In knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a nontrivial knot which cannot be written as the knot sum of two nontrivial 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 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.
The simplest prime knot is the trefoil with three crossings. The trefoil is actually a (2, 3)torus knot. The figureeight knot, with four crossings, is the simplest nontorus 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) are given in the following table.

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Number of prime knots
with n crossings0 0 1 1 2 3 7 21 49 165 552 2176 9988 46972 253293 1388705 Composite knots 0 0 0 0 0 2 1 4 ... ... ... ... Total 0 0 1 1 2 5 8 25 ... ... ... ...
Enantiomorphs are counted only once in this table and the following chart (i.e. a knot and its mirror image are considered equivalent).
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Minimal ropelength prime knots
Transcription
Contents
Schubert's theorem
A theorem due to Horst Schubert states that every knot can be uniquely expressed as a connected sum of prime knots.^{[1]}
See also
References
 ^ Schubert, H. "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". S.B Heidelberger Akad. Wiss. Math.Nat. Kl. 1949 (1949), 57–104.