The first rational Chebyshev knots

Abstract

A Chebyshev knot ${\cal C}(a,b,c,ϕ)$ 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 + ϕ), $ where $a,b,c$ are integers, $T_n(t)$ is the Chebyshev polynomial of degree $n$ and $ϕ\in \R.$ 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.