A polynomial parametrization of torus knots

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