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.

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.

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.

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.

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.

However you draw an unknotted loop you can’t colour it using three colours
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.


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