# 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.

## Key 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.