Sampling Lissajous and Fourier Knots

Boocher, Adam; Daigle, Jay; Hoste, Jim; Zheng, Wenjing

doi:10.1080/10586458.2009.10129057

Cited by 17 publications

(25 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Taking a single cosine, zðtÞ ¼ cosðk z t þ / z Þ defines the well-studied Lissajous knots (Bogle et al 1994;Jones and Przytycki 1998;Lamm 1997;Hoste and Zirbel 2006). In Boocher et al (2009) and Rivin (2016) experiments on random Fourier and Lissajous knots are reported.…”

Section: Smoothed Brownian Motionmentioning

confidence: 99%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

View full text Add to dashboard 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.

show abstract

Section: Smoothed Brownian Motionmentioning

confidence: 99%

Models of random knots

Even-Zohar

2017

J Appl. and Comput. Topology

View full text Add to dashboard Cite

show abstract

“…In (Boocher, Daigle, Hoste, and Zheng, 2009), Lissajous knots have been sampled by numerical experiments, and several knots with relatively small crossing numbers were identified.…”

Section: Motivationsmentioning

confidence: 99%

Computing Chebyshev knot diagrams

Koseleff

Pecker

Rouillier

et al. 2018

Journal of Symbolic Computation

View full text Add to dashboard Cite

A Chebyshev curve C(a, b, c, ϕ) has a parametrization of the form x(t) = Ta(t); y(t) = T b (t); z(t) = Tc(t + ϕ), where a, b, c are integers, Tn(t) is the Chebyshev polynomial of degree n and ϕ ∈ R. When C(a, b, c, ϕ) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when ϕ varies. Let a, b, c be integers, a is odd, (a, b) = 1, we show that one can list all possible knots C(a, b, c, ϕ) in O(n 2 ) bit operations, with n = abc.

show abstract

“…They also conjectured that every knot is a billiard knot in some convex polyhedron. ( [JP], see also [La2,C,BHJS,BDHZ,P]).…”

Section: Introductionmentioning

confidence: 99%