---
tag: GZQ5ZE3X6G
kind: concept
name: Knot invariant
msc: [57K10, 57K14]
sources:
  - url: https://en.wikipedia.org/wiki/Knot_invariant
same_as:
  wikidata: Q1862434
  wikipedia: Knot invariant
---

## Summary

A knot invariant is a quantity attached to each [[knot|QYEAZTYGPT]] that takes the same value on equivalent knots.
Invariants are the main tool to prove that two knots are different: if an invariant differs, the knots are not equivalent.
Examples are the [[crossing number|SWJZ4QRF9W]], the [[knot group|AQVHM3FP0D]], [[tricolorability|KNAGK0W65G]], the [[Alexander polynomial|JKA25KDPQP]] and the [[Jones polynomial|4SPC17FPXE]].

## Explanations

### L1 human

To tell two knots apart, it is not enough to fail to untangle one into the other.
Instead one computes something from each knot that does not change when the knot is moved.
If the two results differ, the knots are surely different.
For example, the trefoil can be coloured with three colours by a certain rule and the unknot cannot, so the trefoil is really knotted.