Knots in self-avoiding walks

Sumners, D. W.; Whittington, S G

doi:10.1088/0305-4470/21/7/030

Cited by 283 publications

(266 citation statements)

References 10 publications

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“…This was observed in numerical simulations (Crippen 1974;Frank-Kamenetskii et al 1975), and proved by Sumners and Whittington (1988) and by Pippenger (1989). Using Kesten's pattern theorem (1963), they showed that the unknot appears with exponentially small probability in n. Moreover, every prime knot appears in the decomposition of K n with multiplicity HðnÞ, except for an exponentially small probability (Soteros et al 1992).…”

Section: Self-avoiding Grid Walkmentioning

confidence: 84%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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The study of knots and links from a probabilistic viewpoint provides insight into the behavior of ''typical'' knots, and opens avenues for new constructions of knots and other topological objects with interesting properties. The knotting of random curves arises also in applications to the natural sciences, such as in the context of the structure of polymers. We present here several known and new randomized models of knots and links. We review the main known results on the knot distribution in each model. We discuss the nature of these models and the properties of the knots they produce. Of particular interest to us are finite type invariants of random knots, and the recently studied Petaluma model. We report on rigorous results and numerical experiments concerning the asymptotic distribution of such knot invariants. Our approach raises questions of universality and classification of the various random knot models.

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Section: Self-avoiding Grid Walkmentioning

confidence: 84%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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“…DNA knots have been the focus of significant study in polymer physics 1,2 and, although they are ubiquitous in nature 3 , their exact role in biological processes is still under investigation 4,5,6 . Despite their interest, knotting remains among the least understood properties of polymers due to a lack of both experimental techniques to observe them as well as rigorous theoretical approaches to describe and characterize them.…”

Section: Abstract: Dna Knots Nanoporementioning

confidence: 99%

Direct observation of DNA knots using a solid-state nanopore

et al. 2016

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“…• The typical annealed topology question is that about ring closure experiment and knot probabilities [25,26,27,28,29]: having a linear polymer with "sticky" ends, what is the probability to obtain a certain type of a knot upon first meeting of the two ends [19,20]? A similar question for the winding model is this: what is the probability that a random walk on the plane links number n (or winding angle 2πn) with an obstacle?…”

Section: Introductionmentioning

confidence: 99%

Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer–Edwards–Prager–Frisch model revisited

Grosberg

2003

J. Phys. A: Math. Gen.

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Using a general Green function formulation, we re-derive, both, (i) Spitzer and his followers results for the winding angle distribution of the planar Brownian motion, and (ii) Edwards-Prager-Frisch results on the statistical mechanics of a ring polymer entangled with a straight bar. In the statistical mechanics part, we consider both cases of quenched and annealed topology. Among new results, we compute exactly the (expectation value of) the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory (= ring polymer). We also consider the generalizations of the problem for the finite diameter (disc-like) obstacle and winding within a cavity.

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Knots in self-avoiding walks

Cited by 283 publications

References 10 publications

Models of random knots

Models of random knots

Direct observation of DNA knots using a solid-state nanopore

Winding angle distribution for planar random walk, polymer ring entangled with an obstacle, and all that: Spitzer–Edwards–Prager–Frisch model revisited

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