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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*crossings0 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).