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.

My understanding of the points being raised regarding algebraic topology is that, without all of the machinery that many mathematicians are now intimately familiar with, it is much easier to conjecture certain statements than to prove them. In other words, if you had asked a mathematician in the 18th century to prove these things, they would’ve understood what you were asking but wouldn’t have been able to do it. The issue is not about whether a modern mathematician, equipped with all of the modern machinery at his/her disposal, can write down complete proofs of everything easily.

Comment by Qiaochu Yuan — January 18, 2011 @ 10:38 pm |

This is certainly a valid point, and I don’t think that we necessarily need to agree. I suppose I’m insisting that it be “unexpectedly hard to prove”, which I don’t think is the case. My first counter-argument would be the same as Ryan’s, which is that a knot would not even have been considered a

mathematical objectby almost any 18th century mathematician. Thus, I’m differentiating betweeen a physical knot, which one might tie in one’s shoelaces, and the topological object which goes by the same name.A poor analogy might be the physical concept of velocity in contrast to the mathematical concept of a derivative. If the velocity is positive then one is moving forward- this is intuitively (and physically) obvious, but you need the MVT to prove the parallel mathematical statement, and the MVT is a rather complicated machine if we think about all that goes into it. And yet, although I see a better argument for “positive velocity means going forward is obvious but (unexpectedly) hard to prove” than for the parallel statement with regard to “the trefoil is unknotted”, I would disagree strongly with that characterization, because it is hard to prove for the

physicalconcept of velocity, but not for the parallelmathematicalconcept of the derivative.I think we must begin by assuming that the 18th century mathematician in question were given the question in mathematical language with all terms properly defined. The argument is now that a trefoil is difficult to distinguish from the unknot, because the most obvious invariant (Euler characteristic, or even homology) fails, because the first homology of all knot complements is the same. So distinguishing a trefoil from an unknot is more difficult than distinguishing, say, a 3-ball from a solid torus. However, I would argue that, when all is said and done, what we have before us is almost the easiest toy problem one could possibly imagine of distinguishing two topological spaces with the same H

_{1}. Therefore, I would argue against it being a more difficult problem than one might expect.Another nice point which you raise is that, indeed, it’s easy to make conjectures about knot diagrams, but hard to prove them. I am not sure that this is unexpected, because the algebraic structure of knot diagrams mod Reidemeister moves is itself rather complicated- a module over a planar algebra generated by crossings modulo Reidemeister moves is not a very easy structure to work with. Not unexpected perhaps; but certainly there are obvious statements which are hard to prove. In this vein, I am therefore tempted to add the Tait Conjectures as another answer.

Comment by dmoskovich — January 19, 2011 @ 9:13 am |

You and Ryan make a very good point about the distinction between mathematical and physical objects. I guess that was a nontrivial issue implicit in the original MO question which may not have been well-addressed.

Comment by Qiaochu Yuan — January 19, 2011 @ 4:34 pm

Hi Daniel,

I was similarly confused by the non-triviality of the trefoil response to that MO thread. I think it’s *far* more difficult to decide on (1) what a mathematical knot should be and (2) what the definition of equivalence of knots should be. Once you’ve decided what those ideas mean, showing a trefoil is non-trivial isn’t so bad, and it’s more or less a “you get what you pay for” deal. If you blur the lines between physical models of reality and reality I suppose one could try to substitute “I haven’t been able to untie the trefoil” for “it’s obviously knotted” but as you observed, that’s a strange statement to make on a mathematics webpage.

Comment by Ryan Budney — January 18, 2011 @ 10:38 pm |

I think that one point Qiaochu is making is that the fundamental group itself might be considered “too hard” of a tool. Homology, or Euler characteristic, are fine, because they use less of a “big machine”, but neither will do the job. That said, I disagree with the above argument, because distinguishing the trefoil from the unknot looks to me like a prototypical toy example of “distinguish two shapes with the same H

_{1}“. So I agree with you that it’s more or less a “you get what you pay for” deal.Comment by dmoskovich — January 19, 2011 @ 9:21 am |

Qiaochu: In some sense an 18th century mathematician didn’t have a language in which to state the conjecture. Analysis Situs was just a twinkle in the eye of a very few.

Comment by Ryan Budney — January 18, 2011 @ 10:46 pm |

I like to tell people that I think of knots as certain morphisms in the braided monoidal category with duals generated by a single object. ^_^

Comment by Eitan Chatav — January 23, 2011 @ 10:01 am |