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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.
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A Chebyshev knot ${cal C}(a,b,c,phi)$ is a knot which has 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 integers, $T_n(t)$ is the Chebyshev polynomial of degree $n$ and $phi in R.$ We show that any two-bridge knot is a Chebyshev knot with $a=3$ and also with $a=4$. For every $a,b,c$ integers ($a=3, 4$ and $a$, $b$ coprime), we describe an algorithm that gives all Chebyshev knots $cC(a,b,c,phi)$. We deduce a list of minimal Chebyshev representations of two-bridge knots with small crossing number.
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 $b+deg C = 3N$. 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.
A Chebyshev curve C(a,b,c,phi) has a parametrization of the form x(t)=Ta(t); y(t)=T_b(t) ; z(t)= Tc(t + phi), where a,b,c are integers, Tn(t) is the Chebyshev polynomial of degree n and phi in RR. When C(a,b,c,phi) has no double points, it defines a polynomial knot. We determine all possible knots when a, b and c are given.
The harmonic knot $H(a,b,c)$ 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 $H(a,b,c)$ for $ a le 4. $ We study the knots $H (2n-1, 2n, 2n+1),$ the knots $H(5,n,n+1),$ and give a table of the simplest harmonic knots.
We show that the Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2. We deduce that, when $ n otequiv 0 Mod 4$ and $(n,j) eq (3,3),$ the Fibonacci knot $ cF_j^{(n)} $ is not a Lissajous knot.
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