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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]

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]

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]