---
tag: AGZSZW7GYE
kind: concept
name: Knot sum
msc: [57K10]
specializes: [Y0K0K5NGWW]
sources:
  - url: https://en.wikipedia.org/wiki/Connected_sum
---

## Summary

The knot sum, or connected sum, of two oriented [[knots|QYEAZTYGPT]] is a new knot.
It is obtained by cutting each knot open at one point and joining the loose ends.
It is the version for knots of the [[connected sum|Y0K0K5NGWW]] of manifolds.
The sum is well defined, commutative and associative, with the [[unknot|1TQ06R4Q66]] as neutral element.
It lets one break knots into [[prime knots|GJ6FJ4G5VC]], much as integers break into primes.

## Explanations

### L1 human

Take two knotted loops of string lying far apart.
Cut each loop once and tie the four loose ends together in pairs, so that you get one loop with both knots on it.
The result is the knot sum.
For example, the sum of a [[trefoil knot|VYTYSVHN4A]] with itself is the granny knot.
The sum of a trefoil with its mirror image is the square knot, which is a different knot.