Over the past 10-12 years, geometric topology has entered a new era. Most of the foundational problems are solved, and there’s a fairly isolated collection of foundational problems remaining. In my mind, the two most representative ones would be the smooth 4-dimensional Poincare hypothesis, and getting a better understanding of the homotopy-type of the group of diffeomorphisms of the n-sphere (especially for n=4, but for n large as well). I want to talk about what I’d call second-order problems in low-dimensional topology, less foundational in nature and more oriented towards other goals, like relating low-dimensional topology to other areas of science. Specifically, this is an attempt to describe the “spaces of knots” subject in a way that might entice low-dimensional topologists to think about the subject.
June 13, 2014
November 7, 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…)
April 4, 2013
Kea, whose actual name is Marni D. Shepheard, is a New Zealand physicist and blogger. Her blog, Arcadian Functor was really interesting and educational, and has morphed into Arcadian Omegafunctor, via blogs with intermediate names.
Kea works on the intersection of higher category theory and particle physics, which is niche mathematics combined with niche physics, and as a result has been out of a job for a long time. Marni’s a survivor though (a famous and celebrated survivor, who, together with Sonja Rendell, survived a mountaineering mishap which would have killed the vast majority of us) and she’s been publishing on viXra and continuing to do physics with no funding and often in total abject poverty. It appears to be taking its toll. (more…)
February 19, 2013
At the “Mathematics of Knots 5″ conference at Waseda University, I attended a most interesting talk by Takefumi Nosaka. Nosaka’s work always gives me the impression of being robust and sophisticated, and this talk was no exception. This time he was in the process constructing new topological invariants of links as images of longitudes in of a ring. (more…)
September 8, 2012
I’ve mentioned before that the fall semester program at ICERM for 2013 will focus on computation in low-dimensional topology, geometry, and dynamics. You can now apply to be a long-term visitor for this as a graduate student, postdoc, or other. The deadline for the postdoctoral positions is January 14, 2013; the early deadline for everyone else is December 1, 2012 and the second deadline March 15, 2013.
There will also be three week-long workshops associated with this, so mark your calendars for these exciting events:
- Exotic Geometric Structures. September 15-20, 2013.
- Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
- Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.
April 13, 2012
Agol’s preprint, which includes a long appendix joint with Groves and Manning, is now on the arXiv.
March 27, 2012
January 3, 2012
Dror Bar-Natan makes the following announcement:
With help from my students, in the next semester I will be running the “wClips Seminar”, which will be a combination of a class, a seminar, and an experiment. We will meeting on Wednesdays at noon starting January 11, 2012 – follow us on http://www.math.toronto.edu/drorbn/papers/WKO/!
The “class” part of this affair is that we will slowly and systematically go over my in-progress joint paper with Zsuzsanna Dancso, “Finite Type Invariants of W-Knotted Objects: From Alexander to Kashiwara and Vergne” (short “WKO”, and again see http://www.math.toronto.edu/drorbn/papers/WKO/), section by section, lemma by lemma, and covering all necessary prerequisites as they arise.
The “seminar” component is the usual. Occasionally people other than me will be telling the story.
The “experiment” part is that every lecture will be video taped and every blackboard will be photographed and everything will be immediately put on the WKO website, so that at the end we will have along with the paper a “video companion” – series of video clips explaining every bit of it. The paper will be mathematically self-contained, yet in addition every section thereof will include a link/reference to the corresponding clip in its video companion. And every video clip will have its written counterpart in one of the sections of the paper.
Feel free to follow almost in real time! Also, please let me know if you want to be added to the wClips mailing list.
June 19, 2011
On Friday, June 17, Japan’s second most-read newspaper, Asahi Shimbun, ran a full-page story on Knot Theory!!! Because of the sudden media exposure, I’ve been getting e-mails from family in Japan like studying Knot Theory makes me some kind of a celebrity. I’m sure it’s the same for every Knot Theorist who’s lived there- we’re enjoying our 15 minutes of fame right now.
The story is occasioned by the release of a fun computer game based on a theorem of Ayaka Shimizu. Ayaka is a member of Akio Kawauchi’s Knot Theory group at OCAMI. The theorem appears in her preprint Region crossing change is an unknotting operation.
January 18, 2011
Recently, there was a soft question on MathOverflow asking for examples of theorems which are `obvious but hard to prove’. There were three responses concerning pre-1930 knot theory, and I didn’t agree with any of them. This led me to wonder whether there might be a bit of a consensus in the mathematical community that knot theory is really much more difficult than it ought to be; and that good knot theory should be all about combinatorics of knot diagrams. And so knot colouring becomes `good knot theory’ for what I think are all the wrong reasons.