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 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…)
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.
I would like to draw attention to a fascinating MO question by Dylan Thurston, originally asked, it seems, by John Conway:
Can a knot be monotonically simplified using under moves?
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:
Algorithm to go from a picture (or pictures) of a string in space, to a piecewise-linear representation of the curve.
A core question to which all of this relates is:
Are there any very hard unknots?
And perhaps more generally, are there any very hard ambient isotopies of knots?
I’ve recently been looking at the following paper in which -TQFT anomalies are treated carefully and various old constructions of Turaev and Walker are elucidated:
Gilmer, P.M. and Masbaum, G., Maslov Index, Mapping Class Groups, and TQFT, Forum Math. 25 (2013), 1067-1106.
It makes me think a lot about just what the anomaly `actually means’… (more…)
Norwegian duo Ylvis have just released a music video about… well, essentially it’s about physical knot theory. It’s about tying “the greatest knot of all”, the Trucker’s hitch.
For those of you who aren’t on regina-announce: Regina 4.96 came out last weekend.
There’s several new features, such as:
- rigorous certification of hyperbolicity (using angle structures and linear programming);
- fast and automatic census lookup over much larger databases;
- much stronger simplification and recognition of fundamental groups;
- new constructions, operations and decompositions for triangulations;
- 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).
- Ben, on behalf of the developers.
A binary operation is associative is . Examples of associative operations include addition, multiplication, connect-sum, disjoint union, and composition of maps.
A binary operation is distributive over another operation if . If then the operation is said to be self-distributive. Examples of self-distributive operations include conjugation , conditioning (assume X and Y are both Gaussian so that such a binary operation makes sense, essentially as covariance intersection), and linear combinations with (say), and elements of a real vector space.
Two nice survey papers about self-distributivity are:
- J. Przytycki, Distributivity versus associativity in the homology theory of algebraic structures. arXiv:1109.4850.
- M. Elhamdadi, Distributivity in Quandles and Quasigroups. arXiv:1209.6518
I won’t survey these paper today- instead I’ll relate some abstract philosphical musings on the topic of associativity vs. distributivity.
Algebraic topology detects information not only about associative structures like groups, but also about self-distributive structures like quandles. I wonder to what extent distributivity can stand in for associativity. Might our associative age give way to a distributive age? Will future science will make essential use of distributive structures like quandles, racks, and their generalizations? At the moment, such structures appear prominently only in low dimensional topology. (more…)
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…)
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.
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 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…)