Here is a quote that captures the ethos against which our maths club militates:
We take other men’s knowledge and opinions upon trust; which is an idle and superficial learning. We must make them our own. We are just like a man who, needing fire, went to a neighbor’s house to fetch it, and finding a very good one there, sat down to warm himself without remembering to carry any back home. What good does it do us to have our belly full of meat if it is not digested, if it is not transformed into us, if it does not nourish and support us? – Montagne, “Of Pedantry”.
This is how mathematics is usually delivered to students: knowledge upon trust. And it is knowledge of a particularly mystical variety, where magical symbols come together in incomprehensible ways. This is the myth we need to dispel, this is the knowledge we need our kids to digest. Sometimes Algebra can be nothing more than a way to write down knots and their properties.
Last session, using the notation we developed together, went well. One unexpected hick-up was the fact that my students had not seen parentheses before, but we took that in our stride. They also hadn’t use variables before, so most of the examples were concrete. But the little use of variables we made was not confusing.
I even introduced some proper maths terminology, the Big Maths Words. First I asked them to give names to the composition operation, and they came up with some reasonable suggestions such as gluing, linking or joining. When I told them the Big Math Word was composition they liked it — it was “like music”, where you “compose” a song from notes. Indeed! They also had reasonable suggestions for the tensor, things such as stacking or building, but they didn’t much like the Big Math Word tensor. I don’t like it very much myself, I confess.
For the first 20 minutes we went over the new and tidied up notation. The rest of the time, 30 minutes or so, we worked out examples of going from a drawn knot to the formula. It is not easy, in fact it is quite a challenge (try the exercises) so they often got things wrong. One pervasive error was getting confused by repeated tensoring (“stacking”) of tangles, K⨂K’⨂K” which often was written as K⨂K’K” — an interesting error.
Tomorrow’s session we will finally reach equations. We have already seen the equation I*=I, which was noticed while developing the notation. First we will start with similar coherence equations, which basically mean that two notations denote the same knot. Things like associativity of composition or tensor are typical coherence equations. But because children have no appreciation of parantheses and operator precedence associativity is perhaps not the best example. Functoriality of the tensor is much more interesting.
- Draw the tangle for (C◦C*)⨂(C◦C*).
- Draw the tangle for (C⨂C)◦(C*⨂C*).
The unit and the co-unit also suggest compact-closed-like coherence axioms, which have a more topological flavour. Try this knot: (I⨂C)◦(C*⨂I). It looks like this:
But this is only a wire with some irrelevant bends. We have the equation
(I⨂C)◦(C*⨂I) = I
There is another similar equation, where the shape is like an ‘S’ rather than like a ‘Z’. Can you guess it?
The trove of equations is deep for this algebraic theory of knots and tangles and there is plenty of scope for children to discover equations once they understand what they represent. In addition to compact-closed we can also find equations from traced, braided and closed-monoidal categories.
But most importantly, I think the point is not to just give such equations to the students in order for them to develop skill in calculating knot equivalences using equational reasoning, so I won’t do that. That’s the kind of mystical maths I want to avoid. What is important are these two points:
- Understanding the idea of equational reasoning, that the same structure can be expressed in several equivalent ways, and that equations are the mathematical way to show this.
- Exploring the space of equations. Ideally, after a while the issues of soundness and completeness should emerge naturally.
Now, a brief comment to the cognoscenti. The categorical structure of this algebra of tangles is not an obvious one. We are using both the unit and the co-unit of compact closed categories, but our category is braided rather than symmetric. However, compact closed categories are symmetric so we are not in an obvious setting. The interaction of braiding and “compact closure” (informally speaking) gives equations like this:
C◦X◦C* = C◦C*
which correspond to this topological isomorphism of tangles:
These are two ways to describing the unknot. The category is clearly non-degenerate because we seem to be able to describe any knot. So combinatorial completeness seems an achievable result. Equational completeness (i.e. any equivalent knots can be proved to be so) however seems like a challenging and interesting question!
In terms of related work, this seems similar enough: Peter Freyd and David Yetter, Braided compact monoidal categories with applications to low dimensional topology, Adv. Math. 77, 156-182, 1989.
If you are interested in another diagrammatic explanation of algebra also have a look at Pawel Sobocinksi’s entertaining and informative blog.