Unlike previous posts, this one precedes the math club meeting. The reason is that we need to tighten some loose ends in our emerging knot theory notation. The first two meetings (I and II) were all about exploring knots and ways to describe them using notations, and we made some impressive strides. But at this stage I need to exercise some authority and steer them firmly in the right direction:
- We are going to use I rather than 0 (zero) to denote the identity of composition. Even though 0 is a neutral element for addition, as correctly spotted by the children, using it as the neutral element for composition is not idiomatic. 1 (one) is more standard, for various reason I will not get into here.
- Using a fraction-like notation for “parallel” (tensorial) composition is also quite unidiomatic, although it is intuitive. We shall introduce the standard ⨂ notation instead.
- We will stick with a generic duality operation _* so that our “unit” and “counit” are C and C* rather than C and D as some of the children insisted on.
These are of course small matters which we could put up with, at the expense of becoming incomprehensible to the rest of the world. The more important thing is that the interplay between “sequential” (“functional”) composition and “parallel” (“tensorial”) composition is not very easy to get right. Look at our overhand knot:
The individual components are well identified but a nice way to put them together is unclear. There is a sequential composition at the top (X◦X◦X) but how to connect that with the C, L and C* underneath is not clear from the decomposition. X “interacts” with C and C with L but not obviously sequentially or in parallel.
The way out is to introduce types. What are types? To paraphrase John Reynolds’s succinct explanation, they are a syntactic discipline that guarantees correct composition. And this is what we need here.
We shall always give knots a type (m, n) ∈ N². The left projection represents how many “loose ends” stick out to the left and the right projection how many to the right. So this is a (4, 6)-knot:
Note the implicit restriction: we allow no loose ends to stick out from the top or from the bottom. Is this a reasonable restriction? Yes, because if a bit of string goes down or up we can always “bend” it left or right then “extend” it until it lines up nicely with the other loose ends. This topological invariance of knots should be intuitively obvious to my young students, I hope.
So the types of our basic knot parts are:
Now we can say that we only allow the composition K◦K’ for knots K:(m,n) and K:(m’, n’) if n=m’, resulting in a knot K◦K’:(m, n’). Here are two composable knots:
And this is their composition, with “composition links” emphasised:
Note that because these are “strings” the loose ends don’t need to be “geometrically” aligned. It is enough to have the right number of loose ends to be able to glue them. And this is their actual composition:
Any two knots K:(m, n) and K’:(m’, n’) can be composed in parallel in a knot K⨂K’:(m+m’, n+n’). The composition of the two knots above is:
So now everything is in place to properly describe algebraically our overhand knot:
Above we introduced -•- as a shortcut for repeated sequential composition, useful for concision. Note that we needed the identity string I not just on the bottom, but also to the left and to the right, in order to make the loose ends match properly in sequential composition. This kind of “padding” with identities will be a useful trick that will allow us to define a lot of knots compositionally!
- (easy) Is this the only way to describe an overhand knot? Give an alternative description.
- (easy) Is this the only kind of overhand knot? If no, write the definition of a different overhand knot.
- (medium) Write the definition of a Granny Knot and its variations: the reef, the thief and the grief knots.
- (medium) Write a definition for the Shoelace Knot.
- (hard) Is there a knot that cannot be described with our notation?