The writhe of permutations and random framed knots

J Appl. and Comput. Topology

2017

Self Cite

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.

Section: Finite Type Invariantsmentioning

confidence: 99%

Section: The Petaluma Modelmentioning

confidence: 99%

Section: The Petaluma Modelmentioning

confidence: 99%

See 1 more Smart Citation

Models of random knots

J Appl. and Comput. Topology

2017

Self Cite

“…This model has the advantage of being based on one random permutation, and it seems related to knotting phenomena arising in biology and elsewhere. In a previous work [EHLN16,Eve16] we investigated the distribution of finite type invariants in the Petaluma model. Here we return to some remaining fundamental questions about this model, such as how many knots can appear and with what probabilities.…”

Section: Introductionmentioning

confidence: 99%

The distribution of knots in the Petaluma model

Hass

Linial

et al. 2018

Algebr. Geom. Topol.

Self Cite

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

“…The second author studies other random knot models in [EZ15] and in work with Hass, Linial, and Nowik [EZHLN14] using petal diagrams of Adams et al [ACD + 15] and random grid diagrams. Other random knot models can be found in work by Dunfield, Obeidin et al [Dun14], by Cantarella, Chapman and Mastin [CCM15], and by Westenberger [Wes16].…”

Section: Introductionmentioning

confidence: 99%

Crossing numbers of random two-bridge knots

Cohen

Topology and its Applications

Krishnan

2018

Self Cite

In a previous work, the first and third authors studied a random knot model for all two-bridge knots using billiard table diagrams. Here we present a closed formula for the distribution of the crossing numbers of such random knots. We also show that the probability of any given knot appearing in this model decays to zero at an exponential rate as the length of the billiard table goes to infinity. This confirms a conjecture from the previous work. 1 knots are parametrized by the already-closed curve (cos(at + φ), cos(bt + θ), cos(ct + ψ)) where t ∈ [0, 2π], and φ, θ, ψ ∈ R are fixed phase shifts [BHJS94, JP98, HZ07, BDHZ09].Not all knots are harmonic or Lissajous. However, the diagrams resulting from the projection of Lissajous curves to the xy-plane give rise to all knots by suitable choice of the crossing information. This was shown by Lamm [Lam99] in the study of Fourier knots.Koseleff and Pecker [KP11b] prove a similar statement for the open harmonic path (cos at, cos bt), in their work on Chebyshev knots. They reparametrize it as (T a (t), T b (t)) using the Chebyshev polynomials T n (cos θ) = cos nθ, and show that any choice for the crossings can be realized by some z(t) = T c (t + φ). Vassiliev represents all knots by polynomials in [Vas90].