Scattering functions of knotted ring polymers.

We discuss the scattering function of a Gaussian random polygon with N nodes under a given topological constraint through simulation. We evaluate the form factor PK(q) of a Gaussian polygon of N = 200 having a fixed knot K for some different knots such as the trivial, trefoil, and figure-eight knots. Here the Gaussian polygons with different knots K have distinct values of the mean-square radius of gyration, R2(G,K). We obtain the Kratky plots of the form factors--i.e., the plots of (qR(G,K))2PK(q) versus qR(G,K)--for the different topological constraints and discuss nontrivial large-q behavior as well as small-q behavior for the scattering functions. We also find that the distinct values of R2(G,K) play an important role in the large-q and small-q properties of the Kratky plots.