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 prime knot or prime link 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 ...

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

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 ...

knot which cannot be written as the

knot 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. This construction plays a key role in the classifica ...

of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links. It can be a nontrivial problem to determine whether a given knot is prime or not. A family of examples of prime knots are the

torus knot In knot theory, a torus knot is a special kind of knot (mathematics), knot that lies on the surface of an unknotted torus in R3. Similarly, a torus link is a link (knot theory), link which lies on the surface of a torus in the same way. Each t ...

s. These are formed by wrapping a circle around a

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 ...

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

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 ...

integers. Knots are characterized by their crossing numbers. The simplest prime knot is the

trefoil A trefoil () is a graphic form composed of the outline of three overlapping rings, used in architecture and Christian symbolism, among other areas. The term is also applied to other symbols with a threefold shape. A similar shape with four rin ...

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

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 ...

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

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 ...

''n'', there are a finite number of prime knots with ''n'' crossings. The first few values are given in the following table. : Enantiomorphs are counted only once in this table and the following chart (i.e. a knot and its

mirror image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...

are considered equivalent). __NOTOC__

Schubert's theorem

A theorem due to

Horst Schubert Horst Schubert (11 June 1919 – 2001) was a German mathematician. Schubert was born in Chemnitz and studied mathematics and physics at the Universities of Frankfurt am Main, Zürich and Heidelberg, where in 1948 he received his PhD under Herb ...

states that every knot can be uniquely expressed as a

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. This construction plays a key role in the classifica ...

of prime knots.Schubert, H. "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". ''S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl.'' 1949 (1949), 57–104.

References

External links

* * {{Knot theory, state=collapsed Knot invariants Knots (knot theory)

Schubert's theorem

See also

References

External links