Colourability

Idea

The colourability of a knot tells one information about its knot group yet has a simple, and visually attractive aspect that seems almost to avoid all mention of groups, presentations, etc., except at a fairly naive level.

The easiest form of colourability to examine is $3$-colourability.

$3$-colourability

Definition

A knot diagram is $3$-colourable if we can assign colours to its arcs such that

1. each arc is assigned one colour;

2. exactly three colours are used in the assignment;

3. at each crossing, either all the arcs have the same colour, or arcs of all three colours meet in the crossing.

Examples and non-examples
• The usual diagram for the trefoil knot is $3$-colourable. (Just do it! Each arc is given a separate colour and it works.)

• The figure-8 knot diagram

category: svg

is not $3$-colourable. (Try it!)

Theorem

$3$-colourability is a knot invariant?.

The proof is amusing to work out oneself. You have to show that if a knot diagram $D$ is $3$-colourable and you perform a Reidemeister move on it then the result is also $3$-colourable. The thing to note is that any arcs that leave the locality of the move must be coloured the same before and after the move is done.

• We can now use phrases such as ‘the trefoil knot is $3$-colourable’ as its validity does not depend on what diagram is used to represent it, (by the above and by Reidemeister's theorem.)

• As the trefoil knot is $3$-colourable and the unknot is not, non-trivial knots exist. Moreover, the trefoil is $3$-colourable and the figure $8$ is not, so these are different. We also get that the bridge number of the trefoil is $2$, as this provides the missing piece of the argument found in that entry.