Computing Chebyshev knot diagrams

Abstract

A Chebyshev curve $\mathcal{C}(a,b,c,\varphi )$ has a parametrization of the form$x(t)=T\_a(t)$; \ $y(t)=T\_b(t)$; $z(t)=T\_c(t+\varphi )$, where $a,b,c$are integers, $T\_n(t)$ is the Chebyshev polynomialof degree $n$ and $\varphi \in \mathbb{R}$. When $\mathcal{C}(a,b,c,\varphi )$ 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 $n=abc$.

References in corpus (3)

• Chebyshev curves, free resolutions and rational curve arrangements
• Chebyshev diagrams for rational knots
• Polynomial knots