Low Dimensional Topology

January 14, 2015

Jones’s new polynomial

Filed under: Knot theory,Quantum topology — dmoskovich @ 11:05 am

Check out this exciting new preprint by Vaughan Jones!

V.F.R. Jones, Some Unitary Representations of Thompson’s Groups F and T, arXiv:1412.7740.

(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…)

January 5, 2015

Topology of musical data

Filed under: Algebraic topology,Computation and experiment,Metric geometry — Ryan Budney @ 5:36 pm

A few years ago a musician friend asked me “there’s this new tool topologists have called Persistent Homology.  I’d like to see what it can do when you apply it to data from music.  Want to help?”

That friend is also an electrical engineer and knows some things about signal processing.  This was important to me — we had some external criterion (from outside of mathematics) for determining whether or not the insights from Persistent Homology were interesting or not.

So I said “okay!”  Not really knowing what I was getting myself into.

(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…)

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

November 3, 2014

Can a knot be monotonically simplified using under moves?

Filed under: Knot theory — dmoskovich @ 12:55 am
Tags: ,

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?

October 22, 2014

Understanding the anomaly

Filed under: 3-manifolds,Mapping class groups,Quantum topology — dmoskovich @ 11:48 am

I’ve recently been looking at the following paper in which 3+1-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…)

September 18, 2014

A song about a knot

Filed under: Knot theory,media — dmoskovich @ 11:43 am
Tags:

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.

(more…)

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.

July 21, 2014

Associativity vs. Distributivity

Filed under: Racks and quandles — dmoskovich @ 7:04 am

A binary operation \star is associative is (a\star b)\star c= a\star (b\star c). Examples of associative operations include addition, multiplication, connect-sum, disjoint union, and composition of maps.

A binary operation \triangleright is distributive over another operation \blacktriangleright if (a\blacktriangleright b)\triangleright c= (a\triangleright c)\blacktriangleright (b\triangleright c). If \triangleright=\blacktriangleright then the operation is said to be self-distributive. Examples of self-distributive operations include conjugation x^y\stackrel{\textup{\tiny def}}{=} y^{-1}xy, conditioning X|Y (assume X and Y are both Gaussian so that such a binary operation makes sense, essentially as covariance intersection), and linear combinations sx+(1-s)y with s\in \mathbb{R} (say), and x,y 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…)

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