## Section: New Results

### Computing Chebyshev knot diagrams

A Chebyshev curve $\mathcal{C}(a,b,c,\phi )$ has a parametrization of the form $x\left(t\right)={T}_{a}\left(t\right)$; $y\left(t\right)={T}_{b}\left(t\right)$; $z\left(t\right)={T}_{c}(t+\phi )$, where $a,b,c$ are integers, ${T}_{n}\left(t\right)$ 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. In [12], 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 $O\left({n}^{2}\right)$ bit operations, with $n=abc$.