A polynomial parametrization of torus knots

For every odd integer N we give explicit construction of a polynomial curve \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal C(t)=(x(t),y(t))}$$\end{document} , where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\rm deg}\, x=3, {\rm deg}\, y=N + 1 + 2[\frac{N}{4}]}$$\end{document} that has exactly N crossing points \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal C(t_i)=\mathcal C(s_i)}$$\end{document} whose parameters satisfy s1 < ⋯ < sN < t1 < ⋯ < tN. Our proof makes use of the theory of Stieltjes series and Padé approximants. This allows us an explicit polynomial parametrization of the torus knot K2,2n+1 with degree (3, 3n + 1, 3n + 2).