The Knot Spectrum Of Random Knot Spaces

It is well known that knots exist in natural systems. For example, in the case of (mutant) bacteriophage P4, DNA molecules packed inside the bacteriophage head are considered to be circular since the two sticky ends of the DNA are close to each other. The DNAs extracted from the capsid, without separating the two ends, can preserve the topology of the (circular) DNAs, and hence are well-defined knots. Furthermore, knots formed within such systems are often varied and different knots occur with different probabilities. Such information can be important in biology. Mathematically, we may view (and model) such a biological system as (by) a random knot space and attempt to obtain information about the system via mathematical analysis and numerical simulation. The question here is to find the probability that a randomly (and uniformly) chosen knot from this space is of a particular knot type. This is equivalent to finding the distribution of all knot types within this random knot space (called the knot spectrum in an earlier paper by the authors). In this paper, we examine the behavior of the knot spectrums for knots up to 10 crossings. Using random polygons of various lengths under different confinement conditions as the random knot spaces (model biological systems), we demonstrate that the relative spectrums of the knots, when divided into groups by their crossing numbers, remain surprisingly robust as these knot spaces vary. For a given knot type K, we let P-K (L, R) be the probability that an equilateral random polygon of length L in a confinement sphere of radius R has knot type K. We give a model for the family of functions P-K (L, R) and show that our model function fits the random polygon data we generated. For a fixed crossing number Cr, 3 <= Cr <= 10, let S-Cr be the subspace consisting of random polygons which form knots that have crossing number Cr. We study the relative distribution of all the different knot types within S-Cr and illustrate how this distribution changes if we keep the length L fixed (or the confinement radius R fixed) and vary the confinement radius R (or the length L). We observe that this distribution is quite robust and remains essentially unchanged under length and confinement radius variation, especially if one concentrates on subfamilies such as alternating prime knots, non-alternating prime knots, or composite knots.