The first rational Chebyshev knots

A Chebyshev knot C(a,b,c,φ) is a knot which has a parametrization of the form x(t)=Ta(t);y(t)=Tb(t);z(t)=Tc(t+φ), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and φ∈R. We show that any rational 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 C(a,b,c,φ). We deduce the list of minimal Chebyshev representations of rational knots with 10 or fewer crossings.