Chebyshev Knots

Abstract

A Chebyshev knot is a knot which admits a parametrization of the form $x(t)=T_a(t); \ y(t)=T_b(t) ; \ z(t)= T_c(t + \phi),$ where $a,b,c$ are pairwise coprime, $T_n(t)$ is the Chebyshev polynomial of degree $n,$ and \phi \in \RR . Chebyshev knots are non compact analogues of the classical Lissajous knots. We show that there are infinitely many Chebyshev knots with $\phi = 0.$ We also show that every knot is a Chebyshev knot.Comment: To appear in Journal of Knot Theory and Ramification