Knots and colours

A knot diagram is a network drawing with crossings, like the ones in the dots-and-lines lesson, except that at each crossing we record which strand goes over. Two diagrams show the same knot exactly when one can be changed into the other by moving the string around — and in 1927 Kurt Reidemeister proved that it is enough to use three simple local moves: adding or removing a twist, sliding one strand over another, and sliding a strand past a crossing.

The colouring rule in the hints is called tricolouring. Each of Reidemeister's moves can be checked to keep a good colouring good (and to keep "at least two colours" true). So whether a knot can be tricoloured does not depend on how it is drawn: it is a knot invariant.

A plain loop drawn with no crossings has a single arc, so it can only ever use one colour: it is not tricolourable. The trefoil is. Therefore the trefoil is genuinely knotted — no wiggling will undo it. That is a proof, not just evidence.

Invariants have limits. The figure-eight knot cannot be tricoloured either, yet it is knotted too; showing that needs stronger invariants, such as colourings with five colours or the Jones polynomial (Vaughan Jones, 1984). Remarkably, there are algorithms that decide whether any given diagram is the unknot (Wolfgang Haken, 1961) — but whether the Jones polynomial alone can always recognise the unknot is still an open question.