Chebyshev diagrams for rational knots

Abstract

We show that every rational knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_a(t),y=T_b(t),z=C(t)$ where $T_k(t)$ are the Chebyshev polynomials, $a=3$ and $b+°C=3N.$ We show that every rational knot also admits a polynomial parametrization with $a=4$. If $C(t)=T_c(t)$ is a Chebyshev polynomial, we call such a knot a harmonic knot. We give the classification of harmonic knots for $a\le 4.$

39p. Submitted

References in corpus (2)

• Curves defined by Chebyshev polynomials
• Sampling Lissajous and Fourier knots

Cited by in corpus (1)

• Chebyshev diagrams for two-bridge knots