Narrow Results

In this paper, we present an algorithm for optimally parametrizing polynomial algebraic curves. Let be a polynomial plane algebraic curve given by a polynomial parametrization , where is a finite field extension of a field of characteristic zero. We prove that if is polynomial over , then Weil's descente variety associated with is surprisingly simple; it is, in fact, a line. Applying this result we are able to derive an effective algorithm to algebraically optimal reparametrize polynomial algebraic curves.

We show that no torus knot of type (2, n), n > 3 odd, can be obtained from a polynomial embedding t ↦ (f(t), g(t), h(t)) where (deg(f), deg(g)) ≤ (3, n + 1). Eventually, we give explicit examples with minimal lexicographic degree.

A Chebyshev knot is a knot which admits a parametrization of the form x(t) = T_a(t); y(t) = T_b(t); z(t) = T_c(t + ϕ), where a, b, c integers, T_n(t) is the Chebyshev polynomial of degree n, and φ ∈ R. Chebyshev knots are non-compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with φ = 0. We also show that every knot is a Chebyshev knot.

The harmonic knot $\mathrm{\text{H}}(a,b,c)$ is parametrized as $K(t)=(T_a(t),T_b(t),T_c(t))$ where $a$, $b$ and $c$ are pairwise coprime integers and $T_n$ is the degree $n$ Chebyshev polynomial of the first kind. We classify the harmonic knots $\mathrm{\text{H}}(a,b,c)$ for $a\le 4.$ We study the knots $\mathrm{\text{H}}(2n-1,2n,2n+1),$ the knots $\mathrm{\text{H}}(5,n,n+1),$ and give a table of the simplest harmonic knots.