Chebyshev diagrams for rational knots

Pierre-Vincent Koseleff; Daniel Pecker

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arXiv · 0906.4083

Chebyshev diagrams for rational knots

Pierre-Vincent Koseleff·Daniel Pecker

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Abstract

We show that every rational knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_a(t), y = T_b(t), z = C(t)$ where $T_k(t)$ are the Chebyshev polynomials, $a=3$ and $b+ °C = 3N.$ We show that every rational knot also admits a polynomial parametrization with $a=4$. If $C (t)= T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a \le 4.$

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BibTeX RIS

Pierre-Vincent Koseleff, Daniel Pecker. 2009-06-22. Chebyshev diagrams for rational knots. https://arxiv.org/abs/0906.4083

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