Lissajous knot
Encyclopedia
In knot theory
, a Lissajous knot is a knot
defined by parametric equations of the form
where
, 
, and 
are integer
s and the phase shifts
, 
, and 
may be any real number
s.
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.
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.
:
There are infinitely many different Lissajous knots, 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, 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. In addition, it is known that every twist knot
with Arf invariant
zero is a Lissajous knot.
, 
, and 
are all odd.
, 
, 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. This is a fairly rare property: only three prime knot
s with twelve or fewer crossings are strongly plus amphicheiral prime knot
, the first of which has crossing number
ten. Since this is so rare, most Lissajous knots lie in the even case.
) 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.
. 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:
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, a Lissajous knot is a knot
Knot (mathematics)
In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations . A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a...
defined by parametric equations of the form
where
, , and are integerInteger
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s and the phase shifts
, , and may be any real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s.
The projection of a Lissajous knot onto any of the three coordinate planes is a Lissajous curve
Lissajous curve
In mathematics, a Lissajous curve , also known as Lissajous figure or Bowditch curve, is the graph of a system of parametric equationswhich describe complex harmonic motion...
, 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
Triangle wave
A triangle wave is a non-sinusoidal waveform named for its triangular shape.Like a square wave, the triangle wave contains only odd harmonics...
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.
Form
Because a knot cannot be self-intersecting, the three integers must be pairwise relatively prime, and none of the quantitiesmay be an integer multiple of pi
Pi
' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
. 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, all of which have:There are infinitely many different Lissajous knots, and other examples with 10 or fewer crossings
Crossing number (knot theory)
In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....
include the 74 knot, the 815 knot, the 101 knot, the 1035 knot, the 1058 knot, and the composite knot 52* # 52, as well as the 916 knot, 1076 knot, the 1099 knot, the 10122 knot, the 10144 knot, the granny knot
Granny knot (mathematics)
In knot theory, the granny knot is a composite knot obtained by taking the connected sum of two identical trefoil knots. It is closely related to the square knot, which can also be described as a connected sum of two trefoils...
, and the composite knot 52 # 52. In addition, it is known that every twist knot
Twist knot
In knot theory, a branch of mathematics, a twist knot is a knot obtained by repeatedly twisting a closed loop and then linking the ends together...
with Arf invariant
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by when he started the systematic study of quadratic forms over arbitrary fields of characteristic2. The Arf invariant is the substitute, in...
zero is a Lissajous knot.
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 reflectionPoint reflection
In geometry, a point reflection or inversion in a point is a type of isometry of Euclidean space...
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. This is a fairly rare property: only three prime knot
Prime knot
In knot theory, a prime knot is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite. It can be a nontrivial problem to determine whether a...
s with twelve or fewer crossings are strongly plus amphicheiral prime knot
Prime knot
In knot theory, a prime knot is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite. It can be a nontrivial problem to determine whether a...
, the first of which has crossing number
Crossing number (knot theory)
In the mathematical area of knot theory, the crossing number of a knot is the minimal number of crossings of any diagram of the knot. It is a knot invariant....
ten. Since this is so rare, most 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 polynomialAlexander polynomial
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923...
. 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
Modular arithmetic
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
2. In addition, the Arf invariant
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by when he started the systematic study of quadratic forms over arbitrary fields of characteristic2. The Arf invariant is the substitute, in...
of a Lissajous knot must be zero. It follows that:
- The trefoil knotTrefoil knotIn topology, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining together the two loose ends of a common overhand knot, resulting in a knotted loop...
and figure-eight knotFigure-eight knot (mathematics)In knot theory, a figure-eight knot is the unique knot with a crossing number of four. This is the smallest possible crossing number except for the unknot and trefoil knot...
are not Lissajous. - No torus knotTorus knotIn knot theory, a torus knot is a special kind of knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime integers p and q. A torus link arises if p and q...
can be Lissajous. - No fiberedFibered knotA knot or link Kin the 3-dimensional sphere S^3 is called fibered or fibred if there is a 1-parameter family F_t of Seifert surfaces for K, where the parameter t runs through the points of the unit circle S^1, such that if s is not equal to t...
2-bridge knot2-bridge knotIn the mathematical field of knot theory, a 2-bridge knot is a knot which can be isotoped so that the natural height function given by the z-coordinate has only two maxima and two minima as critical points....
can be Lissajous.