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A polynomial parametrization of torus knots
Pierre-Vincent Koseleff1, 2, Daniel Pecker2

For every odd integer $N$ we give an explicit construction of a polynomial curve $\cC(t) = (x(t), y (t))$, where $\deg x = 3$, $\deg y = N + 1 + 2\pent N4$ that has exactly $N$ crossing points $\cC(t_i)= \cC(s_i)$ whose parameters satisfy $s_1 < \cdots < s_{N} < t_1 < \cdots < t_{N}$. Our proof makes use of the theory of Stieltjes series and Padé approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$.
1:  IMJ - Institut de Mathématiques de Jussieu
2:  UPMC - Université Pierre et Marie Curie - Paris 6
Polynomial curves – Stieltjes series – Padé approximant – torus knots