Knotting probability of self-avoiding polygons under a topological constraint

We define the knotting probability of a knot $K$ by the probability for a random polygon (RP) 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 $r_{\rm ex}$. 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 $r_{\rm ex}$. 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 relation among the estimates of a fitting parameter for all prime knots, which suggests the local knot picture. Here we remark that the cylindrical SAP gives a model of circular DNA which are negatively charged and semiflexible, where radius $r_{\rm ex}$ corresponds to the screening length.