In
knot theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are ...
, a Lissajous knot is a
knot
A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ' ...
defined by
parametric equations
Parametric may refer to:
Mathematics
* Parametric equation, a representation of a curve through equations, as functions of a variable
*Parametric statistics, a branch of statistics that assumes data has come from a type of probability distribu ...
of the form
:
where
,
, and
are
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
s and the
phase shift
In physics and mathematics, the phase of a periodic function F of some real variable t (such as time) is an angle-like quantity representing the fraction of the cycle covered up to t. It is denoted \phi(t) and expressed in such a scale that it v ...
s
,
, and
may be any
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s.
The projection of a Lissajous knot onto any of the three coordinate planes is a
Lissajous curve
A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations
: x=A\sin(at+\delta),\quad y=B\sin(bt),
which describe the superposition of two perpendicular oscillations in x and y dire ...
, 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
A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.
Like a square wave, the triangle wave contains only odd harmonics. However, the ...
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
must be pairwise
relatively prime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
, 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, all of which have
:
Image:Lissajous 5_2 Knot.png, Three-twist knot
Image:Lissajous Stevedore Knot.png, Stevedore knot
The stevedore knot is a stopper knot, often tied near the end of a rope. It is more bulky and less prone to jamming than the closely related figure-eight knot.
Naming
There is a lack of consensus among knot experts regarding the origin of t ...
Image:Lissajous Square Knot.png, Square knot
Image:Lissajous 8_21 Knot.png, 821 knot
There are infinitely many different Lissajous knots, and other examples with 10 or fewer
crossings
Crossings may refer to:
* ''Crossings'' (Buffy novel), a 2002 original novel based on the U.S. television series ''Buffy the Vampire Slayer''
* Crossings (game), a two-player abstract strategy board game invented by Robert Abbott
* ''Crossings'' ...
include the 7
4 knot, the 8
15 knot, the 10
1 knot, the 10
35 knot, the 10
58 knot, and the composite knot 5
2* # 5
2,
as well as the 9
16 knot, 10
76 knot, the 10
99 knot, the 10
122 knot, the 10
144 knot, the
granny knot
The granny knot is a binding knot, used to secure a rope or line around an object. It is considered inferior to the reef knot (square knot), which it superficially resembles. Neither of these knots should be used as a bend knot for attaching tw ...
, and the composite knot 5
2 # 5
2. In addition, it is known that every
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. (That is, a twist knot is any Whitehead double of an unknot.) The twist knots are an infinite f ...
with
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf i ...
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 reflection
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
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 seven or eight
prime knot
In knot theory, a prime knot or prime link 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 co ...
s with twelve or fewer crossings are strongly plus amphicheiral (10
99, 10
123, 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
) 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 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 1969, John Conway showed a ve ...
. In the odd case, the Alexander
polynomial of the Lissajous knot must be a perfect
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...
. In the even case, the Alexander polynomial must be a perfect square
modulo 2.
In addition, the
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by Turkish mathematician when he started the systematic study of quadratic forms over arbitrary fields of characteristic 2. The Arf i ...
of a Lissajous knot must be zero. It follows that:
* The
trefoil knot
In knot theory, 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. As the simplest kno ...
and
figure-eight knot
The figure-eight knot or figure-of-eight knot is a type of stopper knot. It is very important in both sailing and rock climbing as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under st ...
are not Lissajous.
* No
torus knot
In 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 cop ...
can be Lissajous.
* No
fibered 2-bridge knot can be Lissajous.
References
{{DEFAULTSORT:Lissajous Knot
Knots (knot theory)