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Section: New Results

Computing Chebyshev knot diagrams

A Chebyshev curve 𝒞(a,b,c,φ) has a parametrization of the form x(t)=Ta(t); y(t)=Tb(t); z(t)=Tc(t+φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ. When 𝒞(a,b,c,φ) is nonsingular, it defines a polynomial knot. In [12], we determine all possible knot diagrams when φ varies. Let a,b,c be integers, a is odd, (a,b)=1, we show that one can list all possible knots 𝒞(a,b,c,φ) in O(n2) bit operations, with n=abc.