Lissajous-toric knot

Lissajous-toric knot with parameters 5, 6 and 22 in braid form (with z-axis in horizontal direction)

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

${\displaystyle x(t)=(2+\sin qt)\cos Nt,\qquad y(t)=(2+\sin qt)\sin Nt,\qquad z(t)=\cos p(t+\phi )}$,

where ${\displaystyle N}$, ${\displaystyle p}$, and ${\displaystyle q}$ are integers, the phase shift ${\displaystyle \phi }$ is a real number and the parameter ${\displaystyle t}$ varies between 0 and ${\displaystyle 2\pi }$. ^[1]

For ${\displaystyle p=q}$ the knot is a torus knot.

Braid and billiard knot definitions

Lissajous-toric knot T(4,7,35) as a billiard knot, showing period 7

In braid form these knots can be defined in a square solid torus (i.e. the cube ${\displaystyle [-1,1]^{3}}$ with identified top and bottom) as

${\displaystyle x(t)=\sin 2\pi qt,\qquad y(t)=\cos 2\pi p(t+\phi ),\qquad z(t)=2(Nt-\lfloor Nt\rfloor )-1,\qquad t\in [0,1]}$.

The projection of this Lissajous-toric knot onto the x-y-plane is a Lissajous curve.

Replacing the sine and cosine functions in the parametrization by a triangle wave transforms a Lissajous-toric knot isotopically into a billiard curve inside the solid torus. Because of this property Lissajous-toric knots are also called billiard knots in a solid torus. ^[2]

Lissajous-toric knots were first studied as billiard knots and they share many properties with billiard knots in a cylinder. ^[3] They also occur in the analysis of singularities of minimal surfaces with branch points ^[4] and in the study of the Three-body problem. ^[5]

The knots in the subfamily with ${\displaystyle p=q\cdot l}$, with an integer ${\displaystyle l\geq 1}$, are known as ′Lemniscate knots′. ^[6] Lemniscate knots have period ${\displaystyle q}$ and are fibred. The knot shown on the right is of this type (with ${\displaystyle l=5}$).

Properties

Symmetries of the Lissajous-toric knot T(3,8,7): symmetric union (vertical axis), rotation into mirror image and palindromic property within Q (horizontal axis)

Lissajous-toric knots are denoted by ${\displaystyle K(N,q,p,\phi )}$. To ensure that the knot is traversed only once in the parametrization the conditions ${\displaystyle \gcd(N,q)=\gcd(N,p)=1}$ are needed. In addition, singular values for the phase, leading to self-intersections, have to be excluded.

The isotopy class of Lissajous-toric knots surprisingly does not depend on the phase ${\displaystyle \phi }$ (up to mirroring). If the distinction between a knot and its mirror image is not important, the notation ${\displaystyle K(N,q,p)}$ can be used.

The properties of Lissajous-toric knots depend on whether ${\displaystyle p}$ and ${\displaystyle q}$ are coprime or ${\displaystyle d=\gcd(p,q)>1}$. The main properties are:

• Interchanging ${\displaystyle p}$ and ${\displaystyle q}$:

${\displaystyle K(N,q,p)=K(N,p,q)}$ (up to mirroring).

• Ribbon property:

If ${\displaystyle p}$ and ${\displaystyle q}$ are coprime, ${\displaystyle K(N,q,p)}$ is a symmetric union and therefore a ribbon knot.

• Periodicity:

If ${\displaystyle d=\gcd(p,q)>1}$, the Lissajous-toric knot has period ${\displaystyle d}$ and the factor knot is a ribbon knot.

• Strongly-plus-amphicheirality:

If ${\displaystyle p}$ and ${\displaystyle q}$ have different parity, then ${\displaystyle K(N,q,p)}$ is strongly-plus-amphicheiral.

• Period 2:

If ${\displaystyle p}$ and ${\displaystyle q}$ are both odd, then ${\displaystyle K(N,q,p)}$ has period 2 (for even ${\displaystyle N}$) or is freely 2-periodic (for odd ${\displaystyle N}$).

Example

The knot T(3,8,7), shown in the graphics, is a symmetric union and a ribbon knot (in fact, it is the composite knot ${\displaystyle 5_{1}\sharp -5_{1}}$). It is strongly-plus-amphicheiral: a rotation by ${\displaystyle \pi }$ maps the knot to its mirror image, keeping its orientation. An additional horizontal symmetry occurs as a combination of the vertical symmetry and the rotation (′double palindromicity′ in Kin/Nakamura/Ogawa).

′Classification′ of billiard rooms

In the following table a systematic overview of the possibilities to build billiard rooms from the interval and the circle (interval with identified boundaries) is given:

Billiard room	Billiard knots
${\displaystyle [-1,1]^{3}}$	Lissajous knots
${\displaystyle [-1,1]^{2}\times \mathbb {S} ^{1}}$	Lissajous-toric knots
${\displaystyle [-1,1]\times \mathbb {S} ^{1}\times \mathbb {S} ^{1}}$	Torus knots
${\displaystyle \mathbb {S} ^{1}\times \mathbb {S} ^{1}\times \mathbb {S} ^{1}}$	(room not embeddable into ${\displaystyle \mathbb {R} ^{3}}$)

In the case of Lissajous knots reflections at the boundaries occur in all of the three cube's dimensions. In the second case reflections occur in two dimensions and we have a uniform movement in the third dimension. The third case is nearly equal to the usual movement on a torus, with an additional triangle wave movement in the first dimension.

References

1. See M. Soret and M. Ville: Lissajous-toric knots, J. Knot Theory Ramifications 29, 2050003 (2020).
2. See C. Lamm: Deformation of cylinder knots, 4th chapter of Ph.D. thesis, ‘Zylinder-Knoten und symmetrische Vereinigungen‘, Bonner Mathematische Schriften 321 (1999), available since 2012 as arXiv : 1210.6639.
3. See C. Lamm and D. Obermeyer: Billiard knots in a cylinder, J. Knot Theory Ramifications 8, 353–-366 (1999).
4. See Soret/Ville.
5. See E. Kin, H. Nakamura and H. Ogawa: Lissajous 3-braids, J. Math. Soc. Japan 75, 195--228 (2023) (or arXiv : 2008.00585v4).
6. See B. Bode, M.R. Dennis, D. Foster and R.P. King: Knotted fields and explicit fibrations for lemniscate knots, Proc. Royal Soc. A (2017).