A Statistical Study of Random Knotting Using the Vassiliev Invariants

Employing the Vassiliev invariants as tools for determining knot types of polygons in 3 dimensions, we evaluate numerically the knotting probability PK(N) of the Gaussian random polygon being equivalent to a knot type K. For prime knots and composite knots we plot the knotting probability PK(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 PK(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.

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