# Lissajous knot

(Redirected from Billiard 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]

## 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 ${\displaystyle \phi _{z}=0}$:

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 plus amphicheiral.[7] This is a fairly rare property: only three prime knots with twelve or fewer crossings are strongly plus amphicheiral prime knot, the first of which has crossing number ten.[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

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:

## References

1. ^ a b M.G.V. Bogle, J.E. Hearst, V.F.R. Jones, L. Stoilov, "Lissajous knots", Journal of Knot Theory and Its Ramifications, 3(2), 1994, 121–140.
2. ^ C. Lamm, D. Obermeyer. "Billiard knots in a cylinder", Journal of Knot Theory and Its Ramifications, 8(3), 1999, 353–366.
3. ^ Cromwell, Peter R. (2004). Knots and links. Cambridge, UK: Cambridge University Press. p. 13. ISBN 0-521-54831-4.
4. ^ C. Lamm. "There are infinitely many Lissajous knots." Manuscripta Math., 93:29–37, 1997, Springerlink
5. ^ A. Boocher; J. Daigle; J. Hoste; W. Zheng (2007). "Sampling Lissajous and Fourier knots". arXiv:.
6. ^ Hoste, Jim; Zirbel, Laura (2006). "Lissajous knots and knots with Lissajous projections". arXiv:.
7. ^ Przytycki, Jozef H. (2004). "Symmetric knots and billiard knots". arXiv:.
8. ^ Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks. "The first 1,701,936 knots." Math. Intelligencer, 20(4):33–48, 1998.
9. ^ R. Hartley and A Kawauchi. "Polynomials of amphicheiral knots." Math. Ann., 243:63–70, 1979.
10. ^ K. Murasugi. "On periodic knots." Comment. Math.Helv., 46:162–174, 1971.