Question:What is an alternating knot?

The preprints are:

- Joshua Evan Greene, Alternating links and definite surfaces, arXiv:1511.06329
- Joshua Howie, A characterisation of alternating knot exteriors, arXiv:1511.04945

This post will briefly introduce the problem; I look forward to reading the solutions themselves!

An alternating knot *diagram* is a diagram of a knot in which crossings alternate over and under along the knot. A knot is alternating if it has an alternating diagram. The Tait conjectures are about properties of alternating knots, and there are various other theorems in topology, *e.g.* in Floer homology, in which alternating knots are special. Given that a knot diagram is a mere combinatorial shadow of a topological object (a knot in 3-space), it seems that there ought to be a simple topological characterization of alternating knots in 3-space, with no mention of diagrams. Unfortunately there has not been such a characterization… until now!

Josh G.’s characterization works in a more general setting than Josh H.’s, but they both come down to the same idea- properties of pairings on the first homologies of checkerboard surfaces. In Josh G.’s case (which works also for links), there is a pairing on the first homology of a surface in a homology sphere, and the checkerboard surfaces of the knot are isotopic rel. boundary to two surfaces, one of which has a positive definite pairing, and the other a negative definite pairing.

This is so simple and elegant!! How could it take so long to discover ??

Both solutions yield very similar exponential-time algorithms to determine whether or not a knot is alternating (this would be much harder via the diagram).

The results of the two Joshuas might open the door to a geometric topological proof of Tait’s flyping conjecture. This would very nicely steal the thunder of quantum topology- the great classical triumph of quantum topology has always been the proof of the Tait conjectures, which had no other known proof.

Thanks to Dave Futer for calling my attention to these preprints!

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The main problem that most newly minted math PhDs have is that they don’t know what non-academic jobs are out there, and what they might be well suited to. I certainly had that problem. So the first step is to find out. There are a few companies and a few types of jobs that specifically look to hire math PhDs, and you’ll see some of these advertising on mathjobs.org. But I found this to be too narrow a list and one that I didn’t find particularly appealing – the most obvious ones are the NSA and computerized trading companies.

Instead, I had much better luck investigating jobs that were looking for people from any background who had the types of skills that I thought had made me a good mathematician. This meant thinking in terms of general skills/abilities such as communication, understanding abstract/complex systems and managing complex (collaborative) projects. That opened the door to a much broader range of jobs, including both technical and non-technical jobs. Programming/software engineering (what I ended up with) is on this list, but it’s far from the only one.

Of course, that still leaves the problem of finding the jobs that meet this criteria. What I found most useful for this was something that the book I mentioned above calls an *informational interview*. The idea is simple: you ask someone who has an interesting sounding job to have a short conversation about their career. It helps if someone you know introduces you to them, and LinkedIn can be handy for finding such connections. You ask about what they do during the day, how they like it, how they got the job, etc. I know this sounds a but hoakey, but it’s also really interesting, and it turns out many people like talking about themselves.

The informational interview is explicitly no-strings-attached, i.e. it’s not a job interview and there’s no expectation that the person you talk to will help you get a job. But because it’s no-strings, people tend to be happy to do this. And because it’s a minimal investment for you, it’s a good way to explore options that you might not have thought you’d be interested in. I talked to a lot of different people in this phase, and probably wouldn’t have ended up where I am now otherwise; when I started my job search I was focused on jobs that were directly related to machine learning. But then I talked to an acquaintance from my undergraduate days who works at Google. He introduced me to a former computer science professor who had just started at Google, and who convinced me to apply.

The people that you have informational interviews with may also point you to specific job openings, and may even offer to refer you to someone who’s involved in the hiring decision. Perhaps not surprisingly, it turns out that applying for jobs on job boards is less effective than getting personal referrals, even if it’s from someone that you’ve only talked to over the phone. This may feel strange compared to the academic job search where all applications go through an official process such as mathjob.org. But keep in mind that mathematics is a small world. On the hiring committee at OSU, for the vast majority of applicants, someone on the committee personally knew at least one of the applicant’s letter writers, if not their PhD adviser. Outside of academia, it doesn’t work like that, so many companies use personal referrals to make up for it.

(Unfortunately, this reliance on personal referrals is a factor in the lack of diversity in the tech industry: since people tend to spend their time with others with similar backgrounds, individuals will tend to refer job candidates who are similar to them. My comments above are not intended to justify or defend this system. I’m just describing my understanding of how things work.)

But even without personal referrals, if you present yourself in the right way (particularly for jobs such as software engineer and data scientist that are in sufficient demand) old-fashioned job applications can still be pretty effective. With one job I applied for the old fashioned way, the recruiter e-mailed me back within an hour to schedule a phone call, though it turned out they needed someone to start before my semester was over.

