So I am running a math club for 8 year olds and we are reinventing (an) algebraic knot theory. We are not trying to reinvent existing knot theory, just to make a journey of intellectual and mathematical discovery. In the first meeting I gently shocked the children by showing them that mathematics is not only about numbers, but about patterns in general — such as knots. Being only 8 they are very theatrical in expressing emotion. WHAT!? Maths about KNOTS!? I loved the wild face expressions and hand gestures. We spent most time making and drawing knots and concluded that drawing knots is quite hard. We needed a better notation.
A notation for knots: the knotation
One preliminary observation that I made which resonated well was that there are a lot of numbers out there but we only need 10 symbols to represent all of them: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Of course we could only use 0 and 1, or just 1, but let us not get distracted. It is about having a nice and convenient notation not just any notation. So I drew a couple of knots on the whiteboard and I encouraged the children to try to identify the “smallest pieces the knot is made of”, where by a “piece” I meant this: we draw a small circle on the whiteboard and the “piece” is the bit of knot we see inside the circle. If our circles are small enough it is easy to see that the pieces inside the circles are quite similar. So here is an overhand knot with some parts of it circled.
- We notice that even if we ‘zoom in’ on a knot piece, as in the case of the red circles, the interesting knot piece ‘looks the same’:
- We notice that if a knot piece is too large, as in the case of the blue circle, it can be clearly decomposed into smaller basic knot pieces.
- We also notice that many knot pieces which we select here and there look quite similar, as in the case of the green circles.
I don’t think there is a clear methodology one can use in inventing a good notation for a concept or set of concepts. It is a fun and creative endeavour. It is a design exercise in which many criteria must be balanced. On the one hand we have expressiveness, which means that we can use the notation to represent a large range of concepts. But we also have elegance, which is more subjective but not entirely arbitrary. We want a small (but not too small) set of notations. We want it to be also concise. We want it to be pretty.
So we explored a while and in the end, with some careful guidance, we narrowed in on these constants:
The names C and X were suggested because of the shape of the knot piece they represent. L was short for “line”. These shapes can be flipped over and result in 3 other constants:
Here there was some disagreement between me and the kids. They were happy with the flipping operation and they were happy with X* but they didn’t like C* because it looked more like a D (without the vertical bar). Some of them insisted that we call them D and X*. L* was clearly a non-issue because it was just like L. I had to put my foot down and we settled on C, X, L and the -* operator.
This exercise also presented us with our first equation: L = L*. We didn’t insist on equations because they will become more important in a few weeks.
The final part of the hour was dedicated to developing notations for assembling knots out of these constants. There are two obvious ways to compose knot pieces, horizontally and vertically. Here is how the overhand knot can be split into basic components:
How do we put them together? The bits at the top can be quite clearly written as X⋅X⋅X. The bit below seemed like C⋅C*. And at the bottom we have L. Because the bit at the bottom is quite long many kids wrote it as L⋅L or L⋅L⋅L but they quickly realised our second equation: L = L⋅L. This observation led to an unfortunate turn of events in which the kids decided that L is like 0 because added with itself stays the same, and therefore changed it from L to 0. Next week we will need to change it to the correct one (1) even if I need to exercise my executive power.
This is where we ran out of time. So we didn’t have any good notation for vertical assembly. Because we didn’t have vertical assembly we couldn’t really test our notation — to the trained eye it’s quite obvious it is not quite good yet. The way C, C* and L are composed is not entirely clear because vertical and horizontal composition interact. But it is a start. I want the children to move away from how they are used to study mathematics, as a stipulated set of rules in which answers are either correct or incorrect. I don’t want them to execute the algorithm, but to invent the programming language in which it is written. Coming up with stuff that doesn’t work is OK, as long as we notice it doesn’t work and we improve on it.
But what did we accomplish? Quite a lot. In one hour a dozen 8 year olds rediscovered some of the basic combinators used in monoidal categories, such as the braid, or such as duality, unit and co-unit as used in compact closed categories. We also rediscovered composition and its identity, and some of its equations. We used none of the established names and notations, but we discovered the concepts. We are on our way to inventing a fairly original categorical model of knots and braids. Next week we will quickly figure out the monoidal tensor and move on to the really exciting bit: coherence equations.