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
relatively prime, and none of the quantities
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 :
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
amphicheiral.[7] This is a fairly rare property: only seven or eight
prime knots with twelve or fewer crossings are strongly plus amphicheiral (1099, 10123, 12a427, 12a1019, 12a1105, 12a1202, 12n706).[8] Since this is so rare, ′most′ prime Lissajous knots lie in the even case.
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:
^
abBogle, M. G. V.; Hearst, J. E.; Jones, V. F. R.; Stoilov, L. (1994). "Lissajous knots". Journal of Knot Theory and Its Ramifications. 3 (2): 121–140.
doi:
10.1142/S0218216594000095.
^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". In Stasiak, A.; Katrich, V.; Kauffman, L. (eds.). Ideal Knots. Series on Knots and Everything. Vol. 19. World Scientific. pp. 374–414.
arXiv:math/0405151.
Bibcode:
2004math......5151P.
^Lamm, Christoph (2019). "The Search for Nonsymmetric Ribbon Knots". Experimental Mathematics. 30 (3): 349–363.
arXiv:1710.06909.
doi:
10.1080/10586458.2018.1540313. A complete list of prime strongly positive amphicheiral knots is available in Lamm, Christoph (2023). "Strongly positive amphicheiral knots with doubly symmetric diagrams".
arXiv:2310.05106 [
math.GT].
^Hartley, R.; Kawauchi, A (1979). "Polynomials of amphicheiral knots". Mathematische Annalen. 243: 63–70.
doi:
10.1007/bf01420207.
S2CID120648664.