Graduate Student Solves Decades-Old Conway Knot Problem
quantamagazine.org
quantamagazine.org
I have not read the paper, but I have skimmed it briefly, and it looks pretty exciting. This isn't a case of "here's a new invariant, and, oh, BTW, it works to show the Conway knot isn't slice." It's an actual new technique. And, at first glance, it looks like a pretty simple technique. I didn't immediately see anything here that wasn't just a neat combination of low dimensional topology and basic knot theory techniques.
I'd be interested to see what this technique could do with knots having more than 12 crossings.
One way sliceness is studied is through knot concordance, which is a certain restricted kind of deformation of a knot through 4-d space. All slice knots are concordant to each other. The Rasmussen s-invariant is invariant under concordance. So, the particular pair of knots C and K' cannot be concordant since they have different Rassmussen s-invariants. One consequence is that it's not true that knots with diffeomorphic knot traces are concordant in general.
Another interesting thing is that C and K' are related by positive mutation, which makes them topologically concordant (weaker than concordance). Since K' is slice, this implies C is topologically slice (weaker than sliceness), though this was already known by Freedman's work because the Alexander polynomial of C is 1.
Altogether, C is topologically slice but not slice (not the first example) while also being a positive mutant of a slice knot (which is what made the problem so difficult for so long).
The knot trace X(K) is a 4-dimensional manifold whose boundary is a 3-manifold called the 0-surgery of K. There was a conjecture that knots with diffeomorphic 0-surgeries are concordant (perhaps up to mirror images). While it was already disproved, this pair of knots gives another counterexample. In fact, she gives infinitely many pairs of counterexamples in an earlier paper: https://arxiv.org/abs/1702.03974
Instead of talking to the service/mathematical object (A) directly, you talk to another service/mathematical object (B) that leaks information about (A). Precisely, the information that you want.
The way she leaked that information was through a property called traceness that apparently was underappreciated by knot theorists in terms of sliceness problems. Which makes sense, otherwise it wouldn't be an information leak. Finding an info leak in itself, no matter what discipline your in is already amazing.
As far as I understood the quantamagazine article, mathematical object (B) still had to be constructed which only a person well-versed in knot theory could do. So not only did she find an info leak, she basically created something entirely new that few people can do (yep, the hacker analogy breaks here, this part is the "incredible builder" analogy).
This is so cool. Side channels are everywhere, even in math. Apparently, for knots it's called traceness.
[1] Not sure if side channel is the right word, but I view it as: something that leaks information about another thing. For example, air vibrations leaking information on what instructions the CPU is executing (I'm making this up, one would need very fine-grained air vibration data to see if this would be a side channel).
It's amazing how impactful it can be to take an industry-naïve approach to a problem/project. A lot of disciplines have fundamentally similar challenges, but with solutions that evolved in completely different directions. Intractable problems or recurring issues in one industry can frequently be unblocked by plucking mature solutions or approaches from another, but so many people grow linearly within a single industry/discipline that such cross-seeding of concepts rarely actually have an opportunity to occur.
Originally, it was anything but easy. But at this point in my career, the pattern itself is what sells it:
- I've never left a company having the same title or role I started with. And have always left with substantially greater responsibilities than I started with. Which holds true even for companies I was at for a year or less.
- I've never progressed linearly within the same industry or function when switching companies
- Almost all of my roles from IC to people-manager have involved high levels of trust and autonomy with minimal direct oversight and limited if any pre-existing structure, process, support, infrastructure, or direction.
The above demonstrates over a decade of reliably and successfully adapting between roles and environments, and outlaying it as such when going through my background with a hiring company tends to preempt any reservations/concerns the hiring company has in that regard. I've also now amassed SVP, President, and C-Level recommendations in pretty much every functional area, from individuals that have either directly or indirectly managed me. And can pull in the appropriate reference to address any remaining hesitations around functional capabilities or ramp up expectations. All of which have made my last few transitions significantly easier than earlier in my career.
> And what country do you work, where you are being respected for having industry-naïvety?
I'm in the US. I wouldn't put it as being respected for industry-naïvety, which is one of the reasons you rarely see people grow their career the way I have. I just seek out opportunities where that's not a liability and am fully transparent when I'm in a situation where that may be an expectation. Standard needs get handled (or at least sanity checked) by those with a more standard background if available, abnormal/atypical/ill-defined/unknown needs filter through to me. The value and leverage I provide by bringing to bear my variety of skills, experience, and perspectives in those atypical or rapidly evolving situations more than makes up for the occasional crash course in the basics I need to fill in some industry/functional holes.
