Low Dimensional Topology

May 4, 2014

Low dimensional topology of information

Filed under: Logic,Misc. — dmoskovich @ 5:32 am

Is information geometric, or is it fundamentally topological?

Information theory is a big, amorphous, multidisciplinary field which brings together mathematics, engineering, and computer science. It studies information, which typically manifests itself mathematically via various flavours of entropy. Another side of information theory is algorithmic information theory, which centers around notions of complexity. The mathematics of information theory tends to be analytic. Differential geometry plays a major role. Fisher information treats information as a geometric quantity, studying it by studying the curvature of a statistical manifold. The subfield of information theory centred around this worldview is known as information geometry.

But Avishy Carmi and I believe that information geometry is fundamentally topological. Geometrization shows us that the essential geometry of a closed 3-manifold is captured by its topology; analogously we believe that fundamental aspects of information geometry ought to be captured topologically. Not by the topology of the statistical manifold, perhaps, but rather by the topology of tangle machines, which is quite similar to the topology of tangles or of virtual tangles.

We have recently uploaded two preprints to ArXiv in which we define tangle machines and some of their topological invariants:

Tangle machines I: Concept
Tangle machines II: Invariants (more…)

April 16, 2014

Topology blogs

Filed under: Misc. — Jesse Johnson @ 1:21 pm

Along with not writing many posts over the last year, I also haven’t been reading many math blogs. But I just stumbled across Alex Sisto’s blog, and wanted to share the link. He has a number of really nice posts related to curve complexes, mapping class groups, and even a trefoil knot complement cake. If you haven’t read it before, you should go and read it now.

By the way, if you happen to know of any other good geometry/topology blogs that aren’t in our blog roll (on the right side of the page), please feel free to include the link in a comment so I can add it.

April 4, 2014

The Thin Manifold Conference

Filed under: Misc.,Thin position — Jesse Johnson @ 11:29 am

I just wanted to point everyone’s attention to an upcoming conference The Thin Manifold, being organized by my long-time collaborators Scott Taylor and Maggy Tomova. The main theme of the conference will be thin position for knots and three-manifolds, with many of the talks focusing on the sort of hands-on, cut-and-paste geometric topology that I’ve been writing about on this blog.

There will be some travel funding available for graduate students and early career mathematicians. Before the conference, there will be graduate student workshops, led by Jessica Purcell, who has been doing a lot of very cool work on WYSIWYG geometry/topology and Alex Zupan, who has been proving a lot of nice results about thin position and bridge surfaces. The graduate student workshop is August 5-7, and the conference is August 8-10. I’m looking forward to it and hope to see you there.

March 31, 2014

Train tracks and curve complexes

Filed under: Curve complexes,Surfaces — Jesse Johnson @ 4:14 pm

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 14, 2014

Train tracks on a torus

Filed under: Curve complexes,Surfaces — Jesse Johnson @ 11:47 am

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!

Filed under: 3-manifolds,Computation and experiment,Hyperbolic geometry,Knot theory — Nathan Dunfield @ 11:39 pm

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

Filed under: Knot theory — dmoskovich @ 4:09 am

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

Filed under: 3-manifolds,Triangulations — Ryan Budney @ 12:48 pm

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.

Conference banner

November 19, 2013

What is the Shannon Capacity of a coloured knot?

Filed under: Knot theory,Misc. — dmoskovich @ 10:41 am

I see topological objects as natural receptacles for information. Any knot invariant is information- perhaps a knot with crossing number n is a fancy way of writing the number n, or a knot with Alexander polynomial \Delta(X) is a fancy way of carrying the information \Delta(X). 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…)

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