One of the main ways in which I keep my finger on the pulse of what is hot now in low dimensional topology is to write lots and lots of reviews, both for Zentralblatt MATH and also for MathSciNet. In the last year or so, what has been increasingly coming through the pipe is papers about knot homology and mirror symmetry. There seems to be a lot happening in this field right now. (more…)
May 22, 2015
January 14, 2015
Check out this exciting new preprint by Vaughan Jones!
V.F.R. Jones, Some Unitary Representations of Thompson’s Groups and , arXiv:1412.7740.
December 19, 2014
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 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…)
November 3, 2014
I would like to draw attention to a fascinating MO question by Dylan Thurston, originally asked, it seems, by John Conway:
The question asks whether, rather than searching for Reidemeister moves to simplify a knot diagram, we could instead search for “big Reidemeister moves” in which we view a section which passes underneath the whole knot (only undercrossing) or over the whole knot (only overcrossing) as a single unit, and we replace it by another undersection (or oversection) which has the same endpoints.
This question (or more generally, the question of how to efficiently simplify knot diagrams in practice) loosely relates to a fantasy about being able to photograph a knot with a smartphone, and for the phone to be able to identify it and to tag it with the correct knot type. Incidentally, I’d like to also draw attention to a question by Ryan Budney on the topic of computer vision identification of knots, which is topic I speculated about here:
A core question to which all of this relates is:
And perhaps more generally, are there any very hard ambient isotopies of knots?
September 18, 2014
July 4, 2014
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):
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 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…)
March 2, 2014
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
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.
November 26, 2013
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
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.