Computing Chebyshev knot diagrams

Abstract : A Chebyshev curve $\mathcal{C}(a,b,c,\phi)$ has a parametrization of the form $ x(t)=T_a(t)$; \ $y(t)=T_b(t)$; $z(t)= T_c(t + \phi)$, where $a,b,c$ are integers, $T_n(t)$ is the Chebyshev polynomial of degree $n$ and $\phi \in \mathbb{R}$. When $\mathcal{C}(a,b,c,\phi)$ is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when $\phi$ varies. Let $a,b,c$ be integers, $a$ is odd, $(a,b)=1$, we show that one can list all possible knots $\mathcal{C}(a,b,c,\phi)$ in $\tilde{\mathcal{O}}(n^2)$ bit operations, with $n=abc$.
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Pierre-Vincent Koseleff, Daniel Pecker, Fabrice Rouillier, Cuong Tran. Computing Chebyshev knot diagrams. Journal of Symbolic Computation, Elsevier, 2018, 86, pp.21. ⟨10.1016/j.jsc.2017.04.001⟩. ⟨hal-01232181v2⟩

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