Prime knot

In knot theory, a prime knot or prime link is a knot 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.

Simplest prime link

A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers.

Knots are characterized by their crossing numbers. The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3)-torus knot. The figure-eight knot, with four crossings, is the simplest non-torus knot. For any positive integer n, there are a finite number of prime knots with n crossings. The first few values (sequence A002863 in the OEIS) and (sequence A086825 in the OEIS) are given in the following table.

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19
Number of prime knots with n crossings	0	0	1	1	2	3	7	21	49	165	552	2176	9988	46972	253293	1388705	8053393	48266466	294130458
Composite knots	0	0	0	0	0	2	1	5	...		...		...		...		...		...
Total	0	0	1	1	2	5	8	26	...		...		...		...		...		...

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

A chart of all prime knots with seven or fewer crossings, not including mirror-images, plus the unknot (which is not considered prime).

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.^[1]

See also

• List of prime knots

References

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

External links

• Weisstein, Eric W. "Prime Knot". MathWorld.
• "Prime Links with a Non-Prime Component", The Knot Atlas.