A Statistical Study of Random Knotting Using the Vassiliev Invariants

Abstract

Employing the Vassiliev invariants as tools for determining knot types of polygons in 3 dimensions, we evaluate numerically the knotting probability P K (N) of the Gaussian random polygon being equivalent to a knot type K. For prime knots and composite knots we plot the knotting probability P K (N) against the number N of polygonal nodes. Taking the analogy with the asymptotic scaling behaviors of self-avoiding walks, we propose a formula of fitting curves to the numerical data. The curves fit well the graphs of the knotting probability P K (N) versus N. This agreement suggests to us that the scaling formula for the knotting probability might also work for the random polygons other than the Gaussian random polygon.