Sampling large random knots in a confined space

Abstract

DNA knots formed under extreme conditions of condensation, as in bacteriophage P4, are difficult to analyze experimentally and theoretically. In this paper, we propose to use the uniform random polygon model as a supplementary method to the existing methods for generating random knots in confinement. The uniform random polygon model allows us to sample knots with large crossing numbers and also to generate large diagrammatically prime knot diagrams. We show numerically that uniform random polygons sample knots with large minimum crossing numbers and certain complicated knot invariants (as those observed experimentally). We do this in terms of the knot determinants or colorings. Our numerical results suggest that the average determinant of a uniform random polygon of n vertices grows faster than . We also investigate the complexity of prime knot diagrams. We show rigorously that the probability that a randomly selected 2D uniform random polygon of n vertices is almost diagrammatically prime goes to 1 as n goes to infinity. Furthermore, the average number of crossings in such a diagram is at the order of O(n2). Therefore, the two-dimensional uniform random polygons offer an effective way in sampling large (prime) knots, which can be useful in various applications.

Models of random knots

“…Numerical experiments indicate that this probability vanishes faster than expðÀOðnÞÞ (Millett 2000;Arsuaga et al 2007b). This provides evidence for a strong form of the above-mentioned Delbruck-Frisch-Wasserman conjecture in this model.…”

“…Experiments with the cube model suggest that the expected knot determinant is x exp n 2 ð Þ. It is proposed in Arsuaga et al (2007b) that most knots in this model are prime. It was suggested (Thurston 2011) that the expected crossing number in the spherical case is Hðn 2 Þ.…”

“…Micheletti et al (2008) and Marenduzzo et al (2009). However, the simplicity of the random jump model makes it amenable for rigorous mathematical analysis, while it is arguably a prototype of a polygonal model in spatial confinement (Arsuaga et al 2007b). …”

“…Diao et al (2005), Arsuaga et al (2007b) and Diao et al (2010) and Obeidin (2016) used this observation to construct models for prime alternating knots and links. Except for the (2, n)-torus these are hyperbolic links, whose volume can be read off the diagram up to a multiplicative constant (Lackenby 2004).…”

A fast ergodic algorithm for generating ensembles of equilateral random polygons

“…The radius of gyration estimates the size of a molecule and can be experimentally measured by standard sedimentation assays. In [18] it was found that the squared radius of gyration of a polygonal molecule increases as 1 12 n 2ν with ν = 0.5. Our result is in excellent agreement with that presented in [18] as shown in Figure 9.…”