A polynomial parametrization of torus knots

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

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

For every odd integer $N$ we give an explicit construction of a polynomial curve $\cC(t) = (x(t), y (t))$, where $\deg x = 3$, $\deg y = N + 1 + 2\pent N4$ that has exactly $N$ crossing points $\cC(t_i)= \cC(s_i)$ whose parameters satisfy $s_1 < \cdots < s_{N} < t_1 < \cdots < t_{N}$. Our proof makes use of the theory of Stieltjes series and Padé approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$.

Mots clés: Polynomial curves; Stieltjes series; Padé approximant; torus knots

Domaines: Histoire et perspectives sur les mathématiques [math.HO]; Mathématiques générales [math.GM]

Pierre-Vincent Koseleff, Daniel Pecker. A polynomial parametrization of torus knots. Applicable Algebra in Engineering, Communication and Computing, 2009, 20 (5-6), pp.361-377. ⟨10.1007/s00200-009-0103-7⟩. ⟨hal-00197216⟩