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 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 C(a, b, c, φ). We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number. keywords: Polynomial curves; two-bridge knots; Chebyshev curves; real roots isolation; computer algebra; algorithms Mathematics Subject Classification 2000: 14H50, 57M25, 14P99