured. • The critical exponents governing the correlation function were determined. • A good agreement was observed between the experimental determined critical exponents and the one from numerical simulations. [ABSTRACT FROM AUTHOR]

Abstract: We evaluate numerically the mean-square (MS) radius of gyration and the diffusion coefficient for topological polymers such as ring, tadpole, double-ring, and caged polymers and catenanes. We consider caged polymers with any given number of subchains, and catenanes consisting of two linked ring polymers with a fixed linking number. Through Kirkwood’s approximation we evaluate the hydrodynamic radius, which is proportional to the inverse of the diffusion coefficient, for various topological polymers. Here we take the statistical averages over configurations of topological polymers constructed through the quaternionic algorithm, which generates uniform random walks connecting given two points. It gives ideal chains with no excluded volume. We evaluate numerically the ratio of the square root of the MS radius of gyration to the hydrodynamic radius for several topological polymers, and show for them that the ratio decreases as the topology becomes more complex. [Copyright &y& Elsevier]

We show numerically that critical exponents for two-point intrachain correlation of an infinite chain characterize those of finite chains in selfavoiding walk (SAW) and self-avoiding polygon (SAP) under a topological constraint.We evaluate short-distance exponents ϑ(i, j) through the probability distribution functions of the distance between the ith and jth vertices of N-step SAW (or SAP with a knot) for all pairs (1 ⩽ i, j ⩽ N). We construct the contour plot of ϑ(i, j), and express it as a function of i and j. We suggest that it has quite a simple structure. Here exponents ϑ(i, j) generalize des Cloizeaux's three critical exponents for short-distance intrachain correlation of SAW, and we showthe crossover among them. We also evaluate the diffusion coefficient of knotted SAP for a few knot types, which can be calculated with the probability distribution functions of the distance between two nodes. [ABSTRACT FROM AUTHOR]