# Low Dimensional Topology

## September 29, 2015

### Dispatches from the Dark Side, part 2

Filed under: Uncategorized — Jesse Johnson @ 7:10 pm

## September 27, 2015

### A counterexample to the Strong Neuwirth Conjecture

Filed under: Knot theory,Surfaces — dmoskovich @ 9:30 am
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A few days ago, my co-blogger Nathan Dunfield posted a counterexample to the Strong Neuwirth Conjecture.

N. Dunfield, A knot without a nonorientable essential spanning surface, arXiv:1509.06653

The Neuwirth Conjecture, posed by Neuwirth in 1963, asks roughly whether all knots can be embedded in surfaces in a way analogous to how a torus knot can be embedded in an unknotted torus. A weaker version, the “Weak Neuwirth Conjecture”, asks whether the knot group of any non-trivial knot in the 3-sphere can be presented as a product of free groups amalgameted along some subgroup. This was proven by Culler and Shalen in 1984. But nothing is proven about the ranks of these groups. The Neuwirth Conjecture would give the ranks as the genus of the surface. Thus, the Neuwirth Conjecture is an important conjecture for the structure theory of knot groups.

The Neuwirth Conjecture has been proven for many classes of knots, all via basically the same construction using a nonorientable essential spanning surface. The “Strong Neuwirth Conjecture” of Ozawa and Rubinstein asserts that this construction is always applicable because such a surface always exists.

Dunfield’s counterexample, verified by Snappea, indicates that we will need a different technique to prove the Neuwirth conjecture. Neuwirth’s Conjecture has just become even more alluring and interesting!

## August 25, 2015

### Heisenberg-picture TQFT

Filed under: Quantum topology — dmoskovich @ 8:06 am

This interesting-looking preprint has just appeared on ArXiv:

Theo Johnson-Freyd, Heisenberg-picture quantum field theory, arXiv:1508.05908

It argues for a different category-theoretical formalism for TQFT than the Schroedinger-picture‘ Atiyah-Segal-type axiomatization that we are used to. The Heisenberg-picture‘ functor it proposes has as its target a category whose top level is pointed vector spaces instead of numbers, and whose second to top level is associative algebras instead of vector spaces. The preprint argues that this formalism is better physically motivated, and one might dream that it is better-suited to analyze “semiclassical limit” conjectures such as the AJ conjecture and its variants.

I’m very happy to see this sort of playing-around with the foundations of TQFT, which I am happy to believe are too rigid. I expect there should be a useful Dirac picture also, and that there are other alternative axiomatizations also. Let’s see where this all leads!

## August 19, 2015

### Dongseok Kim 1968-2015

Filed under: Knot theory — dmoskovich @ 11:22 pm

I’m sad to announce the untimely passing of Dongseok Kim, a specialist on Kuperberg spiders and their generalizations. He was also a really nice guy whose conference talks were always well-worth listening to.

Although he’s better known for his quantum $\mathfrak{sl}_3$ and $\mathfrak{sl}_4$ stuff, the part of Kim’s work which was most intriguing for me personally was his work on spiders for Lie superalgebras. His paper on the topic doesn’t seem to be cited, despite superalgebra-related quantum invariants being a hot topic (work of Geer, Patureau-Mirand, Costantino, Turaev…)- has anyone noticed it? (more…)

## July 28, 2015

### Tangle diagram crossings and quantum entanglement

Filed under: Uncategorized — dmoskovich @ 1:55 pm

In low dimensional topology we speak of tangles, while quantum physics speaks of entanglement. Similar words, but is there a deeper connection? Kauffman conjectured that the answer is yes (and I think he’s right, although maybe for other reasons). Glancing through arXiv this morning, I came across the following recent preprint:

Alagic, G., Jarret M., and Jordan S.P. Yang-Baxter operators need quantum entanglement to distinguish knots
arXiv:1507.05979

Their result is what it says in the title. Namely, we comb the knot into a braid, and assign R-matrices to crossings. An R-matrix underlies a linear operator $V\otimes V \rightarrow V \otimes V$. The authors prove that if the this operator maps product states to product states, then it gives rise (via a certain “taking the normalized trace of the operator the braid gives” procedure) to a trivial quantum knot invariant. Thus, entanglement is an essential part of being a nontrivial quantum invariant. Very cool!

There’s a suggestive picture in my head. Entanglement is all about nonlocality, where two non-interacting objects (an overstrand and an understrand?) cannot be described as separate systems (crossing?), but are inseparably intertwined in that they share some sort of coordination. It’s the entanglement which allows the overstrand to “communicate” to the understrand that it is there, making it possible to construct a nontrivial quantum invariant.

I suspect there’s a lot more to this story. Well done Alagic, Jarret, and Jordan!

## July 16, 2015

### Boolean logic with braids

Filed under: Uncategorized — dmoskovich @ 3:03 am

First-off, I’m fairly chuffed that Tangle Machines (arXiv version HERE) was published in Proc. R Soc. A, and they even chose our figure for the cover! Computing with Coloured Tangles has also been accepted for publication. This is good.

One of the constructions of Tangle Machines, which I previously discussed HERE and HERE, is a universal set of logic gates using coloured tangles (and in fact we cheated, because our colouring wasn’t by a quandle but by a more general algebraic structure). It turns out that this idea isn’t new, and actually it’s been done better a long time ago in a different setting, and in a very nice way (thanks anonymous referee!). Boolean logic can be realized using coloured braids! And it’s even potentially useful in quantum computing! So today I’ll discuss this paper and the papers it references:

Alagic, G., Jeffery, S., and Jordan, S. Circuit Obfuscation Using Braids. In 9th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2014) (eds. S.T. Flammia and A.W. Harrow), Vol. 27, pp. 141–160.

## June 19, 2015

### The academic spring fails… or does it?

Filed under: Academic publishing — dmoskovich @ 5:40 am

A few hours ago, Math2.0, the discussion forum for journal publishing reform, closed for good. This is an indicator that the Cost of Knowledge campaign is effectively over. That isn’t to say we shouldn’t still add our names to it if we haven’t yet done so, but the initiative in publishing reform seems to have passed back to the corporations, and it seems to have happened a long time ago.

On the converse side, there are a lot of new OA journals out there, some Green and some Gold. This is a very good thing! Submit there, and we can beat predatory publishers through market competition.

One thing I’m seeing right now is a proliferation of metrics enhancers, promoted by publishers. One which has recently caught my attention is Kudos. It seems a decent enough service and I see no reason not to sign on to it, but the picture it paints of today’s research scene is bleak. Research consists, in that picture, of the production of mountains of papers which nobody reads, with researchers having to promote papers on Twitter and Facebook and through short catchy pop-science paper summaries for anyone to actually read them. And budgets being evaluated on the basis of citation counts, h-indexes, and various altmetrics, which may depend primarily on being in many people’s peripheral vision rather that on actually advancing a research field. I like to think things are not quite that bad in mathematics.

## 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…)

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