Harmonic Knots

Pierre-Vincent Koseleff (UPMC; IMJ; INRIA Paris-Rocquencourt); Daniel; Pecker (UPMC; IMJ)

arXiv:1203.4376·math.GT·September 22, 2014·1 cites

Harmonic Knots

Pierre-Vincent Koseleff (UPMC, IMJ, INRIA Paris-Rocquencourt), Daniel, Pecker (UPMC, IMJ)

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TL;DR

This paper classifies harmonic knots parametrized by Chebyshev polynomials for small parameters, analyzing specific families and providing a comprehensive table of the simplest harmonic knots.

Contribution

It offers a classification of harmonic knots with small parameters and explores particular families, expanding understanding of their structure and properties.

Findings

01

Classified harmonic knots for a ≤ 4.

02

Analyzed knots of the form H(2n-1, 2n, 2n+1).

03

Provided a table of the simplest harmonic knots.

Abstract

The harmonic knot $\overset{˝}{(} a, b, c)$ is parametrized as $K (t) = (T_{a} (t), T_{b} (t), T_{c} (t))$ where $a$ , $b$ and $c$ are pairwise coprime integers and $T_{n}$ is the degree $n$ Chebyshev polynomial of the first kind. We classify the harmonic knots $\overset{˝}{(} a, b, c)$ for $a \leq 4.$ We study the knots $\overset{˝}{(} 2 n - 1, 2 n, 2 n + 1),$ the knots $\overset{˝}{(} 5, n, n + 1),$ and give a table of the simplest harmonic knots.

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Taxonomy

TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · semigroups and automata theory