Low Dimensional Topology

March 9, 2015

Complex hyperbolic geometry of knot complements

Filed under: 3-manifolds,Hyperbolic geometry,Misc. — dmoskovich @ 3:41 am

This morning there was a paper which caught my eye:

Deraux, M. & Falbel, E. 2015 Complex hyperbolic geometry of the figure-eight knot.
Geometry & Topology 19, 237–293.

In it, the authors study a very different geometric structure for the figure-eight knot complement, as the manifold at infinity of a complex hyperbolic orbifold. (more…)

January 13, 2015

Dispatches from the Dark Side

Filed under: Misc. — Jesse Johnson @ 2:49 pm

As some readers of this blog will have already heard, I left my position at Oklahoma State this summer to become a software engineer in Google’s Cambridge/Boston office. My decision to leave academia for the private sector (aka the Dark Side, as certain mathematicians who I won’t name like to call it) was the result of a number of years of soul-searching, research, toe-dipping, etc. In this post, I want to share my experiences for the sake of any young Ph.D.s or current graduate students who are grappling with this same decision. (Disclaimer: The views expressed below are my own and were not endorsed or approved by my employer.)  I’ll focus on software-related jobs, since that’s what I know about, though most jobs for mathematicians these days will probably involve a fair amount of programming anyway. (Also, here’s some additional required reading for anyone finishing up a Ph.D.: The Fame Trap.)

(more…)

December 19, 2014

Concordance Champion Tim Cochran 1955-2014.

Filed under: 4-manifolds,knot concordance,Knot theory,Misc. — dmoskovich @ 8:08 am

Yesterday I received the shocking news of the passing of Tim Cochran (1955-2014), a leader in the field of knot and link concordance. The Rice University obituary is here.

A groundbreaking paper which made a deep impression on a lot of people, including me, was Cochran-Orr-Teichner’s Knot concordance, Whitney towers and L^2 signatures. This paper revealed an unexpected geometric filtration of the topological knot concordance group, which formed the basis for much of Tim Cochran’s subsequent work with collaborators, and the work of many other people.

In this post, in memory of Tim, I will say a few words about roughly what all of this is about. (more…)

July 4, 2014

Could a computer see a knot?

Filed under: Knot theory,Misc. — dmoskovich @ 5:36 am

I don’t know about you, but when I tell non-mathematicians what knot theory is, I often find myself telling a story about identifying a knotted protein by its knottedness- something about different proteins tending to be bendy to differing degrees, so that certain types of protein tend to form knots with higher writhe than others, and that this helps biologists and chemists to distinguish proteins which they would otherwise need a lot of time and money and an electron microscope to tell apart.

One major problem with this story, and with similar stories, is that the knot diagrams have to be photographed (and thus identified) by hand. The pictures are not always easy to interpret (e.g. distinguishing overcrossings from undercrossings):

A real picture of a knot.

Also resolution might be low, objects might be in the way…

This is a computer vision problem as opposed to a math problem- but wouldn’t it be nice if a computer could recognise a knot type from a suboptimal picture? If you could snap a picture of yourself standing in front of an 11n_{24} knot making bunny ears behind it, and your computer would automatically tag it with the correct knot type? Furthermore, wouldn’t it be nice if a computer could recognise your knot on the basis of many noisy pictures, perhaps taken from different angles? (more…)

June 12, 2014

A celebration of diagrammatic algebra

Filed under: 3-manifolds,Combinatorics,Misc.,Quantum topology — dmoskovich @ 5:24 am

Relaxing from my forays into information and computation, I’ve recently been glancing through my mathematical sibling Kenta Okazaki’s thesis, published as:

K. Okazaki, The state sum invariant of 3–manifolds constructed from the E_6 linear skein.
Algebraic & Geometric Topology 13 (2013) 3469–3536.

It’s a wonderful piece of diagrammatic algebra, and I’d like to tell you a bit about it! (more…)

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.

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
(more…)

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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