Eric J. Rawdon

In nature, macromolecular chains often exist under extreme confinement. Current techniques for modeling these chains by sampling random polygons in confined volumes are limited. In previous work, we introduced a simple model for knots under extreme confinement consisting of random polygons restricted to cylinders, where the edges each connect random points on the top and bottom disks of the given confining cylinder. We presented numerical evidence that polygons generated by this cylindrical model exhibit knotting probabilities that match the predicted knotting probabilities of rooted equilateral unit-length edge random polygons confined within a sphere of radius R ≈ 0.62, a stronger degree of confinement than R = 1. In this paper, we show that certain geometric quantities (total curvature, average crossing number, and writhe) of the cylindrical polygons are consistent with the predictions for rooted equilateral unit-length edge random polygons in spherical confinement with radius below R = 1, varying between 0.62 and 0.72 depending on the quantity. This strengthens the case that this simple model can be used to gain insight into the asymptotic behavior of random polygons under strong spherical confinement. Total torsion is the exception: the cylindrical model does not reproduce the spherical total torsion of phantom polygons at any of the confinement radii in the spherical data, an anomaly we explain geometrically and observe directly in the distribution of individual torsion angles. This anomaly disappears when polygons are grouped by crossing number and total torsion is measured relative to each model's own phantom average. Knot complexity then has a similar relative effect on total torsion in both models.

The HOMFLY-PT polynomial is a link invariant which is effective in determining chiral knot and link types with small crossing numbers. In this chapter, we concentrate on knots. We provide a guide for computing the knot types of configurations from 3D coordinates via the HOMFLY-PT polynomial using publicly-available Linux freeware. We include data on the efficacy of HOMFLY-PT for knot types through crossing number 16.

We give a brief introduction to the software KnotPlot. The goals of this chapter are twofold: 1) to help a new user get started with using KnotPlot and 2) to provide veteran users with additional background and functionality available in the software.

Random knot models are often used to measure the types of entanglements one would expect to observe in an unbiased system with some given physical property or set of properties. In nature, macromolecular chains often exist in extreme confinement. Current techniques for sampling random polygons in confinement are limited. In this paper, we gain insight into the knotting behavior of random polygons in extreme confinement by studying random polygons restricted to cylinders, where each edge connects the top and bottom disks of the cylinder. The knot spectrum generated by this model is compared to the knot spectrum of rooted equilateral random polygons in spherical confinement. Due to the rooting, such polygons require a radius of confinement R ⩾ 1. We present numerical evidence that the polygons generated by this simple cylindrical model generate knot probabilities that are equivalent to spherical confinement at a radius of R ≈ 0.62. We then show how knot complexity and the relative probability of different classes of knot types change as the length of the polygon increases in the cylindrical polygons.

Knots in nature are typically confined spatially. The confinement affects the possible configurations, which in turn affects the spectrum of possible knot types as well as the geometry of the configurations within each knot type. The goal of this paper is to determine how confinement, length, and knotting affect the total curvature and total torsion of random polygons. Previously published papers have investigated these effects in the unconstrained case. In particular, we analyze how the total curvature and total torsion are affected by (1) varying the length of polygons within a fixed confinement radius and (2) varying the confinement radius of polygons with a fixed length. We also compare the total curvature and total torsion of groups of knots with similar complexity (measured as crossing number). While some of our results fall in line with what has been observed in the studies of the unconfined random polygons, a few surprising results emerge from our study, showing some properties that are unique due to the effect of knotting in confinement.

This 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 P_c(u)/P_c, where P_c is the probability that a uniformly chosen random polygon has crossing number c, and P_c(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 P_c(A)/P_c (where A stands for “alternating prime”) increases as the confinement radius decreases, and that the average P_c(N) (where N stands for “nonalternating prime”) behaves similar to the average P_{c − 1}(A). We then use our simulations to speculate on limiting behavior.