Lissajous knot
# Lissajous knot

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

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

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

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 or in a (flat) solid torus (Lissajous-toric knot).

## Form

Because a knot cannot be self-intersecting, the three integers $n_{x},n_{y},n_{z}$ must be pairwise relatively prime, and none of the quantities

    $n_{x}\phi _{y}-n_{y}\phi _{x},\quad n_{y}\phi _{z}-n_{z}\phi _{y},\quad n_{z}\phi _{x}-n_{x}\phi _{z}$

may be an integer multiple of pi. Moreover, by making a substitution of the form $t'=t+c$, one may assume that any of the three phase shifts $\phi _{x}$, $\phi _{y}$, $\phi _{z}$ is equal to zero.

## Examples

Here are some examples of Lissajous knots, all of which have $\phi _{z}=0$:

* Three-twist knot $(n_{x},n_{y},n_{z})=(3,2,7)$ $(\phi _{x},\phi _{y})=(0.7,0.2)$
* Stevedore knot $(n_{x},n_{y},n_{z})=(3,2,5)$ $(\phi _{x},\phi _{y})=(1.5,0.2)$
* Square knot $(n_{x},n_{y},n_{z})=(3,5,7)$ $(\phi _{x},\phi _{y})=(0.7,1.0)$
* $8_{21}$ knot $(n_{x},n_{y},n_{z})=(3,4,7)$ $(\phi _{x},\phi _{y})=(0.1,0.7)$

There are infinitely many different Lissajous knots, and other examples with 10 or fewer crossings include the $7_4$ knot, the $8_{15}$ knot, the $10_1$ knot, the $10_{35}$ knot, the $10_{58}$ knot, and the composite knot $5_2^*$ # $5_2,$ as well as the $9_{16}$ knot, $10_{76}$ knot, the $10_{99}$ knot, the $10_{122}$ knot, the $10_{144}$ knot, the granny knot, and the composite knot $5_2$ # $5_2.$ In addition, it is known that every twist knot with Arf invariant zero is a Lissajous knot.

## Symmetry

Lissajous knots are highly symmetric, though the type of symmetry depends on whether or not the numbers $n_{x}$, $n_{y}$, and $n_{z}$ are all odd.

### Odd case

If $n_{x}$, $n_{y}$, and $n_{z}$ are all odd, then the point reflection across the origin $(x,y,z)\mapsto (-x,-y,-z)$ is a symmetry of the Lissajous knot which preserves the knot orientation.

In general, a knot that has an orientation-preserving point reflection symmetry is known as strongly plus amphicheiral. This is a fairly rare property: only seven or eight prime knots with twelve or fewer crossings are strongly plus amphicheiral $(10_{99},$ $10_{123},$ 12a427, 12a1019, 12a1105, 12a1202, 12n706, and an undecided case, 12a435). Since this is so rare, ′most′ prime Lissajous knots lie in the even case.

### Even case

If one of the frequencies (say $n_{x}$) is even, then the 180° rotation around the x-axis $(x,y,z)\mapsto (x,-y,-z)$ is a symmetry of the Lissajous knot. In general, a knot that has a symmetry of this type is called 2-periodic, so every even Lissajous knot must be 2-periodic.

### Consequences

The symmetry of a Lissajous knot puts severe constraints on the Alexander polynomial. In the odd case, the Alexander polynomial of the Lissajous knot must be a perfect square. In the even case, the Alexander polynomial must be a perfect square modulo 2. In addition, the Arf invariant of a Lissajous knot must be zero. It follows that:

* The trefoil knot and figure-eight knot are not Lissajous.
* No torus knot can be Lissajous.
* No fibered 2-bridge knot can be Lissajous.
