Lissajous knot

In knot theory, a Lissajous knot is a knot defined by parametric equations of the form

${\displaystyle x=\cos(n_{x}t+\phi _{x}),\qquad y=\cos(n_{y}t+\phi _{y}),\qquad z=\cos(n_{z}t+\phi _{z}),}$

where ${\displaystyle n_{x}}$, ${\displaystyle n_{y}}$, and ${\displaystyle n_{z}}$ are integers and the phase shifts ${\displaystyle \phi _{x}}$, ${\displaystyle \phi _{y}}$, and ${\displaystyle \phi _{z}}$ may be any real numbers.^[1]

The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve, and many of the properties of these knots are closely related to properties of Lissajous curves.

Replacing the cosine function in the parametrization by a triangle wave transforms every Lissajous knot isotopically into a billiard curve inside a cube, the simplest case of so-called billiard knots. Billiard knots can also be studied in other domains, for instance in a cylinder^[2] or in a (flat) solid torus (Lissajous-toric knot).