A polynomial parametrization of torus knots

For every odd integer N we give explicit construction of a polynomial curve C(t) = (x(t), y(t)), where deg x = 3, deg y = N + 1 + 2[N/4] that has exactly N crossing points C(t(i)) = C(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(2,2n+1) with degree (3, 3n + 1, 3n + 2).