Let $D$ be any elliptic right cylinder. We prove that every type of knot can be realized as the trajectory of a ball in $D.$ This proves a conjecture of Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use Jacobis proof of Poncelets theorem by means of elliptic functions.
We show that every knot can be realized as a billiard trajectory in a convex prism. This solves a conjecture of Jones and Przytycki.
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 n
on 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.
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.
For every odd integer $N$ we give an explicit construction of a polynomial curve $cC(t) = (x(t), y (t))$, where $deg x = 3$, $deg y = N + 1 + 2pent N4$ that has exactly $N$ crossing points $cC(t_i)= cC(s_i)$ whose parameters satisfy $s_1 < ... < s_{N
} < t_1 < ... < t_{N}$. Our proof makes use of the theory of Stieltjes series and Pade approximants. This allows us an explicit polynomial parametrization of the torus knot $K_{2,N}$.
