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 crossings 
0  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

Trefoil knot
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.