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.
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 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) 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).
A theorem due to Horst Schubert states that every knot can be uniquely expressed as a connected sum of prime knots.^{[1]}