Presenting yourself in the right way turns out to be the second tricky part. There are a few simple but essential things like understanding the difference between a resume and a CV, making sure your resume is at most two pages (or better yet one) and focusing it on skills related to the job in question, rather than on unrelated academic merits. It’s easy to get used to the fact that every job in a math department is pretty much the same – some combination of teaching, research and service – making it easy to point to previous experience as evidence that you’ll do well in the job you’re applying to. In the private sector, it’s more common to get a job in a position that you’ve never had before. Fitting an academic background into a non-academic resume takes some creativity, but again the key is to think in terms of broad skills. That means things like communication (teaching, research talks) and managing complex projects (your dissertation, long-distance collaborations). These will help demonstrate that you’ll be able to learn whatever it takes to succeed at the job, even though you’re going to be starting from scratch.

As an example, Google’s interview process consists mostly of working out programming problems where the only background knowledge requirement is (roughly) an undergraduate data structures and algorithms course. (Take a look at the official list of study resources.) What makes the interview hard is that you have to think on your feet about a problem you haven’t seen before, and the interviewers pay close attention to how you think (out loud) through it. So, having lots of experience coding may give you some advantage, but not a huge one. I found that my years of working on hard math problems, plus a few months of computer science cramming, was a surprisingly good preparation. (Standard Disclaimer: The opinions expressed here are my own, and have not been reviewed or approved by my employer.) The process is far from perfect, and has its own biases, but it minimizes the impact of one’s background and experience. And while not all employers have this type of interview process (though many software companies do), many are willing to overlook lack of experience for the sake of potential.

There are also some things you can do to get more direct experience to put on your resume. Many companies are starting to offer internships, even for PhD students (including Google). You can get programming experience by contributing to an open source project. To show off you data science skills, you can compete in a Kaggle competition. (Those are the ones I know of – if you know of any that I missed, leave a comment below!)

OK, so once you’ve worked out all your non-academic skills, and written your resume accordingly, the final step is to determine how you will avoid setting off red flags that some employers look for from academics. In general, the folks who review your resume will have no doubt that you’re smart based on your math background, but they will also be acutely aware that there’s such a thing as being too smart for your (or their) own good. Many employers feel that hiring a “bad” job candidate is much more costly than not hiring a half dozen “good” candidates, so they spend a lot of energy looking for red flags. When considering an academic, there are certain red flags that they may think they see even if you didn’t do anything to indicate them. So, it’s not enough to avoid the red flags – you need to actively provide a counter narrative.

Here are the things I know of that employers may be expecting you to say, and that they may hear even if you don’t say them: (Did I miss any?)

- I couldn’t make it as an academic, so I’ll settle for a private sector job, but I don’t have to like it.
- I just want to think about fun, abstract problems, whether or not they’re useful.
- This job is going to be much easier than being a professor, so I won’t have to work very hard.
- I’m clearly smarter than everyone who currently works for you so I’ll just tell them all what to do.

Now, I know you wouldn’t actually think, let alone say, any of these things. (You wouldn’t, right?) But better than not saying them is to say things that completely refute them (and you have to mean it when you say it!) Have a solid explanation for why you want to leave academia, which focuses on the positive aspects of the private sector. (See my previous post.) Talk about how you want to work on things that have a real world impact, how you’re looking forward to the challenge of adapting to a completely different environment, and how you’ll enjoy being a member of a team and learning from your much more experienced colleagues.

In the end, the process of getting a non-academic job can be long and complex. At the beginning, it feels completely hopeless, but the more you learn and the more non-academics you talk to, the better it gets. And here’s the kicker: There are a lot of jobs out there where the supply and demand dynamics are completely the opposite of academia – where employers are desperately seeking qualified applicants. Once you find your way there, and see what it’s like applying for a job where you’re NOT one of 500 applicants for a single position, it’s completely worth it.

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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!

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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!

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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?

Reshetikhin-Turaev (RT) invariants are constructed in the context of semi-simple categories of representations of quantum groups with some extra structure and satisfying some extra conditions (*modular categories*). For some quantum groups these categories have nice combinatorial descriptions via Kuperberg’s spiders; the most famous is the Temperley-Leib algebra description of a modular category of representations. For superalgebras the categories are no longer semi-simple, although this probably isn’t a major problem. Kim identifies the following fairly manageable-looking description for a category of quantum representations:

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Alagic, G., Jarret M., and Jordan S.P.

Yang-Baxter operators need quantum entanglement to distinguish knotsarXiv: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 . 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!

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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.

We start with a group in which we choose two elements, one which we call `zero’ and the other which we call `one’. Our operation is conjugation . A *computation* is a braid coloured by elements of , with strands incident to endpoints at the bottom each of which is coloured by a zero or a one called the *input* of the computation, and strands incident to endpoints at the top each of which is coloured by a zero or a one called the *output* of the computation. The braid may also have additional strands at the top and bottom (indeed it must if ). These extra strands, called *ancilla*, may be coloured by any element of . The ancilla serve as catalysts for the computation and nothing more.

How does a braid compute? Moving up through the braid from bottom to top, we encounter braid group generators describing strand crossing over strand , and their inverses. If the colour of strand at the bottom of is and the colour of strand at the bottom of is , then the colour of strand at the top is still while the colour of strand at the top is .

