Harmonic Knots

Pierre-Vincent Koseleff (1, 2, 3) , Daniel Pecker (2, 1)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 UPMC - Université Pierre et Marie Curie - Paris 6
3 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs

The harmonic knot $\H(a,b,c)$ 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 $\H(a,b,c)$ for $ a \le 4. $ We study the knots $\H (2n-1, 2n, 2n+1),$ the knots $\H(5,n,n+1),$ and give a table of the simplest harmonic knots.

Mots clés: Rational knots; Continued fractions; Knots; Polynomial curves; Chebyshev curves

Domaines: Topologie géométrique [math.GT]

Pierre-Vincent Koseleff, Daniel Pecker. Harmonic Knots. Journal of Knot Theory and Its Ramifications, 2016, 25 (13), 18p. ⟨10.1142/S0218216516500747⟩. ⟨hal-00680746v2⟩