knot theory In 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 joined so it cannot be und ...

, a prime knot or prime link is a

knot A knot is an intentional complication in Rope, cordage which may be practical or decorative, or both. Practical knots are classified by function, including List of hitch knots, hitches, List of bend knots, bends, List of loop knots, loop knots, ...

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 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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...

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

coprime In number theory, 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 equiv ...

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, Pagan and Christian symbolism, among other areas. The term is also applied to other symbols with a threefold shape. A similar shape with f ...

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 sailing, rock climbing and caving as a method of stopping ropes from running out of retaining devices. Like the overhand knot, which will jam under ...

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

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

''n'', there are a finite number of prime knots with ''n'' crossings. The first few values for exclusively prime knots and for prime ''or'' composite knots are given in the following table. As of June 2025, prime knots up to 20 crossings have been fully tabulated. : 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 reflection (physics), reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical phenomenon, optical effect, it r ...

are considered equivalent). __NOTOC__

Schubert's theorem

A theorem due to Horst Schubert (1919–2001) states that every knot can be uniquely expressed as a connected sum 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