Harmonic Knots

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

1 UPMC - Université Pierre et Marie Curie - Paris 6
2 IMJ - Institut de Mathématiques de Jussieu

The harmonic knot $H(a,b,c)$ is parameterized as $K(t)= (T_a(t) ,T_b (t), T_c (t))$ where $a$, $b$ and $c$ are relatively 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 show that the knot $H(2n-1, 2n, 2n+1)$ is isotopic to $H(4,2n-1, 2n+1)$ (up to mirror symmetry). We study the knots $H(5,n,n+1)$ and give a table of the simplest harmonic knots.

Mots clés: Knots; polynomial curves; Chebyshev curves; rational knots; continued fractions

Domaines: Topologie géométrique [math.GT]

Pierre-Vincent Koseleff, Daniel Pecker. Harmonic Knots. 2012. ⟨hal-00680746v1⟩