Section: New Results

Computing Chebyshev knot diagrams

A Chebyshev curve $𝒞(a,b,c,\phi )$ 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 $\phi \in ℝ$. When $𝒞(a,b,c,\phi )$ 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 $𝒞(a,b,c,\phi )$ in $O\left(n^2\right)$ bit operations, with $n=abc$.