Harmonic Knots

Abstract

The harmonic knot $\stackrel{˝}{(}a,b,c)$ 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 $\stackrel{˝}{(}a,b,c)$ for $a\le 4.$ We study the knots $\stackrel{˝}{(}2n-1,2n,2n+1),$ the knots $\stackrel{˝}{(}5,n,n+1),$ and give a table of the simplest harmonic knots.

18 p., 30 fig