# Low Dimensional Topology

## July 8, 2013

### Tangle Machines- Part 1

Filed under: Combinatorics,Misc. — dmoskovich @ 11:27 am

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 $9_{26}$ 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):

### Tangle Machines- Positioning claim

Filed under: Combinatorics,Knot theory,Misc.,Quantum topology — dmoskovich @ 11:09 am

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:

1. 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.
2. Tangle machines are interesting mathematical objects with rich algebraic structure which present a plethora of new and interesting questions with information theoretic content.
3. 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.

## June 21, 2013

### Lots and lots of Heegaard splittings

Filed under: 3-manifolds,Heegaard splittings,Knot theory — Jesse Johnson @ 12:28 pm

The main problem that I’ve been thinking about since graduate school (so around a decade now) is the following: How does the topology of a three-dimensional manifold determine its isotopy classes of Heegaard splittings? Up until about a year ago, I would have predicted that most three-manifolds probably don’t have many distinct Heegaard splittings, maybe even just a single minimal genus Heegaard splitting and then all of its stabilizations. Sure, plenty of examples have been constructed of three-manifolds with multiple distinct (unstabilized) splittings, but these all seemed a bit contrived, like they should be the exceptions rather than the rule. I even wrote a blog post a couple years back stating what I called the generalized Scharlamenn-Tomova conjecture, which would imply that a “generic” three-manifold has only one unstabilized splitting. However, since writing this post, my view has changed. Partially, this was the result of discovering a class of examples that disprove this conjecture. (I’m hoping to post a preprint about this on the arXiv in the near future.) But it turns out there is an even simpler class of examples in which there appear to be lots and lots of distinct Heegaard splitting. I can’t quite prove that they’re distinct, so in this post I’m going to replace my generalized Scharlemann-Tomova conjecture with a conjecture in quite the opposite direction, which I will describe below.

## May 31, 2013

### The algorithm to recognise the 3-sphere

Filed under: 3-manifolds,Computation and experiment,Triangulations — Ryan Budney @ 10:48 am
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The purpose of this post is to convince you the 3-sphere recognition algorithm is simple.  Not the proof!  Just the statement of the algorithm itself.  I find in conversations with topologists, it’s fairly rare that people know the broad outline of the algorithm.  That’s a shame, because anything this simple should be understood by everyone.

## May 19, 2013

### Flooved

Filed under: Misc. — dmoskovich @ 8:52 am

There’s a startup company in the UK, called Flooved, who are on a mission to revolutionize scientific publishing. What sets them apart from many similar-sounding initiatives is that they seem to have a solid business model and they seem to be doing all of the right things, therefore my bet is that they are going to succeed.

What they do is to compile existing lecture notes, handouts and study-guides, and along the lines of the Open Access movement, to make them freely available online. The advantage to students is clear. The advantage to instructors is that more people read and use the material. The advantage to publishers who contribute content (are you listening, big publishing companies?) is that they get precise and useful information on how the students are using their content, and this helps them make informed decisions to put them ahead of the competition. Beyond this, the Flooved model makes education available to people worldwide, including to people who don’t have access to universities. Now, if only they could also provide assessment and accreditation…

## May 17, 2013

### An old corker on the unknotting of knots

Filed under: Knot theory — Ryan Budney @ 11:13 am
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I imagine many readers of this blog are familiar with the fact that you can knot a circle in 3-space, but not in 4-space.    If you enjoy thinking about why that is true, please read on!

Think of euclidean 3-space, $\mathbb R^3$ as a linear subspace of euclidean 4-space, $\mathbb R^3 \equiv \mathbb R^3 \times \{0\} \subset \mathbb R^4$.  So if you have a knotted circle in 3-space, you can consider it as an embedded circle in 4-space.  And you can unknot it! I think one of the simplest explanations of of this would be the idea to push the knot up into the 4-th dimension every time a strand is close to being an overcrossing (in a planar diagram).   At this stage you could in effect change the crossing to be anything you want, after you’re done modifying the crossings, you could push the knot back into 3-space to get a different knot.

## May 16, 2013

### Organizing knot concordance

Filed under: 3-manifolds,4-manifolds,knot concordance — Ryan Budney @ 10:10 am
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I have a rather naive question for the participants here.  I’m at the Max Planck 4-manifolds semester, currently sitting through many talks about knot concordance and various filtrations of the knot concordance group.

Do any of you have a feeling for how knot concordance should be organized, say if one was looking for some global structure?    In the purely 3-dimensional world there are many very “tidy” ways to organize knots and links.  There’s the associated 3-manifold, geometrization.  There’s double branched covers and equivariant geometrization, arborescent knots and tangle decompositions.  I find these perspectives to be rather rich in insights and frequently they’re computable for reasonable-sized objects.

But knot concordance as a field feels much more like the Vassiliev invariant perspective on knots: graded vector spaces of invariants.  Typically these vector spaces are very large and it’s difficult to compute anything beyond the simplest objects.

My initial inclination is that if one is looking for elegant structure in knot concordance, perhaps it would be at the level of concordance categories.  But what kind of structure would you be looking for on these objects?   I don’t think I’ve seen much in the way of natural operations on slice discs or concordances in general, beyond Morse-theoretic cutting and pasting.   Have you?

## April 23, 2013

### When are two hyperbolic 3-manifolds homeomorphic?

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry — Henry Wilton @ 7:46 am

A preprint of Lins and Lins appeared on the arXiv today, posing a challenge [LL].  In this post, I’m going to discuss that challenge, and describe a recent algorithm of Scott–Short [SS] which may point towards an answer.

The Lins–Lins challenge

The theory of 3-manifolds is now very advanced, and we can even say in a certain sense that we understand ‘all’ 3-manifolds (as I discussed in an earlier post).  But that understanding is very theoretical; the Lins–Lins challenge is to put this theory into practice.

They ask: ‘Are the two closed, hyperbolic 3-manifolds given by Dehn surgery on the following two framed links homeomorphic?’

(I’ve taken the liberty of copying the diagrams from their paper.)

## April 20, 2013

### The next big thing in quantum topology?

Filed under: 3-manifolds,Hyperbolic geometry,Quantum topology,Triangulations — dmoskovich @ 11:02 pm

The place to be in May for a quantum topologist is Vietnam. After some wonderful-sounding mini-courses in Hanoi, the party with move to Nha Trang (dream place to visit) for a quantum topology conference.

I’d like to tell you very briefly about some exciting developments which I expect will be at the centre of the Nha Trang conference, and which I expect may significantly effect the landscape in quantum topology. The preprint in question is $1$-Efficient triangulations and the index of a cusped hyperbolic $3$-manifold by Garoufalidis, Hodgson, Rubinstein, and Segerman (with a list of authors like that, you know it’s got to be good!). (more…)

## April 6, 2013

### New connection between geometric and quantum realms

Filed under: Hyperbolic geometry,Knot theory,Quantum topology — dmoskovich @ 9:41 am

A paper by Thomas Fiedler has just appeared on arXiv, describing a new link between geometric and quantum topology of knots. http://arxiv.org/abs/1304.0970

This is big news!! (more…)

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