Relative Frequencies of Alternating and Nonalternating Prime Knots and Composite Knots in Random Knot Spaces

Abstract

ABSTRACTThis article reports the results of an investigation into the average behavior of the knot spectrum (of knots up to 16 crossings) of a family of random knot spaces. A knot space in this family consists of random polygons of a given length in a spherical confinement of a given radius. The knot spectrum is the distribution of all knot types within a random knot space and is based on the probabilities that a randomly (and uniformly) chosen polygon from this knot space forms different knot types. We show that the relative spectrum of knots, when divided into groups by their crossing number, remains unexpectedly robust as these knot spaces vary. The relative spectrum for a given crossing number c is Pc(u)/Pc, where Pc is the probability that a uniformly chosen random polygon has crossing number c, and Pc(u) is the probability that the chosen polygon has crossing number c and is from the group of knots defined by the characteristic u (such as “alternating prime,” “nonalternating prime,” or “composite”). Specifically, for a fixed crossing number c, the results show that tighter confinement conditions favor alternating prime knots, that is Pc(A)/Pc (where A stands for “alternating prime”) increases as the confinement radius decreases, and that the average Pc(N) (where N stands for “nonalternating prime”) behaves similar to the average Pc − 1(A). We then use our simulations to speculate on limiting behavior.