Non-trivial knot which cannot be written as the knot sum of two non-trivial knots
Simplest prime link
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.
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) 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 chart of all prime knots with seven or fewer crossings, not including mirror-images, plus the unknot (which is not considered prime).
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