The first rational Chebyshev knots

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 $\phi \in \R.$Weshowthatanytwo-bridgeknotisaChebyshevknotwith$a=3$ 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 $\text{cC}(a,b,c,\varphi )$. We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.