Geometrical complexity of conformations of ring polymers under topological constraints.

One measure of geometrical complexity of a spatial curve is the average of the number of crossings appearing in its planar projection. The mean number of crossings averaged over some directions have been numerically evaluated for N-noded ring polymers with a fixed knot type. When N is large, the average crossing number of ring polymers under the topological constraint is smaller than that of no topological constraint. The decrease of the geometrical complexity is significant when the thickness of polymers is small. It is also suggested from the simulation that the relation between the average crossing number and the average size of ring polymers should depend on whether they are under a topological constraint or not.