Lissajous knot
In knot theory, a Lissajous knot is a knot defined by parametric equations of the form where formula_2, formula_3, and formula_4 are integers and the phase shifts formula_5, formula_6, and formula_7 may be any real numbers. 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 or in a (flat) solid torus (Lissajous-toric knot). Form. Because a knot cannot be self-intersecting, the three integers formula_8 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 formula_10, one may assume that any of the three phase shifts formula_5, formula_6, formula_7 is equal to zero. Examples. Here are some examples of Lissajous knots, all of which have formula_14: There are infinitely many different Lissajous  knots, and other examples with 10 or fewer crossings include the 7 knot, the 8 knot, the 10 knot, the 10 knot, the 10 knot, and the composite knot 5 # 5, as well as the 9 knot, 10 knot, the 10 knot, the 10 knot, the 10 knot, the granny knot, and the composite knot 5 # 5. In addition, it is known that every twist knot with Arf invariant zero is a Lissajous knot. Symmetry. Lissajous knots are highly symmetric, though the type of symmetry depends on whether or not the numbers formula_2, formula_3, and formula_4 are all odd. Odd  case. If formula_2, formula_3, and formula_4 are all odd, then the point reflection across the origin formula_21 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. This is a fairly rare property: only seven or eight prime knots with twelve or fewer crossings are strongly plus amphicheiral (10, 10, 12a427, 12a1019, 12a1105, 12a1202, 12n706, and an undecided case, 12a435). Since this is so rare, ′most′ prime Lissajous knots lie in the even case. Even case. If one of the  frequencies (say formula_2) is even, then the 180° rotation around the "x"-axis formula_23 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. In the even case, the Alexander polynomial must be a perfect square modulo 2. In addition, the Arf invariant of a Lissajous knot must be  zero. It follows that: