Prime knot

In knot theory

Knot theory

In mathematics, knot theory is the area of topology that studies 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 to prevent it from becoming undone. In precise mathematical language, a...

, a prime 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...

 that is, in a certain sense, indecomposable. Specifically, it is a non-trivial

Unknot

The unknot arises in the mathematical theory of knots. 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. ambient-isotoped, to the standard unknot, i.e. the embedding of the...

 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 given knot is prime or not.

A nice family of examples of prime knots are the torus knot

Torus knot

In knot theory, a torus knot is a special kind of knot which 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...

s. These are formed by wrapping a circle around a 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 and not touching the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid...

 p times in one direction and q times in the other, where p and q are coprime

Coprime

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

 integers.

The simplest prime knot is the trefoil

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 σ₁³...

 with three crossings.

In knot theory

Knot theory

In mathematics, knot theory is the area of topology that studies 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 to prevent it from becoming undone. In precise mathematical language, a...

, a prime 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...

 that is, in a certain sense, indecomposable. Specifically, it is a non-trivial

Unknot

The unknot arises in the mathematical theory of knots. 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. ambient-isotoped, to the standard unknot, i.e. the embedding of the...

 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 given knot is prime or not.

A nice family of examples of prime knots are the torus knot

Torus knot

In knot theory, a torus knot is a special kind of knot which 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...

s. These are formed by wrapping a circle around a 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 and not touching the circle. Examples of tori include the surfaces of doughnuts and inner tubes. The solid contained by the surface is known as a toroid...

 p times in one direction and q times in the other, where p and q are coprime

Coprime

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

 integers.

The simplest prime knot is the trefoil

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 σ₁³...

 with three crossings. The trefoil is actually a (3, 2)-torus knot. The figure-eight knot

Figure-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...

, with four crossings, is the simplest non-torus knot. For any positive integer

Integer

The integers are natural numbers including 0 and their negatives . They are numbers that can be written without a fractional or decimal component, and fall within the set {.....

 n, there are a finite number of prime knots with n crossings

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....

. The first few values are given in the following table.

n	1	2	3	4	5	6	7	8	9	10
Number of prime knots with n crossings	0	0	1	1	2	3	7	21	49	165

Note that enantiomorphs

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. A chiral object and its mirror image are said to be enantiomorphs...

 are counted only once in this table and the following chart (i.e. a knot and its mirror image

Mirror image

A mirror image is a reflected duplication that appears identical but in reverse. As an optical effect it results from reflection off of substances such as a mirror or water...

 are considered equivalent).

Schubert's theorem

A theorem due to Horst Schubert states that every knot can be uniquely expressed as a connected sum

Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...

of prime knots.