Recently, there was a soft question on MathOverflow asking for examples of theorems which are `obvious but hard to prove’. There were three responses concerning pre-1930 knot theory, and I didn’t agree with any of them. This led me to wonder whether there might be a bit of a consensus in the mathematical community that knot theory is really much more difficult than it ought to be; and that good knot theory should be all about combinatorics of knot diagrams. And so knot colouring becomes `good knot theory’ for what I think are all the wrong reasons.
Today’s story begins in 1956, when Ralph Fox gave an amazingly good talk to undergraduate students at Haverford College. So good was his talk in fact that it actually changed the history of topology (how I dream of giving `the ultimate talk’!). His talk was about coloured knots- but instead of introducing them via homomorphisms from the knot group onto a dihedral group like we did last time, he introduced knot colourings by physically colouring arcs of knot diagrams red, blue, and green subject to Wirtinger rules (of course he didn’t name them that), and he proved invariance by showing that tricolourability is preserved by Reidemeister moves. Thus was born the Fox n-colouring.
It’s quite beautiful and unexpected, and if I were giving an introductory talk to undergrads or to high-school students, surely I would imitate Fox’s approach. Seeing tricolourability introduced as Fox presented it is surely inspiring.
On the other hand, successful popular exposition is always a mixed blessing; it’s wonderful when people find a certain facet of our work inspiring, but painful when they take away oversimplified messages which miss the point. In the case of knot colourings, that point is that a colouring isn’t an arbitrary parlour trick, but has sound mathematical basis as a group homomorphism. As such, the concept of a fundamental group is essential for colourings. Otherwise what possible reason would there be for tricolourability to be a knot invariant? None at all.
I contend that the mathematical public’s impression of knot theory is heavily influenced by Ralph Fox’s oustanding talk of half a certury ago. The impression is that knot theory should be about arbitrary combinatorial games with knot diagrams, which magically happen to work. And any algebraic topology gets deemed a `hard proof of an obvious fact’.
I suppose that the first question really to be asking oneself is what is a knot? I think of a knot as being a smooth (or PL) embedding of a circle into 3-space, modulo ambient isotopy. This is an essentially topological definition, and use of the fundamental group is surely not “over the top” in the study of knots thus defined. Another way of thinking of knots would be as knot diagrams modulo Reidemeister moves. These are combinatorial objects in the plane. Such objects have an operadic nature; anyway, they’re much more complex mathematical objects than (fundamental) groups. I don’t see why one should expect it to be “easy” to distinguish knots by looking at their knot diagrams. Thus, I’m not surprised that I find algebraic topological methods to distinguish knots to be conceptually more natural and easier to understand; although perhaps not quite as much fun as understanding knots using their knot diagrams, especially if we are working in the quantum realm.