Prime knot

{{Short description|Non-trivial knot which cannot be written as the knot sum of two non-trivial knots}}

File:Hopf Link.png]]

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 for exclusively prime knots {{OEIS|A002863}} and for prime or composite knots {{OEIS|A086825}} are given in the following table. As of June 2025, prime knots up to 20 crossings have been fully tabulated. Thistlethwaite, M. "The enumeration and classification of prime 20–crossing knots" University of Tenessee, 2025. https://web.math.utk.edu/~morwen/k20v3.pdf

:

class="wikitable" style="text-align:right;"
n | 1 || 2 || 3 || 4 || 5 || 6 || 7 || 8 || 9 || 10 || 11 || 12 || 13 || 14 || 15 || 16 || 17 || 18 || 19 || 20
Number of prime knots with n crossings | 0 || 0 || 1 || 1 || 2 || 3 || 7 || 21 || 49 || 165 || 552 ||2176 || 9988 || 46972 || 253293 || 1388705 || 8053393 || 48266466 || 294130458 || 1847319428
Composite knots | 0 || 0 || 0 || 0 || 0 || 2 || 1 || 5 || ... || || ... || || ... || || ... || || ... || || ... ||
Total | 0 || 0 || {{#expr:1+0}} || {{#expr:1+0}} || {{#expr:2+0}} || {{#expr:3+2}} || {{#expr:7+1}} || {{#expr:21+5}} || ... || || ... || || ... || || ... || || ... || || ... ||

Enantiomorphs are counted only once in this table and the following chart (i.e. a knot and its mirror image are considered equivalent).

Image:Knot table.svg, not including mirror-images, plus the unknot (which is not considered prime).]]

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