In
knot theory
In the mathematical field of topology, knot theory is the study of 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 joined so it cannot ...
, a torus knot is a special kind of
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 ...
that lies on the surface of an unknotted
torus
In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle.
If the axis of revolution does not tou ...
in R
3. 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
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 equival ...
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 o ...
s ''p'' and ''q''. A torus link arises if ''p'' and ''q'' are not coprime (in which case the number of components is
gcd(''p, q'')). A torus knot is
trivial
Trivia is information and data that are considered to be of little value. It can be contrasted with general knowledge and common sense.
Latin Etymology
The ancient Romans used the word ''triviae'' to describe where one road split or forked ...
(equivalent to the unknot)
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bicond ...
either ''p'' or ''q'' is equal to 1 or −1. The simplest nontrivial example is the (2,3)-torus knot, also known as 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 kn ...
.
Geometrical representation
A torus knot can be rendered geometrically in multiple ways which are
topologically equivalent In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated func ...
(see Properties below) but geometrically distinct. The convention used in this article and its figures is the following.
The (''p'',''q'')-torus knot winds ''q'' times around a circle in the interior of the torus, and ''p'' times around its axis of
rotational symmetry.. If ''p'' and ''q'' are not relatively prime, then we have a torus link with more than one component.
The direction in which the strands of the knot wrap around the torus is also subject to differing conventions. The most common is to have the strands form a right-handed screw for ''p q > 0''.
The (''p'',''q'')-torus knot can be given by the
parametrization
:
where
and
. This lies on the surface of the torus given by
(in
cylindrical coordinates
A cylindrical coordinate system is a three-dimensional coordinate system that specifies point positions by the distance from a chosen reference axis ''(axis L in the image opposite)'', the direction from the axis relative to a chosen reference di ...
).
Other parameterizations are also possible, because knots are defined up to continuous deformation. The illustrations for the (2,3)- and (3,8)-torus knots can be obtained by taking
, and in the case of the (2,3)-torus knot by furthermore subtracting respectively
and
from the above parameterizations of ''x'' and ''y''. The latter generalizes smoothly to any coprime ''p,q'' satisfying