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A polynomial parametrization of torus knots. (English) Zbl 1184.57005

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

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
14H50 Plane and space curves
41A21 Padé approximation
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
41A10 Approximation by polynomials
26C10 Real polynomials: location of zeros

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