Inknot theory, aprime knot orprime link is aknot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as theknot sum of two non-trivial knots. Knots that are not prime are said to becomposite knots orcomposite 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 thetorus knots. These are formed by wrapping a circle around atorusp times in one direction andq times in the other, wherep andq arecoprime integers.

Knots are characterized by theircrossing numbers. The simplest prime knot is thetrefoil with three crossings. The trefoil is actually a (2, 3)-torus knot. Thefigure-eight knot, with four crossings, is the simplest non-torus knot. For any positiveintegern, there are a finite number of prime knots withncrossings. The first few values for exclusively prime knots (sequenceA002863 in theOEIS) and for primeor composite knots (sequenceA086825 in theOEIS) are given in the following table. As of June 2025, prime knots up to 20 crossings have been fully tabulated.^[1]

n	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17	18	19	20
Number of prime knots withn crossings	0	0	1	1	2	3	7	21	49	165	552	2176	9988	46972	253293	1388705	8053393	48266466	294130458	1847319428
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 itsmirror image are considered equivalent).

A theorem due toHorst Schubert (1919–2001) states that every knot can be uniquely expressed as aconnected sum of prime knots.^[2]

• List of prime knots

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

• Knot invariants
• Prime knots and links
• Knots (knot theory)

• Articles with short description
• Short description matches Wikidata