Chebyshev diagrams for rational knots
Abstract
We show that every rational knot K of crossing number N admits a polynomial parametrization x=Ta(t), y = Tb(t), z = C(t) where Tk(t) are the Chebyshev polynomials, a=3 and b+ C = 3N. We show that every rational knot also admits a polynomial parametrization with a=4. If C (t)= Tc(t) is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for a 4.
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