×

On Fibonacci knots. (English) Zbl 1213.57012

A Fibonacci knot or link is a 4-plat with symbol \(C(1,1,\dots, 1)\) in Conway’s notation. This can be generalized to 4-plats with symbol \(C(n,n,\ldots, n)\) for generalized Fibonacci knots. If \(n\) is even then we obtain Fibonacci links. The authors show that the Conway polynomial of a Fibonacci link is a Fibonacci polynomial modulo two. Let \(j\) be the length of the Conway symbol \(C(n,n,\dots, n)\). We denote the Fibonacci knot given by such a Conway symbol as \(\mathcal{F}_j^{(n)}\). The authors conclude that when \((n,j)\neq (3,3)\) and \(n\ncong 0 \pmod 4\), the Fibonacci knot \(\mathcal{F}_j^{(n)}\) is not a Lissajous knot. (Here a Lissajous knot is a knot that admits a parametrization as \(x(t)=\cos(n_x t+\phi_x)\), \(y(t)=\cos(n_y t+\phi_y)\) and \(z(t)=\cos(n_z t+\phi_z)\) and \(0\leq t\leq 2 pi\).)

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
11A55 Continued fractions
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
PDFBibTeX XMLCite
Full Text: arXiv Link