Torus knots are Fourier-(1,1,2) knots

TL;DR

This paper proves that all torus knots can be represented as Fourier-(1,1,2) knots, providing the simplest known Fourier representation and confirming longstanding conjectures in knot theory.

Contribution

It demonstrates that every torus knot admits a Fourier-(1,1,2) representation, resolving a question posed by Kauffman and confirming a conjecture by Boocher et al.

Findings

All torus knots can be parametrized as Fourier-(1,1,2) knots.

The Fourier-(1,1,2) representation is the simplest for these knots.

The paper confirms a conjecture and answers a question in the field.

Abstract

Every torus knot can be represented as a Fourier-(1,1,2) knot which is the simplest possible Fourier representation for such a knot. This answers a question of Kauffman and confirms the conjecture made by Boocher, Daigle, Hoste and Zheng. In particular, the torus knot T(p,q) can be parameterized as x(t)=cos(pt), y(t)=cos(qt+pi/(2p)), and z(t)=cos(pt+pi/2)\cos((q-p)t+pi/(2p)-pi/(4q)).