A polynomial parametrization of torus knots

For every odd integer N we give explicit construction of a polynomial curve ${\mathcal C(t)=(x(t),y(t))}$ , where ${{\rm deg}\, x=3, {\rm deg}\, y=N + 1 + 2\frac{N}{4}]}$ that has exactly N crossing points ${\mathcal C(t_i)=\mathcal C(s_i)}$ whose parameters satisfy s ₁ < ? < s _N < t ₁ < ? < t _N . Our proof makes use of the theory of Stieltjes series and Padé approximants. This allows us an explicit polynomial parametrization of the torus knot K _2,2n+1 with degree (3, 3n + 1, 3n + 2).