We propose a set of simple and fast algorithms for evaluating and using trigonometric expressions in the form, f k ∈ Q, d < n for a fixed n ∈ Z>0: computing the sign of such an expression, evaluating it numerically and computing its minimal polynomial in Q [x]. As critical byproducts, we propose simple and efficient algorithms for performing arithmetic operations (multiplication, division, gcd) on polynomials expressed in a Chebyshev basis (with the same bit-complexity than in the monomial basis) and for computing the minimal polynomial of 2 cos

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A Chebyshev curve C(a, b, c, ϕ) has a parametrization of the form x(t) = Ta(t); y(t) = T b (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 O(n 2 ) bit operations, with n = abc.