Lissajous knot explained

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

where

, , and are integers and the phase shifts , , and 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 Lissajousknot 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

must be pairwise relatively prime, and none of the quantities

may be an integer multiple of pi. Moreover, by making a substitution of the form

, one may assume that any of the three phase shifts , , is equal to zero.

Examples

Here are some examples of Lissajous knots,^[3] all of which have

: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

, , and are all odd.

Odd case

If

, , and are all odd, then the point reflection across the origin 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

) is even, then the 180° rotation around the x-axis 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 Alexanderpolynomial 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.

Notes and References

1. 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. 10.1142/S0218216594000095 .
2. Christoph . Lamm . Daniel . Obermeyer . Billiard knots in a cylinder . Journal of Knot Theory and Its Ramifications . 8 . 3 . 1999 . 353–366. 1998math.....11006L . math/9811006 . 10.1142/S0218216599000225 . 17489206 .
3. Book: Cromwell, Peter R. . Knots and links . Cambridge University Press . Cambridge, UK . 2004 . 13 . 978-0-521-54831-1.
4. Lamm . C. . 1997 . There are infinitely many Lissajous knots . Manuscripta Mathematica . 93 . 29–37 . 10.1007/BF02677455. 123288245 .
5. 0707.4210 . Sampling Lissajous and Fourier knots . 2007 . math.GT . Boocher . Adam . Daigle . Jay . Hoste . Jim . Zheng . Wenjing .
6. Hoste . Jim . Zirbel . Laura . math.GT/0605632. Lissajous knots and knots with Lissajous projections . 2006.
7. Book: Przytycki, Jozef H. . math/0405151 . Symmetric knots and billiard knots . Ideal Knots . Series on Knots and Everything . A. . Stasiak . V. . Katrich . L. . Kauffman . World Scientific . 19 . 374–414 . 2004 . 2004math......5151P .
8. See 2310.05106. Strongly positive amphicheiral knots with doubly symmetric diagrams . 2023 . math.GT . Lamm . Christoph. 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 . 10.1007/bf01420207. 120648664 .
10. Murasugi . K. . 1971 . On periodic knots . . 46 . 162–174 . 10.1007/bf02566836. 120483606 .