A polynomial parametrization of torus knots

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For every odd integer N we give an explicit construction of a polynomial curve (t) = (x(t), y (t)), where °x = 3, °y = N + 1 + 2 N4 that has exactly N crossing points (tᵢ)= (sᵢ) 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,ₙ.