Models of random knots

Even-Zohar, Chaim

doi:10.1007/s41468-017-0007-8

Cited by 18 publications

(17 citation statements)

References 192 publications

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“…We however remark that the resultats are compatible with an exponential decay as for self-avoiding grid walk (see [5] for a nice review) and one may expect that similar arguments such as variants of Kesten pattern theorem may apply in the present case. However, the construction of the curve in our model is different enough so that the proof can not be transposed directly.…”

Section: Number Of Crossingssupporting

confidence: 65%

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Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Crouy-Chanel¹,

Simon²

2019

J Stat Phys

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Three-dimensional three-colour percolation on a lattice made of tetrahedra is a direct generalization of two-dimensional two-colour percolation on the triangular lattice. The interfaces between one-colour clusters are made of bicolour surfaces and tricolour non-intersecting and non-self-intersecting curves. Because of the three-dimensional space, these curves describe knots and links. The present paper presents a construction of such random knots using particular boundary conditions and a numerical study of some invariants of the knots. The results are sources of precise conjectures about the limit law of the Alexander polynomial of the random knots.

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Section: Number Of Crossingssupporting

confidence: 65%

“…There are only a few constructions of random knots in the mathematical literature. The review [5] presents various models of random knots with a list of results about them. In general, the construction of random knots either use directly the combinatorial description of knots -i.e.…”

Section: Contextmentioning

confidence: 99%

Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Crouy-Chanel¹,

Simon²

2019

J Stat Phys

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“…A generic planar connected graph corresponds to a link shadow, where a link is viewed as the embedding of multiple nonintersecting S 1 components. Instead, we employ the random grid walk method [55,56] to sample random knot diagrams, ensuring by construction that the number of components is always one. A grid walk consists of horizontal segments and vertical segments, where vertical segments always pass over horizontal ones.…”

mentioning

confidence: 99%

Evaluating the Jones polynomial with tensor networks

Meichanetzidis

Kourtis

2019

Phys. Rev. E

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We introduce tensor network contraction algorithms for the evaluation of the Jones polynomial of arbitrary knots. The value of the Jones polynomial of a knot maps to the partition function of a q-state Potts model defined as a planar graph with weighted edges that corresponds to the knot. For any integer q, we cast this partition function into tensor network form and employ fast tensor network contraction protocols to obtain the exact tensor trace, and thus the value of the Jones polynomial. By sampling random knots via a grid-walk procedure and computing the full tensor trace, we demonstrate numerically that the Jones polynomial can be evaluated in time that scales subexponentially with the number of crossings in the typical case. This allows us to evaluate the Jones polynomial of knots that are too complex to be treated with other available methods. Our results establish tensor network methods as a practical tool for the study of knots.

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“…1. The results in [EHLN16] and further numerical experiments [Eve17] indicate that our upper bound on P [c 2 (K 2n+1 ) = v] is not expected to be tight. It remains desirable to establish a bound of O(n −2 ) in Theorem 5.…”

Section: Petaluma Knots Are Knottedmentioning

confidence: 84%

“…The study of random knots and links emerges from various perspectives, both theoretical and applied. See [Eve17] for a survey of randomized knot models in the literature. We here pursue the study of the Petaluma model, based on petal diagrams [ACD + 15].…”

Section: Introductionmentioning

confidence: 99%

The distribution of knots in the Petaluma model

Even-Zohar

Hass

Linial

et al. 2018

Algebr. Geom. Topol.

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The representation of knots by petal diagrams (Adams et al. 2012) naturally defines a sequence of distributions on the set of knots. In this article we establish some basic properties of this randomized knot model. We prove that in the random n-petal model the probability of obtaining every specific knot type decays to zero as n, the number of petals, grows. In addition we improve the bounds relating the crossing number and the petal number of a knot. This implies that the n-petal model represents at least exponentially many distinct knots.Past approaches to showing, in some random models, that individual knot types occur with vanishing probability, rely on the prevalence of localized connect summands as the complexity of the knot increases. However this phenomenon is not clear in other models, including petal diagrams, random grid diagrams, and uniform random polygons. Thus we provide a new approach to investigate this question.MSC 57M25, 60B05

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Models of random knots

Cited by 18 publications

References 192 publications

Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Random Knots in 3-Dimensional 3-Colour Percolation: Numerical Results and Conjectures

Evaluating the Jones polynomial with tensor networks

The distribution of knots in the Petaluma model

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