Avishy Carmi and I are in the process of finalizing a preprint on what we call “tangle machines”, which are knot-like objects which store and process information. Topologically, these roughly correspond to embedded rack-coloured networks of 2-spheres connected by line segments. Tangle machines aren’t classical knots, or 2-knots, or knotted handlebodies, or virtual knots, or even w-knot. They’re a new object of study which I would like to market.
Below is my marketing strategy.
My positioning claim is:
- Tangle machines blaze a trail to information topology.
My three supporting points are:
- Tangle machines pre-exist in a the sense of Plato. If you look at a knot from the perspective of information theory, you are inevitably led to their definition.
- Tangle machines are interesting mathematical objects with rich algebraic structure which present a plethora of new and interesting questions with information theoretic content.
- Tangle machines provide a language in which one might model “real-world” classical and quantum interacting processes in a new and useful way.
Next post, I’ll introduce tangle machines. Right now, I’d like to preface the discussion with a content-free pseudo-philosophical rant, which argues that different approaches to knot theory give rise to different `most natural’ objects of study.
In what sense do knots `pre-exist’?
It’s not an uncommon occurance for me to be sitting somewhere, maybe at a synagogue, and for somebody, maybe a lawyer or a doctor or a trader of some sort, to come up to me and to ask me what I do for a living.
“I study knots,” I reply.
It takes them a moment to overcome their surprise, and then they stare at me as though I’ve just told them that I’m a specialist on big toes of dung beetles.
“But… why?” they ask.
They have no difficulty explaining their object of interest. But knots? Why knots? Do I want to tie ships to their moorings more securely? What am I up to?
Why would a mathematician study knots?
The answer doesn’t seem to me to be obvious. I might tell them that knot theory is a part of topology. In topology, we classify shapes up to bla bla bla, and a smooth embedding of in is a shape… a very special shape. And now I need to explain why topology (which requires a lot of mathematical background to set up), and why this special little sub-corner of topology. And I think that I’d fail.
So really, why knots?
Let’s survey the history.
At the beginning of the 20th century, knot theory was a playground for algebraic topology’s new toys like the fundamental group, Riedemeister-Schreier, and homology. It was a “simplest non-degenerate example” of sorts, a family of objects against which new theories could easily and efficiently be tested. I’m not sure that people like Tietze, Seifert, Reidemeister, and Schubert cared about knots for the sake of knots- it seems to me that for them, knots were knot complements and knot complements were baby 3-manifolds, and that is all.
The following generation, it seems to me, viewed knot theory as a baby example in the theory of embeddings and in surgery theory (rather than 3-manifold topology). So they studied knots alongside higher dimensional embeddings of one manifold inside another, including n-knots. Again, knot theory was nothing more than a simplest non-degenerate example. The `real category’ which pre-existed Platonically was much larger, and not necessarily “low dimensional”.
And maybe knots are just links with a single component? Why is the properly of being connected enough to single them out as a prefered object of study? For many problems, it obviously isn’t.
Next comes quantum topology- now here is something new! Suddenly knots were telling us facts about multiple zeta values, Lie algebras, quantum groups, and all sorts of “good math”! But the knots of quantum topology were no longer the familiar smooth embeddings of a circle in 3-space. Instead, a knot was now a network of crossings in a plane- a knot diagram- modulo a set of local combinatorial moves called Reidemeister moves.
The knots of quantum topology emerge as diagrammatic algebraic objects generated by crossings. They are special tangles with one connected component and no loose ends. A tangle is defined inductively- the empty set is a tangle, a crossing is a tangle, a disjoint union of tangles is a tangle, and a “multiplication”, that is, connecting one loose end of a tangle with another loose end by a line in the plane, is also a tangle.
Looked at this way, the planarity constraint is artificial, and we might as well allow arbitrary multiplication- the knot is generated by crossings (that is, by 4-valent vertices whose adjacent edges are ordered into an “in under”, an “out under”, an “in over” and an “out over”, themselves labeled “positive crossing” or “negative crossing”), and the connections between the crossings are combinatorial data (a matching between over/under strands of crossings, like edges of a graph) with no planarity condition. That’s what a virtual knot is. It seems that the “knots” of quantum topology really ought to be virtual knots.
Virtual knots are, I think, a manifestation of the paradigm that algebra is a more general concept than a sequence of symbols. Matrices and tensors are algebraic objects, and they are not rows of symbols. Similarly, graphs are algebraic objects, and maybe simplicial complexes are as well. If we allow algebra to be written in diagrams, then it makes sense to ask about the algebraic content of the concept of a knot (which we know that it has, thanks to quantum topology), so that a knot is a knot diagram, i.e. a special sort of a tangle, i.e. a network of crossings, and then at that rate the natural algebraic objects are the virtual knots, in which no artificial non-algebraic planarity requirement is imposed. Analagously, planar graphs are wonderful objects- but the fundamental category in graph theory is the category of graphs and graph homomorphisms.
Virtual knots have a topological interpretation as knots in thickened surfaces, but it’s topologically unappealing because the surface might have to be stabilized multiple times in the course of an isotopy. So they’re “knots in thickened surfaces up to stabilization”, which is a cumbersome topological object. And yet, algebraically, virtual knots are great. They require much less baggage to define than knots. Except for the fact that their algebra is crazy complicated and that quantum topology uses constructions like the Kontsevich invariant which depend on analysis over a field and need the knots to be topological, from a conceptual point of view it seems that quantum topology should really be studying virtual knots.
But now, go justify such objects to that bloke from law, or from medicine, or from finance. Or to yourself!
The difficulty is that the concept of a knot, while having clear topological content, requires a lot of background (R^n, diffeomorphisms, isotopy, embedding,…) which we also have to believe in, and the concept of a virtual knot, while having clear algebraic content, selects one diagrammatic algebra from among infinitely many possibilities (why crossings as generators? Why Reidemeister moves?) and then claims that this is somehow natural. I don’t really understand why, on a conceptual level; not why knots pre-exist and not why virtual knots pre-exist. But, at least for knots, I believe that they do. One sees knots all around- knotted shoelaces, knotted wires, knotted hair- and it seems to me that these things have an existence independent of smooth or PL topology, in the same way that numbers have an existence independent of the particular axiomatic framework used to define them. Knots are certainly more fundamental than any of the plethora of fine tools that we use to study them (algebraic topological constructions of various sorts, Lie algebras, and so on). For virtual knots, I’m still not convinced.
My first supporting point for tangle machines is that they pre-exist. This isn’t a statement which one can hope to prove, and certainly it can’t be tested. But it is a selling point for the idea- I want to care about tangle machines because they exist, not because they help to solve problems about objects whose existence I am less convinced of.
And next post, I’ll tell you what a tangle machine is.