lacunary functions and intersecting three dimensional cobordism categories

I have made it a habit of keeping really good notes and records the past few years. I notice when I go along certain routes. In particular, one sort of frequently traveled path is starting to look extremely interesting, except that I can’t say why it is, or speak much to why one thing reflects on the other.

I’m talking here about lacunary functions and intersecting three dimensional cobordism categories. They oddly seem to demand adjacent parts of mental machinery.

I’ve taken to making a large list of lacunaries and making phase portraits of them. Here’s a good one:

\prod_{n=1}^{\infty} \left(1 - \frac{q^{2^{n}}+q^{n^{2}}}{1-q^{2^{n}}}\right)
\prod_{n=1}^{\infty} \left(1 - q^{n^2} - 2q^{2^{n}}\right)

Both of these lacunary functions have exactly the same roots!

At the same time I’ve been thinking about intersecting three dimensional cobordism categories, with
freehand illustrations below. (I haven’t figured out how to automate the production of these
diagrams with tikz yet). And what is striking is that I’ve been in about this mental territory where
I am thinking about both lacunary functions and these sorts of intersecting cobordism diagrams at the same time.

One thing I might try next is to (a la Arthur Dent and scrabble tiles style), draw one intersecting cobordism
diagram, and then make a phase portrait for a lacunary function while looking at the cobordism diagram that I have
just drawn. Perhaps there is a functor between the two, but it’s not one I can get at by tedious algebraization
of the diagrams (I am not going to do this, I really don’t care about the egregious algebraization of everything), but
I prefer a synaesthetic approach.

The green and blue ones are knotted as in the Alexander Horned Sphere:


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