Lissajous and Fourier Knots

TL;DR

This paper demonstrates that any knot in three-dimensional space can be represented as a Fourier knot of type (1,1,2), showing a deformation from Lissajous knots, thus expanding the understanding of knot representations.

Contribution

It proves that all knots can be realized as Fourier knots of a specific type through deformation of Lissajous knots, establishing a new universal representation.

Findings

Any knot in  can be represented as a Fourier (1,1,2) knot.

Deformation from Lissajous to Fourier knots is always possible.

Provides a new method for knot representation and classification.

Abstract

We prove that any knot of $\mathbb{R}^3$ is isotopic to a Fourier knot of type $(1,1,2)$ obtained by deformation of a Lissajous knot.