Simple knot (knot theory)

In knot theory, a simple knot or simple link is a knot that, in a sense, is indecomposable. More precisely, it is a nontrivial node that cannot be represented as a concatenation of two nontrivial nodes. Knots that are not simple are said to be compound knots or compound gears . Determining whether a given node is simple or not can be challenging.

Content

1 Examples
2 Schubert Theorem
3 See also
4 notes
5 Literature
6 References

Examples

A good example of a family of simple knots is toric knots . These nodes are formed by twisting the circle a torus p times in one direction and q times in the other, where p and q are coprime integers .

The simplest simple knot is a trefoil with three intersections. The trefoil is, in fact, a (2, 3) -toric knot. The G8 node with four intersections is the simplest non-toric node. For any positive integer n, there are finitely many simple nodes with n intersections . The first few values of the number of simple nodes (sequence A002863 in OEIS ) are given in the following table.

n	one	2	3	four	5	6	7	8	9	10	eleven	12	13	fourteen	fifteen	16
The number of simple nodes with n intersections	0	0	one	one	2	3	7	21	49	165	552	2176	9988	46 972	253,293	1,388,705
Compound nodes	0	0	0	0	0	2	one	four	...		...		...		...
Total	0	0	one	one	2	5	8	25	...		...		...		...

Note that the antipodes were considered in this table and in the figure below only once (i.e., the node and its mirror reflection are considered equivalent).

Images of all simple nodes with seven or less intersections without specular reflections. (The trivial knot is not considered simple)

Schubert's Theorem

A theorem by Horst Schubert states that any knot can be uniquely represented as a concatenation of simple knots ^[1] .

Notes

↑ Schubert, 1949 , p. 57-104.

Literature

H. Schubert. Die eindeutige Zerlegbarkeit eines Knotens in Primknoten // S.-B Heidelberger Akad. Wiss. Math.-Nat. Kl. - 1949.

Links

Weisstein, Eric W. Prime Knot on Wolfram MathWorld .
[Prime Links with a Non-Prime Component] Knot Atlas

[_aad37f29e5f3ff17-1] Schubert, 1949 , p. 57-104.