Summary: The authors discuss probabilities of random knotting and linking through numerical simulations using topological invariants of knots and links. They define knotting probability \(P_K(N)\) by the probability of an \(N\)-noded polygon having knot type \(K\). They introduce a universal fitting formula for the knotting probability and show that the formula gives good fitting curves to the numerical estimates of knotting probabilities for different models of random polygon. The authors consider linking of two \(N\)-noded random polygons which have fixed knot types \(K_1\) and \(K_2\), respectively. They define linking probability \(P_L^{K_1,K_2} (R;N)\) by the probability that a given link \(L\) is formed when one puts a pair of \(N\)-noded random polygons of \(K_1\) and \(K_2\) in distance \(R\). For \((L,K_1,K_2) =(0,0,0)\) and \((2^2_1,0,0)\), the authors numerically evaluate the linking probabilities \(P_L^{K_1K_2} (R;N)\) where 0 denotes the trivial knot or the trivial link, \(2^2_1\) the simplest nontrivial link (the Hopf link). They also discuss a formula which approximates the linking probability. Applying the numerical result of the linking probability they calculate the second virial coefficient of a ring polymer solution at the \(\theta\) temperature.
For the entire collection see [Zbl 0901.00045].