Chebyshev diagrams for two-bridge knots

Abstract

We show that every two-bridge knot $K$ of crossing number $N$ admits a polynomial parametrization $x=T_3(t),y=T_b(t),z=C(t)$ where $T_k(t)$ are the Chebyshev polynomials and $b+°C=3N$. 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 $a\le 3.$ Most results are derived from continued fractions and their matrix representations.

25p. 17 figures, submitted. This is a part of "Chebyshev diagrams for rational knots", arXiv:0906.4083

References in corpus (4)

• Curves defined by Chebyshev polynomials
• Sampling Lissajous and Fourier knots
• Chebyshev diagrams for rational knots
• A polynomial parametrization of torus knots