Chebyshev diagrams for rational knots

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

1 SALSA - Solvers for Algebraic Systems and Applications
2 UPMC - Université Pierre et Marie Curie - Paris 6

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 $b+ \deg C = 3N.$ 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 $a \le 4.$

Mots clés: Polynomial curves; Chebyshev curves; rational knots; continued fractions

Domaines: Topologie géométrique [math.GT]

Pierre-Vincent Koseleff, Daniel Pecker. Chebyshev diagrams for rational knots. 2009. ⟨hal-00397617⟩