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 $(n,j) \neq (3,3),$ 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 $x(t)=T_a(t); \ y(t)=T_b(t) ; \ z(t)= T_c(t + \phi),$ 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.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 ${\cal C}(a,b,c,\phi)$ 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.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