A polynomial parametrization of torus knots

Abstract

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).