Chebyshev Knots

TL;DR

Chebyshev knots are a class of knots parametrized by Chebyshev polynomials, with the paper demonstrating their abundance and that every knot can be represented as a Chebyshev knot, highlighting their universality.

Contribution

The paper introduces Chebyshev knots, proves their existence in infinite families, and shows that all knots can be represented as Chebyshev knots, expanding the understanding of knot parametrizations.

Findings

Infinitely many Chebyshev knots with zero phase shift.

Every knot can be represented as a Chebyshev knot.

Chebyshev knots generalize classical knot parametrizations.

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.