Knots and colours
The question
Tie a knot in a piece of string and glue the two ends together. How could you prove that it really is knotted — that no amount of wiggling, without cutting, will turn it into a plain loop?
Try the experiment before reading on. There’s no wrong way to explore.
Experiment
Colouring knots with three colours
Colour every arc of the diagram. At each crossing, the three arcs that meet must be all the same colour or all different.
Click an arc to change its colour (or focus it and press Enter, or 1, 2, 3). An arc runs from one under-crossing to the next.
- Crossings in this diagram
- 3
- Crossing number (fewest possible)
- 3
Arcs coloured: 0 of 3.
Why this rule? Any two diagrams of the same knot are related by three kinds of local moves, the Reidemeister moves: adding or removing a twist, sliding one strand over another, and sliding a strand past a crossing. Each move turns a valid colouring with two or more colours into another one.
The trefoil can be tricoloured and the unknot cannot. Because tricolourability is an invariant, no amount of wiggling turns the trefoil into the unknot: the trefoil really is knotted.
So tricolourability is a knot invariant: if one diagram of a knot can be tricoloured, every diagram of it can. A plain circle has just one arc, so the unknot cannot be tricoloured.
Hints, if you want them
What’s going on
What’s going on
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.
Where this leads
- Advanced undergraduate · Lesson availableKnot theoryKnots and links in space and their diagrams, Reidemeister moves, invariants that tell knots apart (tricolourability and Fox colourings, linking number, the Kauffman bracket and Jones polynomial), Seifert surfaces and genus, and an honest map of what is known and open.
- Research pathways · Lesson availableLow-dimensional topology: 3- and 4-manifoldsOptional research direction: 3-manifolds through Heegaard splittings, surgery and the eight Thurston geometries; the Poincaré conjecture and geometrisation (Perelman); hyperbolic 3-manifolds; and the strange world of 4-manifolds (Freedman, Donaldson, exotic ℝ⁴), separating established results from open questions.
- Research pathways · Lesson availableCategorical representation theoryOptional research direction: representations on categories rather than vector spaces, including tensor and fusion categories, module categories, categorical actions and categorification, such as Chuang–Rouquier sl₂-categorification and Khovanov homology.