Yesterday afternoon the University of Edinburgh had a bumper edition of the Scottish Topology seminar series. The theme was knot concordance (which I shall explain presently), and given that this is Julia’s thesis topic she was invited to be one of the speakers. (Well, she was only invited to be a speaker once one of the other speakers had dropped out, but let’s not hurt her feelings by dwelling on that…) What has this all got to do with me? Well, I was bored with sitting at home and decided that I wanted a piece of the action, so I managed to persuade Julia to write me a part in her talk.
To understand my part, you’ll need to know a bit about what knot concordance is. It’s the theory of slice knots, which I talked about a bit in a previous blog post. Every knot is the boundary (i.e. edge) of a surface, like this:
But no knot, except for the unknotted circle, can bound a surface that doesn’t have any holes in it. (In the trefoil picture above, the ‘hole’ is that bit in the middle where you can see through to the background.) However, for some knots we can make the holes go away by pushing them into the 4th dimension. The problem is: how do we tell which knots?!
We can ask the same question in higher dimensions. A high-dimensional knot is an n-dimensional sphere living in (n+2)-dimensional space. And we say that it is slice if it is the boundary of an (n+1)-dimensional disk (i.e. surface without holes) living in (n+3)-dimensional space.
That’s all a bit difficult to imagine. Mathematicians think so too, so they have developed a range of algebraic techniques to help them answer the question. That means they can play with matrices (which they understand) rather than high-dimensional space (which they don’t). The matrices which they use basically contain information about how all the different holes in a surface interact with each other. This information helps the mathematicians to tell whether the holes can be got rid of.
Here comes the interesting bit. (And it’s where I come in!) Imagine the algebra as a creature (played to perfection in the seminar by McHaggis). The algebra (i.e. the matrices) is clever, but it is also very gullible. If a knot came up to it in the street and said “Hey baby, I’m slice”, the algebra would go “Wa-hey, let’s have a party!”.
Luckily for the algebra, in high dimensions the knots always tell the truth. In other words, the information in the matrices is always enough to be able to tell if the knot is slice.
But when n=1 (i.e. with our own familiar 3D knots) we have a problem. Knots can lie! People didn’t figure this out until the 1970s and it really threw them. They needed to find a bigger and badder creature to deal with the problem and spot the deception. I played the role of this topology monster, protecting McHaggis from those sneaky knots which he thought were slice but actually weren’t.
Julia’s thesis is all about developing these topological techniques to spot the sneaky knots, and she’s managed to catch quite a few out! However, there’s a long way to go before we will be able to spot all of them. There are different levels of sneakiness – infinitely many levels in fact! Julia can only deal with the first level of sneakiness, but there are mathematicians (for example, Julia’s mathematical brother Mark Powell) who have their ambitions set much higher.
I find it especially interesting that all the knots are simple and truthful except for the knots that live in dimension 3, and which are being pushed into the 4th dimension. Is it a coincidence that we live in a 3 (+ time = 4) dimensional universe? Surely not! I certainly like to believe that we live in 3 (+1) dimensions precisely because those are the dimensions which are the most complex and intricate mathematically.
Next week Julia and McHaggis will be heading off to sunny Indiana in the USA to visit the famous knot theorist Chuck Livingston. Beware, any sneaky knots that are out there! You will be found!