Torus knot

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Torus knot

In knot theory

Knot theory

In mathematics, knot theory is the area of topology that studies knot s. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs drastically in that the ends are joined together to prevent it from becoming undone....
, a torus knot is a special kind of knot

Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations ....
which lies on the surface of an unknotted torus

Torus

In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle, which does not touch the circle....
in R³. Similarly, a torus link is a link

Link (knot theory)

In mathematics, a link is a collection of knot s which do not intersect, but which may be linked together. A knot can be described as a link with one component....
which lies on the surface of a torus in the same way. Each torus knot is specified by a pair of coprime

Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common divisor other than 1 or, equivalently, if their greatest common divisor is 1....
integer

Integer

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Encyclopedia

In knot theory

Knot theory

Knot (mathematics)

In mathematics, a knot is an embedding of a circle in 3-dimensional Euclidean space, R3, considered up to continuous deformations ....
which lies on the surface of an unknotted torus

Torus

Link (knot theory)

Coprime

In mathematics, the integers a and b are said to be coprime or relatively prime if they have no common divisor other than 1 or, equivalently, if their greatest common divisor is 1....
integer

Integer

The integers are natural numbers including 0 and their negative and non-negative numberss . They are numbers that can be written without a fractional or decimal component, and fall within the set ....
s p and q. The (p,q)-torus knot winds q times around a circle inside the torus, which goes all the way around the torus, and p times around a line through the hole in the torus, which passes once through the hole, (usually drawn as an axis of symmetry). If p and q are not relatively prime, then we have a torus link with more than one component.

The (p,q)-torus knot can be given by the parameterization where 0 ≤ t < 2πp. This lies on the surface of the torus given by (in cylindrical coordinates).

Torus knots are trivial

Unknot

The unknot arises in the knot theory. Intuitively, the unknot is a closed loop of rope without a knot in it. A knot theorist would describe the unknot as an image of any embedding that can be deformed, i.e....
iff

IFF

IFF, Iff or iff can stand for:* Identification Friend or Foe, an electronic radio-based identification system utilizing transponders...
either p or q is equal to 1. The simplest nontrivial example is the (2,3)-torus knot, also known as the trefoil knot

Trefoil knot

In knot theory, the trefoil knot is the simplest nontrivial knot . It can be obtained by joining the loose ends of an overhand knot. It can be described as a -torus knot, and is the closure of the 2-stranded braid group s1?....
.

Properties

Each torus knot is prime

Prime knot

In knot theory, a prime knot is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-unknot knot which cannot be written as the knot sum of two non-trivial knots....
and chiral

Chirality (mathematics)

In geometry, a figure is chiral if it is not identical to its mirror image, or more particularly if it cannot be mapped to its mirror image by rotations and translations alone....
. Any (p,q)-torus knot can be made from a closed braid

Braid theory

In topology, braid theory is an abstract geometry theory studying the everyday braid concept, and some generalisations. The idea is that braids can be organised into group s, in which the group operation is 'do the first braid on a set of strings, and then follow it with a second on the twisted strings'....
with p strands. The appropriate braid word is The crossing number

Crossing number

In mathematics, crossing numbers arise in two related contexts: in knot theory and in graph theory.*In knot theory, crossing number of a knot refers to the minimal number of crossings in any knot diagram for the knot....
of a torus knot is given by

c = min((p−1)q, (q−1)p).

The genus of a torus knot is The Alexander polynomial

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....
of a torus knot is The Jones polynomial

Jones polynomial

In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1983. Specifically, it is an knot invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients....
of a (right-handed) torus knot is given by

The complement of a torus knot in the 3-sphere

3-sphere

In mathematics, a '3-sphere' is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space....
is a Seifert-fibered manifold

Seifert fiber space

A Seifert fiber space is a 3-manifold together with a "nice" decomposition as a disjoint union of circles. In other words it is a -bundle over a 2-dimensional orbifold....
, fibred over the disc with two singular fibres.

Let Y be the p-fold dunce cap

Dunce hat (topology)

For the item of clothing designed to be humiliating, now rarely, practiced, see dunce cap.In topology, the dunce hat is a compact space topological space formed by taking a solid triangle and quotient space all three sides together, with the orientation of one side reversed....
with a disk removed from the interior, Z be the q-fold dunce cap with a disk removed its interior, and X be the quotient space obtained by identifying Y and Z along their boundary circle. The knot complement of the (p, q)-torus knot deformation retract

Deformation retract

In topology, a retraction, as the name suggests, "retracts" an entire space into a subspace. A deformation retraction is a function which captures the idea of continuous function shrinking a space into a subspace....
s to the space X. Therefore, the knot group

Knot group

In mathematics, a knot is an embedding of a circle into 3-dimensional Euclidean space. The knot group of a knot K is defined as the fundamental group of the knot complement of K in R3,...
of a torus knot has the presentation

Presentation of a group

In mathematics, one method of defining a group is by a presentation. One specifies a set S of generating set of a group so that every element of the group can be written as a product of some of these generators, and a set R of relations among those generators....

Torus knots are the only knots whose knot groups have non-trivial center (which is infinite cyclic, generated by the element in the presentation above).

Torus knot

Properties

See also

External links