Lissajous knot

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

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

A Lissajous 8₂₁ knot

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
  $(n_{x},n_{y},n_{z})=(3,5,7)$
  $(\phi _{x},\phi _{y})=(0.7,1.0)$
• 
  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). 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

A Lissajous knot with three factors: $(n_{x},n_{y},n_{z})=(4,5,41)$,
$(\phi _{x},\phi _{y})=(0.01,0.16)$

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.