Daniel Moskovich has started a thread on what he calls: “Tangle Machines.”
The series started with Tangle Machines- Positioning claim.
From that post:
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.
You know where Daniel lost me in his sales pitch. 😉
But, it’s Daniel’s story so read on “as though” knots “pre-exist,” even though he later concedes the pre-existing claim cannot be proven or even tested.
Tangle Machines- Part 1 by Daniel Moskovich begins:
In today’s post, I will define tangle machines. In subsequent posts, I’ll realize them topologically and describe how we study them and more about what they mean.
To connect to what we already know, as a rough first approximation, a tangle machine is an algebraic structure obtained from taking a knot diagram coloured by a rack, then building a graph whose vertices correspond to the arcs of the diagram and whose edges correspond to crossings (the overcrossing arc is a single unit- so it “acts on” one undercrossing arc to change its colour and to convert it into another undercrossing arc). Such considerations give rise to a combinatorial diagrammatic-algebraic setup, and tangle machines are what comes from taking this setup seriously. One dream is that this setup is well-suited to modeling mutually interacting processes which satisfy a natural `conservation law’- and to move in a very applied direction of actually identifying tangle machine inside data.
To whet your appetite, below is a pretty figure illustrating a 926 knot hiding inside a synthetic collection of phase transitions between anyons (an artificial and unrealistic collection; the hope is to find such things inside real-world data):
Hard to say if in the long run this “new” data structure will be useful or not.
Do stay tuned for future developments.