---
tag: KNAGK0W65G
kind: concept
name: Tricolorability
msc: [57K10]
sources:
  - url: https://en.wikipedia.org/wiki/Tricolorability
same_as:
  wikidata: Q7841270
  wikipedia: Tricolorability
---

## Summary

A [[knot|QYEAZTYGPT]] is tricolorable if the arcs of a diagram can be colored with three colors under a simple rule.
At each crossing the three meeting arcs must show one color or all three colors, and at least two colors must appear.
Whether this is possible does not depend on the chosen diagram, so tricolorability is a property detected by a [[knot invariant|GZQ5ZE3X6G]].
It is the simplest tool to prove that a knot is really knotted.

## Explanations

### L1 human

Draw a knot and break each under-strand at the crossings, so the drawing falls into arcs.
Color each arc red, green or blue.
The rule is that at every crossing you see either one color or all three colors.
The plain circle has a single arc, so it can only use one color.
The [[trefoil knot|VYTYSVHN4A]] has three arcs, and giving them three different colors satisfies the rule at all three crossings.
So the trefoil is tricolorable and the circle is not, which proves that the trefoil cannot be untied.