Low Dimensional Topology

October 13, 2013

A noteworthy knot simplification algorithm

Filed under: Computation and experiment,Knot theory — dmoskovich @ 8:26 am

This post concerns an intriguing undergraduate research project in computer engineering:

Lewin, D., Gan O., Bruckstein A.M.,
TRIVIAL OR KNOT: A SOFTWARE TOOL AND ALGORITHMS FOR KNOT SIMPLIFICATION,
CIS Report No 9605, Technion, IIT, Haifa, 1996.

A curious aspect of the history of low dimensional topology are that it involves several people who started their mathematical life solving problems relating to knots and links, and then went on to become famous for something entirely different. The 2005 Nobel Prize winner in Economics, Robert Aumann, whose game theory course I had the honour to attend as an undergrad, might be the most famous example. In his 1956 PhD thesis, he proved asphericity of alternating knots, and that the Seifert surface is an essential surface which separates alternating knot complements into two components the closures of both of which are handlebodies.

Daniel Lewin is another remarkable individual who started out in knot theory. His topological work is less famous than Aumann’s, and he was murdered at the age of 31 which gives his various achievements less time to have been celebrated; but he was a remarkable individual, and his low dimensional topology work deserves to be much better known. (more…)

October 2, 2013

Regina 4.94

Filed under: 3-manifolds,Computation and experiment,Triangulations — Benjamin Burton @ 4:00 pm

It’s the season for it!  For those of you who work with normal surfaces, Regina 4.94 also came out last week.  It adds triangulated vertex links, edge drilling, and a lot more speed and grunt.

Take the new linear/integer programming machinery for a spin with the pre-rolled triangulation of the Weber Seifert dodecahedral space.  Regina can now prove 0-efficiency in just 10 seconds, or enumerate all 1751 vertex surfaces in ~10 minutes, or (with a little extra code to coordinate the slicing and searching for compressing discs) prove the entire space to be non-Haken in ~2 hours.

Read more of what’s new, or download and tinker at regina.sourceforge.net.

September 30, 2013

SnapPy 2.0 released

Marc Culler and I pleased to announce version 2.0 of SnapPy, a program for studying the topology and geometry of 3-manifolds. Many of the new features are graphical in nature, so we made a new tutorial video to show them off. Highlights include
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July 11, 2013

Smooth proof of Reidemeister-Singer

Every construction I know of 3-manifold invariants from Heegaard splittings factors through the Reidemeister-Singer Theorem:

Reidemeister-Singer Theorem: For any two Heegaard splittings H_1 and H_2 of a 3-manifold M, there exists a third Heegaard splitting H which is a stabilization of both.

This theorem is definitely part of the big story in 3-manifold topology, and is usually proven in the PL category, as for example in Nikolai Saveliev’s Lectures on the Topology of 3-manifolds. There is another nice PL proof due to Craggs, Proc. Amer. Math. Soc. 57, n 1 (1976), 143-147.

I think of a Heegaard splitting as being intrinsically a smooth topology construction (a level set of a Morse function), and so I would really like the proof of Reidemeister-Singer to live in the smooth category. I think that there should be consistent smooth and PL stories of 3-manifold topology living side by side. In the 1970′s, Bonahon wrote a smooth proof of Reidemeister-Singer, which uses Cerf Theory (naturally, because we’re investigating paths between Morse functions). Unfortunately, Bonahon’s proof was never published, and it is lost.

A year ago (but I only saw it this morning), François Laudenbach posted a smooth proof of Reidemeister-Singer to arXiv: http://arxiv.org/abs/1202.1130. I think that this is wonderful! There are too few papers like this- there is insufficient incentive to streamline the storylines of foundations. I am very happy to have found this proof, and I want such a proof to be a part of my smooth 3-manifold topology foundations.

Edit: Thanks to George Mossessian and to Ryan Budney, who point out in the comments that Jesse Johnson proved Reidemeister-Singer using Rubinstein and Scharlemann’s sweep-outs, which involves singularity theory which is much less sophisticated that Cerf Theory: http://front.math.ucdavis.edu/0705.3712
Perhaps that should be the “smooth proof from The Book” (or the “proof from The Smooth Book”)!

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):

9_26 example

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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.

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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.

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May 31, 2013

The algorithm to recognise the 3-sphere

Filed under: 3-manifolds,Computation and experiment,Triangulations — Ryan Budney @ 10:48 am
Tags: ,
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.   

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May 19, 2013

Flooved

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

Exciting news in academic publishing!

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
Tags:

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. 

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