Sampling Lissajous and Fourier Knots

Abstract

A Lissajous knot is one that can be parameterized as

$K(t)=\left (\cos (n_x t+\phi_x), \cos (n_y t+\phi_y), \cos (n_z t+\phi_z)\right )$,

where the frequencies $n_x$, $n_y$, and $n_z$ are relatively prime integers and the phase shifts $\phi_x$, $\phi_y$, and $\phi_z$ are real numbers. Lissajous knots are highly symmetric, and for this reason, not all knots are Lissajous. We prove several theorems that allow us to place bounds on the number of Lissajous knot types with given frequencies and to efficiently sample all possible Lissajous knots with a given set of frequencies. In particular, we systematically tabulate all Lissajous knots with small frequencies and as a result substantially enlarge the tables of known Lissajous knots.

A Fourier-$(i, j, k)$ knot is similar to a Lissajous knot except that the $x$, $y$, and $z$ coordinates are now each described by a sum of $i$, $j$, and $k$ cosine functions, respectively. According to Lamm, every knot is a Fourier-$(1,1,k)$ knot for some $k$. By randomly searching the set of Fourier-$(1,1,2)$ knots we find that all 2-bridge knots with up to 14 crossings are either Lissajous or Fourier-$(1,1,2)$ knots. We show that all twist knots are Fourier-$(1,1,2)$ knots and give evidence suggesting that all torus knots are Fourier-$(1,1,2)$ knots.

As a result of our computer search, several knots with relatively small crossing numbers are identified as potential counterexamples to interesting conjectures.