On Fibonacci Knots

TL;DR

This paper explores the properties of Fibonacci knots, revealing their Conway polynomials relate to Fibonacci polynomials modulo 2 and identifying conditions under which these knots are not Lissajous knots.

Contribution

It establishes a connection between Fibonacci polynomials and Conway polynomials of Fibonacci links, and determines when Fibonacci knots are not Lissajous knots.

Findings

Conway polynomials of Fibonacci links are Fibonacci polynomials modulo 2.

Fibonacci knots are not Lissajous knots for certain parameters.

Conditions involving n and j determine the Lissajous knot status of Fibonacci knots.

Abstract

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.