The linking probability that two closed random walks (i.e., random polygons: RPs) are mutually entangled, P link , is investigated numerically quite accurately. Here, P link is a function of the distance between two RPs, R, and the number of polygonal segments, N . In a previous paper, we numerically estimated P link 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 R g : ξ = R/R g . We have also shown that the N -and R-dependence of P link can be well approximated by a simple function:, where κ 1 , ν 1 , κ 2 , ν 2 and C are fitting parameters which depend on N . In this paper we extract the general N -dependence of P link . We evaluate numerically the five fitting parameters as functions of N , i.e., κ 1 (N ), ν 1 (N ), κ 2 (N ), ν 2 (N ) and C(N ). Considering physical requirements of P link , 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 P link for an arbitrary N at least roughly, but also to predict the possible asymptotic behavior of P link at N = ∞. §1. IntroductionRing polymers are fascinating objects in polymer science. By simply connecting both ends of a linear polymer, some statistical properties of the polymer become nontrivial 1)-9) because of the crucial condition, namely the topological condition that the initial topology of the ring polymer never changes when it is moving and fluctuating under thermal noise. The topological condition makes the phase space of a ring polymer decompose into smaller subspaces with different topological states and any ring polymer in the system does not move from the phase subspace of a given initial topology to another subspace with a different topology. The restriction of possible phase spaces leads to a considerable amount of decrease in the entropy; consequently we have the entropic force derived from the topological constraint acting on the ring polymer. Hereafter, we call such an entropic force that is derived from topological conditions as a topological force.The topological force may strongly affect several physical properties of ring polymers in dilute solution 1)-6) and melt. 7)-9) Furthermore, we may regard polymer netDownloaded from https://academic.oup.