In my last post, I described how a train track on a surface determines a collection of loops in a surface, namely the loops that are carried by the track. Looking at these loops from the perspective of the the Farey graph for the torus, this set consists of the loops corresponding to vertices in one of the components that results from cutting the Farey graph along a certain edge. In the curve complex, train tricks define partitions that are almost as simple, though they are necessarily more complicated because there is no one simplex that separates separates the complex. Still, this type of partition comes in very useful for calculating distances in the curve complex (and was central to my recent preprint with Yoav Moriah) but to see how that works, we need something a bit stronger. In this post, I’ll explain how we can turn the partition defined by a train track into two sets of curves with a buffer between them. By placing these buffers next to each other, we can build larger gaps that imply a lower bound on the distance between certain loops in the curve complex.

## March 31, 2014

## March 14, 2014

### Train tracks on a torus

A little over a year ago, I started writing a series of posts on train tracks and normal loops, then got distracted by other things. In the mean time, I wrote a paper with Yoav Moriah involving train tracks and curve complex distances, which gave me a whole new perspective on what train tracks really mean, more in line with much of Masur and Minsky’s work [1]. So, I want to resuscitate the series of posts on train tracks, but in a slightly different direction than where I was headed before. I’ll start by looking at a very simple case: train tracks on a torus. If you need a review of what train tracks are (the mathematical object, not the literal ones), you can reread my earlier post.

## March 2, 2014

### SnapPy 2.1: Now with extra precision!

Marc Culler and I released SnapPy 2.1 today. The main new feature is the ManifoldHP variant of Manifold which does all floating-point calculations in quad-double precision, which has four times as many significant digits as the ordinary double precision numbers used by Manifold. More precisely, numbers used in ManifoldHP have 212 bits for the mantissa/significand (roughly 63 decimal digits) versus 53 bits with Manifold.

## January 24, 2014

### Distinguishing the left-hand trefoil from the right-hand trefoil by colouring

This morning, I’ve been looking through a very entertaining paper in which Roger Fenn distinguishes the left-hand trefoil from the right-hand trefoil in a way that could be explained to elementary school children.

R. Fenn, Tackling the trefoils. (more…)

## December 12, 2013

### Banker finds a duplication in a 3-manifold table

Daniel Moskovich recently wrote about the discovery by a lawyer of a duplication in the knot tables called the “Perko pair”.

Now a banker has found another duplicate in yet another table of 3-manifolds. This time it was Ben Burton, and the duplicate appears in the Hildebrand-Weeks cusped hyperbolic census.

## November 26, 2013

### What’s Next? A conference in question form

Mark your calendars now: in June 2014, Cornell University will host “What’s Next? The mathematical legacy of Bill Thurston”. It looks like it will be a very exciting event, see the (lightly edited) announcement from the organizers below the fold.

## November 19, 2013

### What is the Shannon Capacity of a coloured knot?

I see topological objects as natural receptacles for information. Any knot invariant is information- perhaps a knot with crossing number is a fancy way of writing the number , or a knot with Alexander polynomial is a fancy way of carrying the information . A few days ago, I was reading Tom Leinster’s nice description of Shannon capacity of a graph, and I was wondering whether we could also define Shannon capacity for a knot. Avishy Carmi and I think that we can (and the knots I care about are coloured), and although the idea is rather raw I’d like to record it here, mainly for my own benefit.

For millenea, the Inca used knots in the form of quipu to communicate information. Let’s think how we might attempt to do the same. (more…)

## November 7, 2013

### Debunking knot theory’s favourite urban legend

The following post recycles Richard Elwes’s lovely blog post and this MathOverflow answer. It is dedicated to the memory of the greatest knot-shaker I have met, Kumar Pallana (1918-2013).

Yesterday I received correspondence from a certain Kenneth A. Perko Jr., whose name perhaps you have heard before. Its contents are too delicious not to share- knot theory’s favourite urban legend is completely false!

**Myth:**Ken Perko, a New York lawyer with no formal mathematical training, was having a slow day at the office. Bored and in-between troublesome clients, he toyed with a long piece of rope, which he had tangled up to represent knot in Rolfsen’s table (Rolfsen, like Kuga, was popular among non-mathematicians at the time). As Perko played with it, the knotted rope began to change before his eyes, and glancing back at the book, he suddenly realized that what he was holding in his hands was the ! Was it magic? Ken Perko shook the rope, and did it again. Sure enough, the and were the same knot!

Excited, Ken Perko shot off a paper to PAMS, containing only a title and a list of figures demonstrating an ambient isotopy. His paper entered the Guiness Book of World Records as the “shortest mathematics paper of all time”, and Ken Perko obtained immortality.

This is the Perko pair:

What a story! The human drama, the “math for the masses” aspect that a complete amateur could make a massive mathematical discovery by playing with some string, the beautiful magenta pair of knots, the importance of attention to detail and using all your senses (not just your head)! What a shame that virtually everything written above turns out to be false! (more…)

## October 13, 2013

### A noteworthy knot simplification algorithm

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

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.