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Why knot?

You can un-knot a knot, by cutting it not?

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Image: Flickr user clickykbd, CC BY-NC-SA 2.0.

1 February 2018

As any fule kno…

you can’t tie a knot in a length of string without letting go of the ends.

Let’s be a bit more mathematical: you can’t tie a knot in a loop of string without cutting it. (We can think about the string together with your body as forming a loop, and cutting the loop as letting go of an end.)

Let’s be specific: by a knot in a length of string, let’s agree that we’re talking about a trefoil knot, so the thing that everybody knows is that you can’t convert this:

Loop

to this:

Trefoil knot

without cutting the loop.

We know from  experience that no matter how complicated a tangle we make with a loop, it never becomes a trefoil (and vice versa). But this is just saying that we haven’t managed to do it. It might be that a really, really complicated tangle, which is impossible to do with the kind of string that we can physically tangle, might be a stage on the route from a loop to a trefoil. Basically, proof by incompetence is not very convincing.  Moreover, it is not what we rely on to say that other mathematical jobs are impossible, such as expressing $\sqrt{2}$ as the ratio of two integers, or finding a construction using straight edge and compasses to trisect an arbitrary angle.

Of course, there is a more convincing argument than one from incompetence, and that’s what this article is about. The argument has three ingredients:

  1. a mathematical model of a knot
  2. a way of describing how the knot can be manipulated
  3. a reason for believing that an unknotted loop cannot be manipulated into a trefoil.

Knots in three dimensions

I define a knot to be a simple, closed, smooth curve in three-dimensional space. Simple means that it doesn’t intersect itself, and smooth means that there are no corners. Making this rigorous involves a bit of calculus, but doesn’t add anything to the discussion.

We say that two knots are equivalent if it is possible to deform one into the other in such a way that every stage in the deformation is also a knot. Making this rigorous involves a lot of calculus, and would add a huge amount to the discussion. Unfortunately, most of it would not be helpful.

A knot is trivial if it can be deformed to the unit circle in the $xy$-plane. This is also (just to be confusing) called the un-knot, and it’s what we would usually regard as not being a knot at all.

But in order to get anywhere, we have to get a handle on this. Fortunately, it is possible to avoid the third dimension almost completely by looking at the shadow cast by a knot on the $xy$-plane. And remarkably, any possible deformation can be broken down into a combination of a very small number of archetypical deformations. So let’s see how that works.

Knot projections

The shadow of a knot on the $xy$-plane is just what we get if we forget about the $z$ coordinate of each point on the knot. This is pretty much what we get any time we think of drawing a picture of a knot.

Projected knot. Image:Flickr user fdecomite, CC BY 2.0.

There are however a couple of snags!

The first is that the shadow won’t in general be simple (because it will have intersections) and it also might not be smooth (because even if the three dimensional curve has no corners, the projection might do).

For the second issue, I claim (and it should be very plausible) that by deforming the knot slightly, you can get rid of any corners.

For the first, and much more important issue, I admit that we don’t just look at the shadow: we also have to keep track of which strand goes over, and which goes under. In other words, this really is just what we usually think of when we think of a picture of a knot. We also insist that no more than two strands cross at any given point, and that they point in different directions wherever they cross: it is easy to believe that this can always be arranged by a slight deformation.

The two pictures we saw above of a circle and the trefoil are examples of knot projections. One of them is obviously the un-knot. The claim is that the other is not the un-knot in a cunning disguise.

Invariants

Now we have something of a problem. I want to argue that it is impossible to deform a knot so that we start off with its projection being the un-knot, and end up with the projection of a trefoil. But how can one do this? I can’t put a limit on how long and complicated the deformation must be, so there is no way of showing that no matter what deformation we try, it doesn’t do the job.

We attack the problem indirectly, by considering a quantity called an invariant. Generally speaking, an invariant is a property of some kind of mathematical object which does not change when allowable operations are performed. Here, we are going to think about a knot projection invariant: this will be a property of a knot projection which does not change when the (full three dimensional) knot is deformed.

But first, we need to get a better handle on how deformations of knots are reflected in their projections.

Reidemeister moves

In 1932 Kurt Reidemeister published Knotentheorie, in which he described how any deformation of a knot could be built up from a combination of three basic moves applied to knot projections as described above, now called the Reidemeister moves. These are:

Type I

Type II

Type III

It is easy to believe that, at some intermediate point between the endpoints of each of these latter two, the projection is illegal. By illegal we mean that either it must acquire a corner, three strands cross at a point, or strands are tangential at the crossing. It is harder to believe (but fortunately Reidemeister did all the work already) that only these moves are required.

If you have ever done one of those puzzles where you look at a picture of a tangle and have to decide whether it is a knot (or not), you have almost certainly used this, without ever realising that you were using a theorem from topology. The good news is that the intuitive approach is in fact correct.

If you want to chase this up, Reidemeister’s book has been translated into English, and is available here.

Actually, if you do this you will find that I have been lying slightly. His analysis is for knots built out of straight line segments, and though it seems obvious (in the way that topology is often obvious) it still takes more work to show that this is equivalent to the curvy version that I have been describing.

Tri-colouring

We are getting there. All we need now is to find an invariant, which will mean a property of a knot projection that is unchanged by a Reidemeister move, and is different for the trefoil knot and the unknot. If we manage to do this, we are done!

The property is that of tricolouring.

We say that a knot is tri-colourable, if its projection can be coloured so that it has strands of three different colours, and at each intersection either just one colour, or all three colours are involved.

We can now check explicitly that if a projection is tri-colourable, then applying a Reidemeister move does not alter this. There is no better way of seeing this than looking at pictures.

Moves of type I

This is the easiest one. Everything stays the same colour.

Type I tri-colouring

Moves of type II

There are two possibilities here. Either the two initial strands are the same colour, and after the overlapping move all strands are the same colour, or the two initial strands are different colours, in which case we have

Type II tri-colouring

Moves of type III

This time there are several possible cases. Everything the same colour before and after the move is the easiest. One other is given by

Type III tri-colouring

Your mission, should you choose to accept it, is to figure out the other possibilities and check each of them preserves the tri-colourability of the projection.

This tells us that being tri-colourable is an invariant of the Reidemeister moves, and so is an invariant of knots, since two knots are actually the same if their projections are related by a sequence of Reidemeister moves.

Ta-da!

Obviously, the un-knot, represented by a circle cannot be tri-coloured: there is only one strand.

On the other hand, the trefoil knot clearly can be tri-coloured. Here is a proof:

Trefoil knot tri-colouring

Since the Reidemiester moves preserve tri-colourability, it is not possible to convert a knot whose projection is the circle, to one whose projection is the trefoil knot: in other words, it is impossible to convert an unknotted loop into a trefoil knot.

But that’s just what we mean when we say that you cannot tie a knot in a length of string without letting go of an end. So in conclusion, it isn’t incompetence that prevents us from doing so, it really can’t be done!

Rob teaches mathematics at Coventry University.
@RobJLow   robjlow.blogspot.co.uk    All articles by Robert

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