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## Section: New Results

### Computing Chebyshev knot diagrams

A Chebyshev curve $𝒞\left(a,b,c,\phi \right)$ has a parametrization of the form $x\left(t\right)={T}_{a}\left(t\right)$; $y\left(t\right)={T}_{b}\left(t\right)$; $z\left(t\right)={T}_{c}\left(t+\phi \right)$, where $a,b,c$ are integers, ${T}_{n}\left(t\right)$ is the Chebyshev polynomial of degree $n$ and $\phi \in ℝ$. When $𝒞\left(a,b,c,\phi \right)$ is nonsingular, it defines a polynomial knot. In [12], we determine all possible knot diagrams when $\phi$ varies. Let $a,b,c$ be integers, $a$ is odd, $\left(a,b\right)=1$, we show that one can list all possible knots $𝒞\left(a,b,c,\phi \right)$ in $O\left({n}^{2}\right)$ bit operations, with $n=abc$.