Low Dimensional Topology

March 30, 2015

MOO is classical

Filed under: 3-manifolds,Dehn surgery,Quantum topology — dmoskovich @ 9:43 am

The simplest quantum 3-manifold invariant is the Murakami-Ohtsuki-Okada (MOO) invariant. It comes from $\mathrm{U}(1)$ Chern-Simons theory in the way that the $\mathrm{SU}(2)$ Reshetikhin-Turaev invariant comes from $\mathrm{SU}(2)$ Chern-Simons Theory. It has a closed formula in terms of the order of the first cohomology class of the $3$-manifold $M$ and an eighth root of unity. Witten’s Chern-Simons theory for gauge group $\mathrm{U}(1)$ shows that the MOO invariant can be reformulated in terms of classical Riemann theta functions with characteristic, but the relationship is by way of quantum field theory.

A recently published paper by Gelca and Uribe, which is also the topic of a book by Gelca and some nice slides, constructs the MOO invariant from theta functions completely classically essentially without using anything quantum at all (although the representation theory behind it was originally developed for quantum mechanical purposes). Thus, like the Alexander polynomial and the linking number, MOO is seen to be quantum but also classical.

There is also a more analytic, heat-equation-based way of seeing the same thing due to Andersen, but I haven’t read Andersen’s paper and therefore I can’t say anything about that. (more…)

March 22, 2015

SnapPy 2.3 released

Marc Culler and I are pleased to announce version 2.3 of SnapPy. New features include:

• Major improvements to the link and planar diagram component, including link simplification, random links, and better documentation.
• Basic support for spun normal surfaces.
• New extra features when used inside of Sage:
• Better compatibility with OS X Yosemite and Windows 8.1.
• Development changes:
• Major source code reorganization/cleanup.
• Source code repository moved to Bitbucket.
• Python modules now hosted on PyPI, simplifying installation.

All available at the usual place.

March 9, 2015

Complex hyperbolic geometry of knot complements

Filed under: 3-manifolds,Hyperbolic geometry,Misc. — dmoskovich @ 3:41 am

This morning there was a paper which caught my eye:

Deraux, M. & Falbel, E. 2015 Complex hyperbolic geometry of the figure-eight knot.
Geometry & Topology 19, 237–293.

In it, the authors study a very different geometric structure for the figure-eight knot complement, as the manifold at infinity of a complex hyperbolic orbifold. (more…)

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.

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.)

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.

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.

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:

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?

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…)

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