Computing Chebyshev knot diagrams

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