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.^[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 $n_x, n_y, n_z$ 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,^[3] 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,^[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 $n_x$ , $n_y$ , 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.^[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 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

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

↑ 1.0 1.1 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.
↑ C. Lamm, D. Obermeyer. "Billiard knots in a cylinder", Journal of Knot Theory and Its Ramifications, 8(3), 1999, 353–366.
↑ Cromwell, Peter R. (2004). Knots and links. Cambridge, UK: Cambridge University Press. p. 13. ISBN 0-521-54831-4.
↑ C. Lamm. "There are infinitely many Lissajous knots." Manuscripta Math., 93:29–37, 1997, Springerlink
↑ A. Boocher, J. Daigle, J. Hoste, W. Zheng (2007). "Sampling Lissajous and Fourier knots". arXiv:0707.4210.
↑ Hoste, Jim; Zirbel, Laura (2006). "Lissajous knots and knots with Lissajous projections". arXiv:math.GT/0605632.
↑ Przytycki, Jozef H. (2004). "Symmetric knots and billiard knots". arXiv:math/0405151.
↑ Jim Hoste, Morwen Thistlethwaite, and Jeff Weeks. "The first 1,701,936 knots." Math. Intelligencer, 20(4):33–48, 1998.
↑ R. Hartley and A Kawauchi. "Polynomials of amphicheiral knots." Math. Ann., 243:63–70, 1979.
↑ K. Murasugi. "On periodic knots." Comment. Math.Helv., 46:162–174, 1971.