Lissajous-toric knot explained

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

x(t)=(2+\sinqt)\cosNt, y(t)=(2+\sinqt)\sinNt, z(t)=\cosp(t+\phi),

where

, and

are integers, the phase shift

\phi

is a real numberand the parameter

varies between 0 and

2\pi

.^[1]

For

p=q

the knot is a torus knot.

Braid and billiard knot definitions

In braid form these knots can be defined in a square solid torus (i.e. the cube

[-1,1]³

with identified top and bottom) as

x(t)=\sin2\piqt, y(t)=\cos2\pip(t+\phi), z(t)=2(Nt-\lfloorNt\rfloor)-1, 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-toricknot 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

p=q ⋅ l

, with an integer

l\ge1

, are known as ′Lemniscate knots′.^[6] Lemniscate knots have period

and are fibred. The knot shown on the right is of this type (with

l=5

Properties

Lissajous-toric knots are denoted by

K(N,q,p,\phi)

. To ensure that the knot is traversed only once in the parametrization the conditions

\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

\phi

(up to mirroring). If the distinction between a knot and its mirror image is not important, the notation

K(N,q,p)

can be used.

The properties of Lissajous-toric knots depend on whether

and

are coprime or

d=\gcd(p,q)>1

. The main properties are:

Interchanging

and

K(N,q,p)=K(N,p,q)

(up to mirroring).

Ribbon property:

and

are coprime,

K(N,q,p)

is a symmetric union and therefore a ribbon knot.

Periodicity:

d=\gcd(p,q)>1

, the Lissajous-toric knot has period

and the factor knot is a ribbon knot.

Strongly positive amphicheirality:

and

have different parity, then

K(N,q,p)

is strongly positive amphicheiral.

Period 2:

and

are both odd, then

K(N,q,p)

has period 2 (for even

) or is freely 2-periodic (for odd

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

5₁\sharp-5₁

).It is strongly positive amphicheiral: a rotation by

\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
[-1,1]³	Lissajous knots
[-1,1]² x S¹	Lissajous-toric knots
[-1,1] x S¹ x S¹	Torus knots
S¹ x S¹ x S¹	(room not embeddable into R³ )

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.

Notes and References

See M. Soret and M. Ville: Lissajous-toric knots, J. Knot Theory Ramifications 29, 2050003 (2020).
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 .
See C. Lamm and D. Obermeyer: Billiard knots in a cylinder, J. Knot Theory Ramifications 8, 353–-366 (1999).
See Soret/Ville.
See E. Kin, H. Nakamura and H. Ogawa: Lissajous 3-braids, J. Math. Soc. Japan 75, 195--228 (2023) (or).
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).