Fourier Knots

TL;DR

This paper demonstrates that all knots can be represented as Fourier knots with specific parametric forms, providing a new perspective on knot representations and their mathematical properties.

Contribution

It introduces Fourier knots of type (n_1, n_2, n_3) and proves that every knot can be expressed as a Fourier knot of type (1, 1, n), expanding the understanding of knot parametrizations.

Findings

Every knot has a checkerboard diagram.

Every knot is the closure of a rosette braid.

Every knot can be represented as a Fourier knot of type (1, 1, n).

Abstract

We show that every knot has a checkerbord diagram and that every knot is the closure of a rosette braid. We define Fourier knots of type (n_1, n_2, n_3) as knots which have parametrizations where each coordinate function x_i(t) is a finite Fourier series of length n_i, and conclude that every knot is a Fourier knot of type (1, 1, n) for some natural number n.