Chebyshev Knots

Pierre-Vincent Koseleff (1, 2) , Daniel Pecker (2)

1 SALSA - Solvers for Algebraic Systems and Applications
2 UPMC - Université Pierre et Marie Curie - Paris 6

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 $\phi \in \mathbb{R} .$ 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.

Mots clés: Polynomial curves; Chebyshev polynomials; Chebyshev curves; Lissajous knots; long knots; braids

Domaines: Topologie géométrique [math.GT]

Pierre-Vincent Koseleff, Daniel Pecker. Chebyshev Knots. Journal of Knot Theory and Its Ramifications, 2011, 20 (4), pp.575-593. ⟨10.1142/S0218216511009364⟩. ⟨hal-00344501v2⟩