Lissajous knots

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 $\phi_z=0$:

Image:Lissajous 5_2 Knot.png, Three-twist knot
$(n_x,n_y,n_z)=(3,2,7)$
$(\phi_x,\phi_y)=(0.7,0.2)$
Image:Lissajous Stevedore Knot.png, Stevedore knot
$(n_x,n_y,n_z)=(3,2,5)$
$(\phi_x,\phi_y)=(1.5,0.2)$
Image:Lissajous Square Knot.png, Square knot
$(n_x,n_y,n_z)=(3,5,7)$
$(\phi_x,\phi_y)=(0.7,1.0)$
Image:Lissajous 8_21 Knot.png, 8₂₁ 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 7₄ knot, the 8₁₅ knot, the 10₁ knot, the 10₃₅ knot, the 10₅₈ knot, and the composite knot 5₂^* # 5₂, as well as the 9₁₆ knot, 10₇₆ knot, the 10₉₉ knot, the 10₁₂₂ knot, the 10₁₄₄ knot, the granny knot, and the composite knot 5₂ # 5₂. 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 (10₉₉, 10₁₂₃, 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.

References

{{DEFAULTSORT:Lissajous Knot
Knots (knot theory)