In knot theory, a prime knot is a knot which is, in a certain sense, indecomposable. Specifically, it is one which cannot be written as the knot sum of two nontrivial knots. Knots which are not prime are said to be composite. It can be a nontrivial problem to determine whether a given knot is prime or not.
The simplest prime knot is the trefoil with 3 crossings. The trefoil is actually a (2,3)-torus knot. The figure-eight knot, with 4 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 are given in the following table.
|Number of prime knots
with n crossings
- Prime Knot at MathWorld.