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

,

where , , and are integers, the phase shift is a real number and the parameter varies between 0 and .[1]

For 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 with identified top and bottom) as

.

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]

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 . To ensure that the knot is traversed only once in the parametrization the conditions 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 (up to mirroring). If the distinction between a knot and its mirror image is not important, the notation can be used.

The properties of Lissajous-toric knots depend on whether and are coprime or . The main properties are:

  • Interchanging and :
(up to mirroring).
  • Ribbon property:
If and are coprime, is a symmetric union and therefore a ribbon knot.
  • Periodicity:
If , the Lissajous-toric knot has period and the factor knot is a ribbon knot.
  • Strongly-plus-amphicheirality:
If and have different parity, then is strongly-plus-amphicheiral.
  • Period 2:
If and are both odd, then 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 ). It is strongly-plus-amphicheiral: a rotation by 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 roomBilliard knots
Lissajous knots
Lissajous-toric knots
Torus knots
(room not embeddable into )

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).
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