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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. 

The goal of this chapter is to describe different techniques used to measure knotting in open curves. Note that there is no "agreed upon" definition for describing knotting in open curves. As a result, we describe the context motivating each definition and then describe some advantages and disadvantages of the different approaches.