Computing Chebyshev knot diagrams

P.-V Koseleff (1, 2, 3) , D Pecker (2, 3) , Fabrice Rouillier (1, 2, 3) , C Tran (3, 2)

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

A Chebyshev curve $\cC(a,b,c,\phi)$ has a parametrization of the form $ x(t)=T_a(t)$; \ $y(t)=T_b(t)$; $z(t)= T_c(t + \phi)$, where $a,b,c$ are integers, $T_n(t)$ is the Chebyshev polynomial of degree $n$ and $\phi \in \RR$. When $\cC(a,b,c,\phi)$ is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when $\phi$ varies. Let $a,b,c$ be integers, $a$ is odd, $(a,b)=1$, we show that one can list all possible knots $\cC(a,b,c,\phi)$ in $\tcO(n^2)$ bit operations, with $n=abc$.

Mots clés: minimal polynomial; factorization of Chebyshev polynomials; Zero dimensional systems; Chebyshev curves; Lissajous knots; polynomial knots; Chebyshev forms

Domaines: Calcul formel [cs.SC]

P.-V Koseleff, D Pecker, Fabrice Rouillier, C Tran. Computing Chebyshev knot diagrams. 2015. ⟨hal-01232181v1⟩