Low Dimensional Topology

June 11, 2015

Slice-ribbon progress

Filed under: Uncategorized — dmoskovich @ 10:29 am

There has been some recent interesting progress around the Slice Ribbon Conjecture. In particular, Yasui is giving talks on an infinite family counterexamples to the Akbulut-Kirby Conjecture (1978) that he has constructed:

Akbulut-Kirby Conjecture: If 0-surgeries on two knots give the same 3-manifold, then the knots with relevant orientations are concordant.

Note that some knots are not concordant to their reverses (Livingston), but the 0-surgery of a knot and its reverse are homeomorphic, so Akbulut-Kirby had to revise their original formalism to allow for arbitrary orientations. Abe and Tagami recently showed that if the Slice-Ribbon Conjecture is true then the Akbulut-Kirby Conjecture is false. Thus Yasui has eliminated an avenue to falsify the Slice-Ribbon Conjecture.

June 7, 2015

Eynard-Orantin Theory enters Quantum Topology

Filed under: Uncategorized — dmoskovich @ 9:18 am

I’m now reading the following paper:

G. Borot and B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative topological recursion of A-polynomials, Quantum Topol. 6 (2015), 39-138.

In it, the authors apply the Eynard-Orantin topological recursion to conjecture an all-order asymptotic expansion of the coloured Jones polynomial of the complement of a hyperbolic knot, extending the volume conjecture.

To get an overview of Eynard-Orantin Theory, I’m looking at:

  1. The original paper.
  2. Eynard’s own overview– an expanded version of an ICM talk.
  3. Some superb slides on the topic by Mulase.

(more…)

November 8, 2014

Regina goes mobile

Filed under: Uncategorized — Benjamin Burton @ 9:25 pm

For those of you with iThings, you can now run Regina on the iPad – just follow this App Store link.

Feedback is very welcome (as are “how do I…?” questions), especially for a brand new port such as this.

Triangulation gluings  Viewing normal surfaces  Enumerating normal surfaces

September 1, 2014

Regina 4.96 and hyperbolic manifolds

Filed under: Uncategorized — Benjamin Burton @ 11:29 pm

For those of you who aren’t on regina-announce: Regina 4.96 came out last weekend.

There’s several new features, such as:

  1. rigorous certification of hyperbolicity (using angle structures and linear programming);
  2. fast and automatic census lookup over much larger databases;
  3. much stronger simplification and recognition of fundamental groups;
  4. new constructions, operations and decompositions for triangulations;
  5. and more—see the Regina website for details.

You will find (1) and (2) on the Recognition tab, (3) on the Algebra tab, and (4) in the Triangulation menu.

If you work with hyperbolic manifolds then you may be happy to know that Regina now integrates more closely with SnapPy / SnapPea.  In particular, if you import a SnapPea triangulation then Regina will now preserve SnapPea-specific data such as fillings and peripheral curves, and you can use this data with Regina’s own functions (e.g., for computing boundary slopes for spun-normal surfaces) as well as with the in-built SnapPea kernel (e.g., to fill cusps or view tetrahedron shapes).  Try File -> Open Example -> Introductory Examples, and take a look at the figure eight knot complement or the Whitehead link complement for examples.

Finally, a note for Debian and Ubuntu users: the repositories have moved, and you will need to set them up again as per the installation instructions (follow the relevant Install link from the GNU/Linux downloads table).

Enjoy!

– Ben, on behalf of the developers.

June 13, 2014

Spaces of knots and low-dimensional topology

Filed under: Uncategorized — Ryan Budney @ 1:10 pm
Tags: , ,

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.

(more…)

November 7, 2013

Debunking knot theory’s favourite urban legend

Filed under: Uncategorized — dmoskovich @ 11:04 pm

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 10_{161} 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 10_{162}! Was it magic? Ken Perko shook the rope, and did it again. Sure enough, the 10_{161} and 10_{162} 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:
Weisstein 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

Save Kea!

Filed under: Uncategorized — dmoskovich @ 11:25 am

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

It came from K2

Filed under: Uncategorized — dmoskovich @ 11:34 am

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 K_2 of a ring. (more…)

September 8, 2012

ICERM Fall 2013: Topology, geometry, and dynamics

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:

  1. Exotic Geometric Structures. September 15-20, 2013.
  2. Topology, Geometry, and Group Theory: Informed by Experiment. October 21-25, 2013.
  3. Geometric Structures in Low-Dimensional Dynamics. November 18-22, 2013.

April 13, 2012

The virtual Haken conjecture

Agol’s preprint, which includes a long appendix joint with Groves and Manning, is now on the arXiv.

Next Page »

The Rubric Theme. Create a free website or blog at WordPress.com.

Follow

Get every new post delivered to your Inbox.

Join 247 other followers