One measure of geometrical complexity of a spatial curve is the number of
crossings in a planar projection of the curve.
For N-noded ring polymers with a fixed knot type, we evaluate numerically
the average of the crossing number over some directions. We find that the
average crossing number under the topological constraint are less than that of
no topological constraint for large N. The decrease of the geometrical
complexity is significant when the thickness of polymers is small.
The simulation with or without a topological constraint also shows that the
average crossing number and the average size of ring polymers are independent
measures of conformational complexity.Comment: 8 pages, 4figure
