# 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.

## May 22, 2015

### Recent coloured HOMFLYPT-related stuff

Filed under: Knot theory,Quantum topology — dmoskovich @ 10:41 am

One of the main ways in which I keep my finger on the pulse of what is hot now in low dimensional topology is to write lots and lots of reviews, both for Zentralblatt MATH and also for MathSciNet. In the last year or so, what has been increasingly coming through the pipe is papers about knot homology and mirror symmetry. There seems to be a lot happening in this field right now. (more…)

## 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…)

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