Chebyshev diagrams for two-bridge knots

Pierre-Vincent Koseleff; Daniel Pecker

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arXiv · 0909.3281

Chebyshev diagrams for two-bridge knots

Pierre-Vincent Koseleff·Daniel Pecker

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Abstract

We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t), y = T_b(t), z =C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+°C = 3N$. 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 $a \le 3.$ Most results are derived from continued fractions and their matrix representations.

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BibTeX RIS

Pierre-Vincent Koseleff, Daniel Pecker. 2009-09-17. Chebyshev diagrams for two-bridge knots. https://arxiv.org/abs/0909.3281

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