---
tag: AQVHM3FP0D
kind: concept
name: Knot group
msc: [57K10, 57M05]
sources:
  - url: https://en.wikipedia.org/wiki/Knot_group
same_as:
  wikidata: Q1777486
  wikipedia: Knot group
---

## Summary

The knot group of a [[knot|QYEAZTYGPT]] is the [[fundamental group|S1XK3HXHFX]] of its complement in $\R^3$.
It is a [[knot invariant|GZQ5ZE3X6G]]: equivalent knots have isomorphic knot groups.
It is a powerful one, since it distinguishes the unknot from every other knot.
A diagram of the knot gives an explicit presentation of it, the [[Wirtinger presentation|P6GR8ZQCK3]].

## Explanations

### L1 human

Imagine the knot as a fixed wire in space, and consider loops of string that start and end at a fixed point and may wind around the wire.
Two loops count as the same if one can be slid to the other without crossing the wire.
Following one loop and then another gives a way to combine loops, and this makes them a group.
For the unknot, a loop is described by how many times it winds through the circle, so the group is the integers $\Z$.
For the trefoil, the group is not commutative, which proves that the trefoil cannot be untied.