Tangled Up in Low-Dimensional Topology

Knots and tangles. In a post about “tangle machines,” Daniel Moskovich imagines trying to explain to an educated non-mathematician what he studies: “Why knots? Do I want to tie ships to their moorings more securely?”
Image: public domain, from Nordisk Familjebok, via Projekt Runeberg, and Wikimedia Commons.

Low Dimensional Topology is, logically enough, a blog about low-dimensional topology. Authors Ryan Budney, Nathan Dunfield, Jesse Johnson, Daniel Moskovich, and Henry Wilton write about 2-, 3-, and 4-manifolds, knot theory, quantum topology, and more Heegaard splittings than you can shake a stick at, if you are in the habit of shaking sticks at Heegaard splittings. Posts are expository but aimed at other topologists, and the authors often cover recent results in their fields. They’ve also written about a number of open problems and keep an up-to-date list of relevant conferences.

Last month Moskovich started a series of posts about “tangle machines,” the subject of a paper he is working on with Avishy Carmi. “Tangle machines aren’t classical knots, or 2-knots, or knotted handlebodies, or virtual knots, or even w-knots. They’re a new object of study which I would like to market,” he writes. This idea of a mathematical idea as a product to be marketed is a bit foreign, but it’s an interesting exercise to think about what your own “marketing strategy” might be for your specific research topic.

Before diving into the glories of tangle machines, Moskovich writes, “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.” (Personally, I don’t think what follows is cranky enough to be called a rant, but that’s just me.) The rest of the post helped me understand how knots became such an important topic in low-dimensional topology in the first place. In the next post in the series, Moskovich actually defines tangle machines and talks a bit about why he finds them more natural than knots. In future posts, I’m interested in seeing what’s up with Reidemeister moves in tangle machines versus knots.

The LDT archives go all the way back to 2007, but I just want to point out a couple other recent articles I found interesting. In one, Henry Wilton asks, “When are two hyperbolic 3-manifolds homeomorphic?” Although the two manifolds in question, which came from an arXiv paper by Lins and Lins, ended up being relatively easy to tell apart, the advertisement for the Scott-Short algorithm and the comments are quite interesting. I also liked Ryan Budney’s post about the algorithm to recognize the 3-sphere. On many LDT posts, the mathematician whose work is being discussed will often chime in in the comments, clarifying a point or expanding on an idea. So you can read the comments without fearing for your sanity!

It’s always good to see research-level math written in a way that gives more of a big picture overview than most journal articles do and sometimes even offers glimpses into what the hard parts were. The blog is definitely geared toward the research topologist, but other mathematicians can listen in and get a feel for what’s going on in this corner of the mathematical world.