Defining topology through interviews. Interview four with Tom Hockenhull.

The next of my  Defining topology through interviews  series is with Tom Hockenhull. Tom is a PhD student in mathematics at Imperial College London and his research is in knot theory. You should also check out the cool topology themed comics he does for chalkdust magazine!

1. What do you say when trying to explain your work to non-mathematicians?

I’m in the odd position that my field gets more intelligible for non-mathematicians the more specific I am (well, to a point). It’s actually quite hard to capture the idea of ‘topology’ in enough generality, I find – but it’s rather easy to explain the subfield of topology I work in, which is knot theory. I usually talk about unknot recognition. If I get a piece of string, tie a knot in it, and then fuse the ends together, I get a knot which is trapped in the string — I can’t untie it without cutting the string again.

creating-a-knot
Tie a knot and fuse the ends

The question is, if you give me a big mess of string, can I tell easily whether I can just straighten it out into a big unknotted loop without cutting the string at all? People can understand quite easily that this is a hard question in general.

knot-tangle
Can you make this into a simple loop of unknotted string without cutting it?

So how might I go about doing it? Well, one way to recognise if two things are different in general is to look at some easy to discern property of them and see if they’re different. For instance, if I’m trying to work out whether two insects are of the same species, I might look at how many legs they have.

bugs
Are these insects from the same species?

If I have a big tangle of string, I might try and work out some property of it that is different from that of a plain loop of string – then I can tell that it’s a different knot altogether. What’s the analogous notion for knots? Well, this is sort of what my work is in: there are a whole bunch of different properties we might try and use to compare knots. One which is easy to understand is the property of being able to colour a picture of the knot using exactly three colours (no less!), so that at each crossing I have three different colours, or all the colours are the same.

coloured-crossing
Colouring a crossing using exactly three colours

You can see that I can’t colour a simple loop of string with three colours (only one) – but I can colour the picture of the knot below.

colouring
However you draw an unknotted loop you can’t colour it using three colours
colouring-trefoil
You can colour this one with exactly three colours!

It follows that I can’t turn one into the other without cutting the string!

2. What would your own personal description of ‘topology’ be?

There’s the standard ‘rubber sheet geometry’ or ‘the study of spaces up to deformation’, which are probably the most generally accurate, although they don’t really tell you much about what doing topology looks like or feels like. I suppose, though, that the problem is that ‘topology’ now encompasses a whole bunch of different areas that are rooted in the same place but are vastly different in their techniques, language and flavour.

3. How does your work relate, if at all, to the Nobel Prize work?

I’m not aware of any direct relevance – although that could be down to my ignorance! The word ‘quantum’ tends to pop up in descriptions of the Nobel work and in relation to a number of things to do with my work, but in my experience the use of the word quantum in my area seems to carry little relevance to its meaning in the world of physics.

Grid Diagrams for knots

Think of a knot as a twisted up loop of string. You can draw this as a picture with a line representing the string and the line breaking when one part of the string goes underneath another. Knot theory is the study of the topological properties of knots, and has been around since the late 1700s. In the 1860s Lord Kelvin conjectured that atoms were knots of aether and this led Peter Guthrie Tait to start drawing classification tables for knots. In fact Peter Guthrie Tait’s original research into the theory of these knot diagrams is said to have led to the birth of topology, now one of the main fields of pure mathematics. Here are some knot drawings from Tait’s classification of knots:

Tait knots

you can find some more here.

I was recently on a choir trip in Princeton and I decided to go to some math(s) classes. I went to a class by Peter Ozsváth titled ‘Algebraic Topology’ in which he taught the snake lemma and 5-lemma extremely well (lots of coloured chalk and enthusiastic diagram chasing). He then applied them to something the class had been learning: grid homology of knots. I wont try to explain this all in one post but I’ll show you how a grid diagram relates to a knot.

A grid diagram is a square grid, with boxes either filled in with an X, an O, or left blank such that:

  • every column has exactly one X and one O,
  • every row has exactly one X and one O.

The number of boxes in each row/column is called the grid number of the grid diagram. When all the Xs and Os are in place we join the X and O in the same row and the X and O in the same column with lines, letting the horizontal lines cross under the vertical lines. We will do an example to show how this represents a knot!

Example : a grid diagram with grid number 7.

Start with a grid (7 by 7) with Xs and Os as specified:

KN1

we see that really it looks like a very simple sudoku.

Join the Xs and Os with lines in each row/column, letting the horizontal lines cross under the vertical lines:

KD2

Forget the grid (!):

KD3

Forget the Xs and Os (!):

KD4

Morph to make it look a bit more ‘knotty’:

KD5

So there we have an example which illustrates the general transition from knot diagram to knot! You can trace this backwards to get a knot diagram for every knot but this was is a bit trickier as you have to get the Xs and Os to be one per row/column. In fact many different looking knot diagrams represent the same knot. Next time I’ll explain the grid homology you can define with these diagrams.

Below are some references for those interested: