46 research outputs found
On Fibonacci Knots
We show that the Conway polynomials of Fibonacci links are Fibonacci
polynomials modulo 2. We deduce that, when n \not\equiv 0 \Mod 4 and the Fibonacci knot \cF_j^{(n)} is not a Lissajous knot.Comment: 7p. Sumitte
Chebyshev Knots
A Chebyshev knot is a knot which admits a parametrization of the form where are
pairwise coprime, is the Chebyshev polynomial of degree 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
We also show that every knot is a Chebyshev knot.Comment: To appear in Journal of Knot Theory and Ramification
Poncelet's theorem and Billiard knots
Let be any elliptic right cylinder. We prove that every type of knot can
be realized as the trajectory of a ball in This proves a conjecture of
Lamm and gives a new proof of a conjecture of Jones and Przytycki. We use
Jacobi's proof of Poncelet's theorem by means of elliptic functions
The first rational Chebyshev knots
A Chebyshev knot is a knot which has a parametrization
of the form where
are integers, is the Chebyshev polynomial of degree and We show that any two-bridge knot is a Chebyshev knot with and also
with . For every integers ( and , 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.Comment: 22p, 27 figures, 3 table
Computing Chebyshev knot diagrams
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.Comment: 8