Lissajous knot
In
where , , and are
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] or in a (flat) solid torus (Lissajous-toric knot).
Form
Because a knot cannot be self-intersecting, the three integers must be pairwise
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 :
-
821 knot
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 plus
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 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
- ^ .
- S2CID 17489206.
- ISBN 978-0-521-54831-1.
- S2CID 123288245.
- ].
- arXiv:math.GT/0605632.
- Bibcode:2004math......5151P.
- arXiv:2310.05106 [math.GT].
- S2CID 120648664.
- S2CID 120483606.