Today we are going to learn some basic theory about slice knots. (Don’t worry – the relationship between this and the first two posts will become clear in due course!)
Every knot can be constructed as the edge of a 2-dimensional surface. For example, the trefoil:
(The wonderful program SeifertView is free to download and will produce a surface for any knot you give it.)
You will notice that this surface has a hole in the middle of it. It turns out that only an unknotted piece of string can bound a surface with no holes in it (such a surface is called a disc). But what if we had another dimension of space in which to play? Would that affect the result?
The 4th dimension is hard, if not impossible, to picture, so let’s start with a 3-dimensional analogy. If we have a curve in 2 dimensions that intersects itself, it creates a kind of ‘hole’ in space. But if we push the curve into the third dimension then the intersection point goes away and so does the hole:
The question is: is it possible to remove holes in surfaces by pushing them from 3 dimensions into 4 dimensions? In the case of surfaces bounded by knots, the answer is: sometimes. These special knots are called slice knots.
It is, in general, very difficult to decide whether a knot is slice or not. Mathematicians have worked out one way of telling if a knot is not slice: compute a number, called a signature, and if it is not zero then the knot is not slice. If you know a little bit about matrices, read on and I will tell you how to compute the signature.
First, take your knot and find a surface that it bounds. (This is called a Seifert surface). Then, draw a curve on the surface around every hole that you see, like this:
We want to make a matrix that contains all the data of how these different curves link and knot around each other. First, give each curve a number, then push each curve off the surface in turn and calculate how many times it links with all the other curves. We enter the linking number of the pushed off curve i with another curve j into the (i,j)th entry of our matrix. For example:
For this knot, our matrix is
Now, whenever you have a square matrix you can compute a set of numbers called eigenvalues. These are numbers λ that satisfy the equation Mv=λv, where M is our matrix and v is some vector. (If you would like to learn more about eigenvalues, take a look at the Wikipedia article.)
The signature of a matrix is
(the number of positive eigenvalues) – (the number of negative eigenvalues)
and it is a deep theorem that slice knots will always have signature equal to zero.
Next time we will learn about the signature of torus knots, and we will see how they are tied in to the number theory we learnt about in the first two parts of this blog-series. Stay tuned!