The first rational Chebyshev knots

Abstract

A Chebyshev knot $\mathcal{C}(a,b,c,\varphi )$ is a knot which 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 polynomial of degree $n$ and $\varphi \in \mathbb{R}.$ We show that any two-bridge knot is a Chebyshev knot with $a=3$ and also with $a=4$. For every $a,b,c$ integers ($a=3,4$ and $a$, $b$ coprime), we describe an algorithm that gives all Chebyshev knots $\cC(a,b,c,ϕ)$. We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.

22p, 27 figures, 3 tables

References in corpus (3)

• Curves defined by Chebyshev polynomials
• Sampling Lissajous and Fourier knots
• Chebyshev diagrams for rational knots