Topological sum rules in the knotting probabilities of DNA

We revisit a topological sum rule that the sum of the coefficients (or the amplitudes) of the knotting probabilities of self-avoiding polygons (SAP) over all prime knots is equal to 1. Here we define the coefficient as a fitting parameter of a formula expressing the knotting probability as a function of the number of segments, where the estimates of the fitting parameters are shown to be very close to those of the asymptotic expansion of the knotting probability. We numerically show these results for a model of semi-flexible ring polymers such as circular DNA consisting of cylindrical segments with radius rex of unit length by simulation with several values of rex. From the sum rule we argue that SAPs with the trefoil knot and its descendants are dominant among all SAPs if the excluded volume is large. We also suggest that the knot exponent of a prime knot is smaller than 1 if the excluded volume is very small and it gradually increases to 1 as the excluded volume increases.