Lissajous knot

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

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

A Lissajous 8₂₁ knot

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).

Form

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.

Examples

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]

Symmetry

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.

Odd case

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.

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.

Consequences

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

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. ^[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.

References

1. Bogle, M. G. V.; Hearst, J. E.; Jones, V. F. R.; Stoilov, L. (1994). "Lissajous knots". Journal of Knot Theory and Its Ramifications. 3 (2): 121–140. doi:10.1142/S0218216594000095.
2. Lamm, Christoph; Obermeyer, Daniel (1999). "Billiard knots in a cylinder". Journal of Knot Theory and Its Ramifications. 8 (3): 353–366. arXiv: math/9811006 . Bibcode:1998math.....11006L. doi:10.1142/S0218216599000225. S2CID   17489206.
3. Cromwell, Peter R. (2004). Knots and links. Cambridge, UK: Cambridge University Press. p. 13. ISBN   978-0-521-54831-1.
4. Lamm, C. (1997). "There are infinitely many Lissajous knots". Manuscripta Mathematica. 93: 29–37. doi:10.1007/BF02677455. S2CID   123288245.
5. Boocher, Adam; Daigle, Jay; Hoste, Jim; Zheng, Wenjing (2007). "Sampling Lissajous and Fourier knots". arXiv: 0707.4210 [math.GT].
6. Hoste, Jim; Zirbel, Laura (2006). "Lissajous knots and knots with Lissajous projections". arXiv: math.GT/0605632 .
7. Przytycki, Jozef H. (2004). "Symmetric knots and billiard knots". In Stasiak, A.; Katrich, V.; Kauffman, L. (eds.). Ideal Knots. Series on Knots and Everything. Vol. 19. World Scientific. pp. 374–414. arXiv: math/0405151 . Bibcode:2004math......5151P.
8. See Lamm, Christoph (2023). "Strongly positive amphicheiral knots with doubly symmetric diagrams". arXiv: 2310.05106 [math.GT]. This article contains a complete list of prime strongly positive amphicheiral knots up to 16 crossings.
9. Hartley, R.; Kawauchi, A (1979). "Polynomials of amphicheiral knots". Mathematische Annalen. 243: 63–70. doi:10.1007/bf01420207. S2CID   120648664.
10. Murasugi, K. (1971). "On periodic knots". Commentarii Mathematici Helvetici . 46: 162–174. doi:10.1007/bf02566836. S2CID   120483606.