Random space and plane curves

Abstract

We study random knots, which we define as a triple of random periodic functions (where a random function is a random trigonometric series, \[f(\theta) = \sum_{k=1}^\infty a_k \cos (k \theta) +b_k (\sin k \theta),\] with $a_k, b_k$ are independent gaussian random variables with mean $0$ and variance $\sigma(k)^2$ - our results will depend on the functional dependence of $\sigma$ on $k.$ In particular, we show that if $\sigma(k) = k^\alpha,$ with $\alpha < -3/2,$ then the probability of getting a knot type which admits a projection with $N$ crossings, decays at least as fast as $1/N.$ The constant $3/2$ is significant, because having $\alpha < -3/2$ is exactly the condition for $f(\theta)$ to be a $C^1$ function, so our class is precisely the class of random \emph{tame} knots. We also find some suprising experimental observations on the zeros of Alexander polynomials of random knots (with slowly and non-decaying coefficients), and even more surprising observations on their coefficients. Our observations persist in other models of random knots, making it likely that the results are universal.