---
tag: 878B9A4D9B
kind: definition
name: Tricolorability
msc: [57K10]
defines: KNAGK0W65G
uses: [QYEAZTYGPT, T3HAGWYKK7]
sources:
  - url: https://en.wikipedia.org/wiki/Tricolorability
---

## Definition

Let $D$ be a diagram of a [[knot|QYEAZTYGPT]] $K$.
The *arcs* of $D$ are the pieces into which $D$ falls when each under-strand is cut at each crossing.
A *tricoloring* of $D$ is a map from the set of arcs of $D$ to a set of three colors such that:

1. at each crossing, the over-arc and the two under-arcs have either one color or three distinct colors;
2. at least two colors are used.

The diagram $D$ is *tricolorable* if it has a tricoloring.
The knot $K$ is *tricolorable* if some diagram of $K$ is tricolorable.

## Remarks

Each [[Reidemeister move|T3HAGWYKK7]] turns a tricoloring into a tricoloring, so if one diagram of $K$ is tricolorable then every diagram of $K$ is.
Example: the standard diagram of the [[trefoil knot|VYTYSVHN4A]] has three arcs and three crossings, and each crossing meets all three arcs; three distinct colors satisfy both clauses.
Non-example: the round diagram of the [[unknot|1TQ06R4Q66]] has one arc, so clause 2 fails; hence the trefoil is not the unknot.
The [[figure-eight knot|C01XBGAHC6]] is not tricolorable.
With the colors read as elements of $\Z/3$, clause 1 says $2a \equiv b + c \pmod 3$ at a crossing with over-arc $a$ and under-arcs $b, c$; this is Fox $3$-coloring.
A knot is tricolorable if and only if $3$ divides its determinant $\abs{\Delta_K(-1)}$, where $\Delta_K$ is the [[Alexander polynomial|JKA25KDPQP]].