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).

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

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

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

Here are some examples of Lissajous knots,^[3] all of which have ${\displaystyle \phi _{z}=0}$:

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

There are infinitely many different Lissajous knots,^[4] 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,^[1] 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.^[5] In addition, it is known that every twist knot with Arf invariant zero is a Lissajous knot.^[6]

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

If ${\displaystyle n_{x}}$, ${\displaystyle n_{y}}$, and ${\displaystyle n_{z}}$ are all odd, then the point reflection across the origin ${\displaystyle (x,y,z)\mapsto (-x,-y,-z)}$ 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 positive amphicheiral.^[7] This is a fairly rare property: only seven prime knots with twelve or fewer crossings are strongly positive amphicheiral (10₉₉, 10₁₂₃, 12a427, 12a1019, 12a1105, 12a1202, 12n706).^[8] Since this is so rare, ′most′ prime Lissajous knots lie in the even case.

If one of the frequencies (say ${\displaystyle n_{x}}$) is even, then the 180° rotation around the x-axis ${\displaystyle (x,y,z)\mapsto (x,-y,-z)}$ 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.

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.^[9] In the even case, the Alexander polynomial must be a perfect square modulo 2.^[10] 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.