This is just a short post to draw attention to a new preprint by Friedl and Powell The presentation of the Blanchfield pairing of a knot via a Seifert matrix.
The Blanchfield pairing on the Alexander module occurs in various places in knot theory, including in quantum topology. Levine’s 1977 argument for its expression in terms of the Seifert matrix doesn’t make easy reading (the authors suggest it’s incomplete- I can’t judge), and it is notoriously difficult to prove that the Blanchfield pairing is Hermitian. The authors deal deftly with both problems using a more modern but clearly sensible toolbox. Time to rewrite the textbooks.
I wish there were more papers like this. Some aspects of low dimensional topology could use a careful, sensible, modern reboot such as that of this paper.
Drew Zemke, who is a grad student of Jason Manning, posted a proof of the Simple Loop Conjecture for 3-manifolds modeled on Sol last week.
The Simple Loop Conjecture fits into that family of statements such as Dehn’s Lemma and the Sphere Theorem which translate statements about fundamental groups into statements about 3-manifolds. Such theorems allow us to trade 3-manifolds for their fundamental groups (which are much simpler mathematical objects). (more…)
A few days ago, two people named Joshua (one Howie and one Greene) independently posted to arXiv a similar solution to an old question of Ralph Fox:
Question: What is an alternating knot?
The preprints are:
This post will briefly introduce the problem; I look forward to reading the solutions themselves! (more…)
Back in January, I wrote a post about my experience with the differences and trade-offs between academic careers and private sector careers. In this post, I want to present some practical advice for anyone with an academic math background who might be seeking a non-academic job. This advice is based on both my own experience and the advice I found while trying to make the transition. Much of it is very similar to the advice you’ll find (in more detail) in a book called What Are You Going To Do With That? which I read early on and found very helpful. (The book is about finding a non-academic job with any type of PhD.)
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!
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!
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 and 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…)
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
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 . 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!
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.
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.