Fourier Knots

Abstract

This paper introduces the concept of a Fourier knot. A Fourier knot is a knot that is represented by a parametrized curve in three dimensional space such that the coordinate functions are finite Fourier series in the parameter. The previously studied Lissajous knots constitute a case of Fourier knots with single frequencies in each coordinate direction. Not all knots are Lissajous. In fact the trefoil knot and the figure eight knot are the first examples of non- Lissajous knots. This paper gives robust and simple Fourier representations for these and other examples. This new version of the paper contains a crucial reference to the beautiful work of Aaron Trautwein on Harmonic Knots. See <http://www.carthage.edu/~trautwn>. See also <http://math.uic.edu/~kauffman/>.