Prime knot

Definition

Let $K$ be a tame knot in $S^3$ . The knot $K$ is prime if:

$K$ is not the unknot;
whenever $K$ is isotopic to a knot sum $K_1 \# K_2$ , 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 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.

Reading path

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Knot concept
Knot sum concept
Unknot example
Prime knot definition

Tag	`Y8KSN16X06`
URI	`m2://Y8KSN16X06`
Kind	definition
Writing	W0 stub
MSC	57K10
Defines	Prime knot
Uses	Knot, Unknot, Knot sum
Changed	2 November 2026

Prime knot

Definition

Remarks

Reading path

Further reading

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