Computing Chebyshev knot diagrams

A Chebyshev curve C(a, b, c, φ) 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. When C(a, b, c, φ) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when φ varies. Let a, b, c be integers, a is odd, (a, b) = 1, we show that one can list all possible knots C(a, b, c, φ) in Õ(n) bit operations, with n = abc.