As we go about our days at university, between tutorials, practicals, labs, lectures, seminars and miscellaneous other classes, it’s all too easy to forget that teaching is only one part of the job performed by the various academics from whose expertise we are so lucky to benefit. It’s easy to think of professors and lecturers as nothing but the people who stand up at the front of a room and talk. For some, they’re a disembodied voice accompanying a set of recorded PowerPoint slides.

A miraculous discovery one might make upon taking up the offer of office hours, is that these staff do not simply walk off and cease existing once class is done; they have offices where, yes, they might write us an email and mark our exams, but, more importantly, they carry out their research.

But what does that mean, carrying out research? What does one do research in? And how do you do it? That’s what I set out to answer, at least in part, by taking part in the Winter Research Program at UQ this past mid-year break.

It all started with perusing the list of projects available and deciding to apply for one entitled “Knots and their Grid Diagrams”, supervised by Dr Daniele Celoria. I had, prior to this, a passing familiarity with knots due to having tied my own shoes for almost two decades and thought that a grid might be a good gift idea for a square enthusiast.

Of course, being a student of mathematics, I’d had enough run-ins with strange definitions that I knew well enough to look up what these words meant precisely before I went through with the project.

To arrive at a mathematical knot, begin with a string (or rope, or twine, or anything else you’d like to inevitably spend an afternoon untangling) and twist it into whatever
‘knotted’ shape you’d like. Now, join the two loose ends together to make a continuous loop, and make the whole strand have zero thickness.

At a certain point there we did leave the realm of the physical, but in some ways that’s the point; we mathematicians aren’t concerned with any physical properties like strength or likelihood of coming undone just when you’re walking eight and a half bags of groceries up two flights of stairs with an excited puppy yapping at your ankles.

Then, a grid diagram is one way of representing a (mathematical) knot. The specifics here aren’t too important, but the gist is that we lay the knot flat and so that every time the string passes over itself, it does so in a vertical direction, and every time it passes under, it goes horizontally. This can now be very conveniently drawn in a grid.

There are all sorts of questions that a mathematically minded person might ask at this point, like ‘how can you tell two knots apart?’ or ‘how many different kinds of knots are there?’. These are very well-studied questions, with surprisingly complex answers. With only four weeks in the program, Daniele and I were interested in a smaller, simpler question: how many changes to places where the string crosses over itself are necessary to turn one knot into another?

There are, broadly speaking, two kinds of research in maths. In the first, one sits at one’s desk and thinks about the problem, potentially writing out some notes and scribbling away with tedious algebra trying to gain some insight that makes the problem a bit easier. In the second, one asks a computer very nicely to spend several days rattling through various possibilities until it finds an interesting result.

We used the latter strategy. That’s not to say that I didn’t spend several days sitting down and thinking about the problem — I had to write out a very nice request that the computer was capable of reading, of course, and spend far too long persuading it to function, but I stretched my critical thinking and creativity in ways I can’t say I’ve had to before. I had to very quickly acquire the skill of reading a lot of dense academic text, be it a textbook outlining the ideas behind Knot Theory and grid diagrams, or a paper with the most up-to-date results. I learned a lot about how to ask questions, and how to seek support from a mentor.

Doing research was really quite different from coursework. Typically, in an assignment or an exam we’re given an outline to follow or a specific goal to reach. ‘Find the maximum x such that y’ and ‘show that z is w’ are both really saying ‘use this procedure that you learned in class’. ‘Discuss the implications of a, b, and c’ is just ‘regurgitate and apply what we taught you four weeks ago, ideally adhering to the writing structures we used’.

That is, we found that to turn one specific knot into a different one, you only need to perform one crossing change — it was previously only known that it could be done with two
such changes. As I type that out it almost sounds pitiful, but that’s research. Yeah, sometimes you get a huge breakthrough, but we can’t all be Einstein.

I’m so glad I took part in Winter Research. I learned a great deal: about knots and grid diagrams, about Mathematics more broadly, about research, and about myself. It pushed me to properly think about pursuing Honours once I finish my degree at the end of 2025, which has now become a real goal I’m working towards. Who knows where the journey will go after that, but I’m now prepared to chart some new small mathematical territory of my own.