Linking of uniform random polygons in confined spaces

Arsuaga, Javier; Blackstone, T; Diao, Yuanan; Karadayi, Enver; Saito, Masahico

doi:10.1088/1751-8113/40/9/001

Cited by 34 publications

(74 citation statements)

References 21 publications

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“…. ; Y m independently with the same 3-dimensional distribution, the above extends to two-component links (Arsuaga et al 2007a), and similarly for any number of components (Fig. 5).…”

Section: Random Jumpmentioning

confidence: 61%

“…Consider the linking number L mn of a random two-component link with n and m segments. It is known (Arsuaga et al 2007a;Flapan and Kozai 2016) that its variance is HðnmÞ, and it is conjectured that L mn = ffiffiffiffiffiffi nm p converges in distribution to a Gaussian (Panagiotou et al 2010;Karadayi 2010). Based on our analysis of the Petaluma model (Even-Zohar et al 2016) we tend to doubt this conjecture.…”

Section: Random Jumpmentioning

confidence: 99%

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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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“…. ; Y m independently with the same 3-dimensional distribution, the above extends to two-component links (Arsuaga et al 2007a), and similarly for any number of components (Fig. 5).…”

Section: Random Jumpmentioning

confidence: 61%

Section: Random Jumpmentioning

confidence: 99%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

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“…Theorem 4 resolves a difficulty that was encountered in the uniform random polygon model [5]. In fact, it gives the first explicit description of the asymptotic probability distribution for any knot or link invariant.…”

Section: Theoremmentioning

confidence: 73%

“…Common models are based on random 4-valent planar graphs with randomly assigned crossings, random diagrams on the integer grid in R 2 , Gaussian random polygons [10,5,29], and random walks on lattices in R 3 [35,33]. While many interesting numerical studies have been performed, and interesting results obtained in these models, there have been few rigorous derivations of associated statistical measures.…”

Section: Introductionmentioning

confidence: 99%

Invariants of Random Knots and Links

Even-Zohar

Hass

Linial

et al. 2016

Discrete Comput Geom

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We study random knots and links in R 3 using the Petaluma model, which is based on the petal projections developed in [2]. In this model we obtain a formula for the limiting distribution of the linking number of a random two-component link. We also obtain formulas for the expectations and the higher moments of the Casson invariant and the order-3 knot invariant v3. These are the first precise formulas given for the distributions and higher moments of invariants in any model for random knots or links. We also use numerical computation to compare these to other random knot and link models, such as those based on grid diagrams. MSC: 57M25¨60B05 IntroductionIn this paper we study the distribution of finite type invariants of random knots and links. Our purpose is to investigate properties of typical knots, avoiding biases caused by focusing attention on a limited set of commonly studied examples. While tables of knots with up to 16 crossings have been compiled [18], and much is understood about infinite classes of knots, such as torus and alternating knots, we suspect that our view of the collection of all knots is distorted by the choices that simplicity and availability have given us. We have little knowledge of the distribution of knot invariants such as the Jones polynomial, or the linking number, among highly complicated knots and links. Studying a model of random knots allows us to probe for typical behavior beyond the familiar classes. As we elaborate below, the spectacular success of the probabilistic method in combinatorics makes us hopeful that it has much to offer in topology as well.A variety of models for random knots and links have been studied by physicists and biologists, as well as mathematicians. Common models are based on random 4-valent planar graphs with randomly assigned crossings, random diagrams on the integer grid in R 2 , Gaussian random polygons [10,5,29], and random walks on lattices in R 3 [35,33]. While many interesting numerical studies have been performed, and interesting results obtained in these models, there have been few rigorous derivations of associated statistical measures.In this paper we study a model of random knots and links called the Petaluma model, based on the representation of knots and links as petal diagrams that was introduced by Adams and studied in [2]. The Petaluma model has the advantage of being both universal, in that it represents all knots and links, and combinatorially simple, so that knots have simple descriptions in terms of a single permutation. We obtain here what appears to be the first precise formulas in any random model for the distributions of knot and link invariants.

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“…Let D be a knot diagram with n crossings. Then there are at least 2 3 √ n distinct unknot diagrams that can be obtained by performing crossing exchanges on D.…”

Section: Our Main Resultsmentioning

confidence: 99%

On the Number of Unknot Diagrams

Medina¹,

Ramírez-Alfonsín²,

Salazar³

2019

SIAM J. Discrete Math.

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Let D be a knot diagram, and let D denote the set of diagrams that can be obtained from D by crossing exchanges. If D has n crossings, then D consists of 2 n diagrams. A folklore argument shows that at least one of these 2 n diagrams is unknot, from which it follows that every diagram has finite unknotting number. It is easy to see that this argument can be used to show that actually D has more than one unknot diagram, but it cannot yield more than 4n unknot diagrams. We improve this linear bound to a superpolynomial bound, by showing that at least 2 3 √ n of the diagrams in D are unknot. We also show that either all the diagrams in D are unknot, or there is a diagram in D that is a diagram of the trefoil knot. arXiv:1710.06470v1 [math.CO]

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Linking of uniform random polygons in confined spaces

Cited by 34 publications

References 21 publications

Models of random knots

Models of random knots

Invariants of Random Knots and Links

On the Number of Unknot Diagrams

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