What computations can a braid perform? A-priori, we might suspect that maybe it can’t do all that much, because conjugations are rather special operations. In particular, it doesn’t seem like we should be able to realize an AND gate with a braid. To remind you, , pronounces ` and ‘, returns . So it returns if both and are one, and zero otherwise. The reason AND seems really hard to realize using coloured braids (try it if you don’t believe me!) is that it responds differently to ones than to zeros, so there is an `if’ concept built in; and how could a braid do that? More formally, it’s recovering a product from conjugation, which seems impossible.

It really looks like one should be able to prove that AND can’t be realized by braids, and this is what people indeed believed for a long time. This is one case in which our intuition is wrong, however. Kitaev showed that if the group is the symmetric group , then AND can indeed be realized by a -coloured braid (note: ancilla are important- this, and the fact that is `sufficiently rich’, is what comes to the rescue). The proof is pretty similar to a celebrated result of Barrington and of Krohn-Maurer-Rhodes about representing maps using conjugations. Ogburn and Preskill showed that the alternating group , which is half as large as , is enough.

Monchon constructed a Toffoli gate braid, which is explicitly drawn in Appendix A of Alagic-Jeffery-Jordan. It has 132 crossings and 14 strands (11 of which are ancilla), so it isn’t an easy construction.

To remind the reader, a Toffoli gate is a universal reversible logic gate which contains within it a universal set of gates. Thus, an AND gate is a subcomputation of a Toffoli gate. A Toffoli gate accepts 3 bits , , and , returns and , and inverts if and only if .

Explicitly, encodes zero and encodes one. You can see the construction there.

There are some obvious questions which this raises- as any binary boolean function can be realized by a coloured braid (this is what the above result shows), and because the boolean circuit model is Turing complete, we know that any computable function can be represented by an -coloured braid with ancilla. It’s pretty clear, however, that it would be tremendously suboptimal to realize individual Toffoli gates and to string them all together- the braid language can recover boolean circuits, but not in a simple or intuitive way. How could a simple braid computation be performed? What is the simples braid that would realize some given binary boolean function, and how would we set-out to find it? Is the group the best possible, or are there more reasonable groups (in whatever sense) which do the same job?

How is all of this practical? Well, an element in the braid group can be thought of as a motion of points in a disc (more accurately, as an element of the fundamental group of a configuration space on the disc). The timelines of these moving points trace out the braid- for instance, if we have three points, one of which is stationary and two of which change places, we obtain (or its inverse) in the braid group on 3 strands.

Now imagine 2-dimensional particles- anyons they are called, and they are just-about physical- which move around one another in a disc. Say they have 60 states, and when one moves around another it picks up a phase which corresponds to the conjugation operation. Again, this is just about physical. Then voilá! The above construction gives rise to a Toffoli gate -coloured braid!

There’s more to the story. We can use braid relations, AKA Reidemeister moves on braids, to obfuscate or to disguise our computation for security purposes, so an eavedropper with access to the system would have trouble figuring out which computation was being performed.

Computing seems to have a definite topological side, which is only just now slowly emerging into the light. The visionary who first saw this was Kauffman. There are many sides to the story, and I’m sure that topology appears in many different and even independent ways; a Toffoli gate braid is a particularly pretty one.

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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.

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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.

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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:

- The original paper.
- Eynard’s own overview– an expanded version of an ICM talk.
- Some superb slides on the topic by Mulase.

Eynard-Orantin Theory studies a certain universal recursion formula based on a plane analytic curve (a Riemann surface) which Eynard calls the `spectral curve’, the Cauchy differentiation kernel, and the residue calculus on it. Like quantum topology itself, the main question about Eynard-Orantin Theory is what it is exactly that it is computing.

The recursion associates to a Riemann surface with extra structure a family of differential forms called `invariants’. The initial terms and are canonically given, and remaining terms are defined by the universal recursive formula by `removing pairs of pants’.

Examples of such recursions had occured previously *e.g.* in work of Mirzakhani.

In low dimensional topology, the curve is canonically associated with the A-polynomial. The form essentially determines the hyperbolic volume of the knot complement, while corresponds to the Reidemeister torsion.

Mirror symmetry relates these quantities with quantum invariants such as coloured Jones polynomials. Dijkgraaf, Fuji, and Manabe pointed out that mirror symmetry relates not only with the coloured Jones polynomial, but also the higher terms. They required them to add some ad-hoc terms which Borot-Eynard eliminate by chosing a different and perhaps more suitable wave function.

The topological recursion lives on the B-model side of mirror symmetry. To trully understand it, however, and to prove it in the cases of interest in low dimensional topology, the goal is to find an A-model proof of it.

More and better conjectures are a sure indicator for the vibrancy of the topic. Eynard-Orantin Theory is coming into play in many places in mathematics, and now it is coming to topology. Have the famous conjectures concerning asymptotics of quantum invariants at last met their match?

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