Computing Chebyshev knot diagrams

12/24/2015

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by   P. -V Koseleff, et al.

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A Chebyshev curve C(a,b,c,ϕ) has a parametrization of the form x(t)=T_a(t); y(t)=T_b(t); z(t)= T_c(t + ϕ), where a,b,care integers, T_n(t) is the Chebyshev polynomialof degree n and ϕ∈R. When C(a,b,c,ϕ) is nonsingular,it defines a polynomial knot. 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) inÕ(n^2) bit operations, with n=abc.