Relaxing from my forays into information and computation, I’ve recently been glancing through my mathematical sibling Kenta Okazaki’s thesis, published as:
K. Okazaki, The state sum invariant of 3–manifolds constructed from the linear skein.
Algebraic & Geometric Topology 13 (2013) 3469–3536.
It’s a wonderful piece of diagrammatic algebra, and I’d like to tell you a bit about it! (more…)
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 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):
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.
Gil Kalai, my old Graph Theory professor at Hebrew University, and a great mathematical inspiration, who won the Rothschild Prize a few weeks ago (congratulations Gil!), wrote a very nice blog post about another massive recent result in low dimensional topology.
There is a recent interesting question on MO regarding a paper by Benedetti and Ziegler which I found most interesting.
Upon reading the question, I right away downloaded and printed Benedetti and Ziegler’s paper, after which I sat down to glance through it. My first impression is that it constitutes a really high-class piece of mathematics; the exposition is clear enough that a non-specialist can sit down and enjoy it, and the results are deep and interesting. There’s something really inspirational about a paper like that. (more…)
A Jacobi diagram is surely the most tantalizing object in all of quantum topology. Its definition is combinatorial and easy to visualize, to understand, and to manipulate; and the smallest insight into the structure of the graded space of Jacobi diagrams would lead to huge rewards in quantum topology and in the study of Lie algebras. And yet, despite the concerted efforts of some of the finest minds in mathematics (for instance and in no particular order: Greg Kuperberg, Dror Bar-Natan, Alex Stoimenow, Oliver Dasbach, Pierre Vogel, Jerry Levine, David Yetter, Tomotada Ohtsuki, Justin Sawon, David Broadhurst, Maxim Kontsevich, and many others), they remain forever teasing, tempting, but forever maddeningly beyond reach.