---
tag: Y8KSN16X06
kind: definition
name: Prime knot
msc: [57K10]
defines: GJ6FJ4G5VC
uses: [QYEAZTYGPT, 1TQ06R4Q66, AGZSZW7GYE]
sources:
  - url: https://en.wikipedia.org/wiki/Prime_knot
---

## Definition

Let $K$ be a tame [[knot|QYEAZTYGPT]] in $S^3$.
The knot $K$ is *prime* if:

1. $K$ is not the [[unknot|1TQ06R4Q66]];
2. whenever $K$ is isotopic to a [[knot sum|AGZSZW7GYE]] $K_1 \# K_2$, one of $K_1$ and $K_2$ is the unknot.

## Remarks

Equivalently, $K$ is nontrivial and every $2$-sphere in $S^3$ that meets $K$ transversely in two points bounds a ball meeting $K$ in an unknotted arc.
Example: the [[trefoil knot|VYTYSVHN4A]] has Seifert genus $1$.
Genus is additive under knot sum and only the unknot has genus $0$, so a decomposition $K_1 \# K_2$ forces one summand to be trivial.
Non-example: the granny knot, the sum of two trefoils, fails clause 2.
The unknot fails clause 1.
By Schubert's theorem, every nontrivial knot is a knot sum of finitely many prime knots, unique up to order and isotopy.