40 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
Solving the Triangular Ising Antiferromagnet by Simple Mean Field
Few years ago, application of the mean field Bethe scheme on a given system
was shown to produce a systematic change of the system intrinsic symmetry. For
instance, once applied on a ferromagnet, individual spins are no more
equivalent. Accordingly a new loopwise mean field theory was designed to both
go beyond the one site Weiss approach and yet preserve the initial Hamitonian
symmetry. This loopwise scheme is applied here to solve the Triangular
Antiferromagnetic Ising model. It is found to yield Wannier's exact result of
no ordering at non-zero temperature. No adjustable parameter is used.
Simultaneously a non-zero critical temperature is obtained for the Triangular
Ising Ferromagnet. This simple mean field scheme opens a new way to tackle
random systems.Comment: 14 pages, 2 figure
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