The caveat being that I'm pretty useless or even detrimental[1] in many environments - such as companies that are highly regimented and a cross-disciplinary mindset would be antithetical to the culture and cause undesired friction, or in highly stable and mature environments where deviating from standard practices by introducing foreign concepts would destabilize the steady-state of the existing situation. In those situations, I'm far more of a liability than an asset. And will directly probe during the interview process to understand the context of the role and whether it's a good fit for what I bring to the table.
[1] While usually considered detrimental, I've also been placed in those situations with the express intent of introducing such a destabilizing factor or internal friction, as a way to shake things up and allow for change.
It's almost as if I feel my role is more about changing constantly and "advising" like a consultant, which I usually despise, than staying 10 years maintaining an old rotting software.
There's definitely value in rotation, and my newest company, a 150 yo venerable institution, is amazed at what one can bring from small startups. They have had me rotate from team to team to "bring the mojo" since I started, where usually you pop a profiler, show that that's not how you use a hashmap, and save a few millions here and there for the users / clients... Now I'm the profiler guy, something I learned to do in my previous company to fix a few mem leaks... And each job was like that, they never let me dust in a team doing ticket after ticket in soul sucking sprints :D
Are you someone maybe who like to give smartass-like suggestions on absolutely everything until someone let you try ?
I feel this hard. Any relatively well known long standing problem I (wrongly?) assume must be impossible for me to solve because so many much more experienced that me people have put many hours of work into it already. But maybe I'm overestimating the actual effort being put into these sorts of things, or the amount of benefit fresh eyes can bring.
It's not the first time this has happened.
I know some (very bright) graduate students do get assistant professor positions at top universities right out of their PhDs; I think those are tenure-track?
My guess is that these are "named postdocs" ("[some name] Assistant Professor", e.g. https://www.mathjobs.org/jobs/jobs/14065 or https://www.mathjobs.org/jobs/jobs/15707 ). Despite their names, they're limited to 3 years and only get renewed in exceptional circumstances.
You would have to have the right connections, or publish papers that were undeniably brilliant. (ie. be a celebrity.)
It's true that, in general, there are far more graduating PhD students every year than tenure-track faculty positions. But I think "no career path" drastically overstates the difficulty. It's way easier than making it as a pro athlete, for example.
> A lot of assistant professors, etc. sleep in their cars.
Are you thinking of adjunct professors, who are not tenure-track? Assistant professors typically make at least 50k+, and in the more lucrative fields (computer science, business schools, biopharm I think) >100k is common.
> You would have to have the right connections, or publish papers that were undeniably brilliant. (ie. be a celebrity.)
You're correct that an average PhD student will probably not get any tenure-track faculty interviews, let alone offers. This is especially true in more resource-constrained fields like the humanities.
I always hear about it and topology. Makes me want to read a book on it...
I can recommend a book on topology though. Robert Ghrist’s book:
https://www.math.upenn.edu/~ghrist/notes.html
Do you like physics? If so maybe try John Baez’s book for more knot-centric inspiration:
https://www.amazon.com/GAUGE-FIELDS-KNOTS-GRAVITY-Everything...
In the 1980s, Vaughan Jones discovered the Jones polynomial, which is a property of knots which remarkably turned out to have deep connections to all sorts of things including quantum field theory! This led to 3 decades and counting of intense study into the relationship between knots and fundamental physics. I'd like to say more, but I'm knot really qualified to speak about the connections to other fields. So that's basically the tl;dr of why so many people care about knots!
Does this mean that the strings in string theory are knots?
In particular, every 3-manifold is the boundary of a 4-manifold obtained in a way that's reminiscent of knot traces from the article. You take a disjoint collection of knots in the boundary of a 4-ball, then glue in the 1-handles ("caps" in the article) along these knots, but with slight change: you glue in 1-handles with any framing whatsoever, not just the 0-framing like in knot traces (and actually using just +1-framing and -1-framing is sufficient). It's actually a remarkable fact in its own right that every 3-manifold bounds a 4-manifold; this is saying the 3-dimensional cobordism group is trivial. Other-dimensional cobordism groups are not trivial in general.
Every 3-manifold has a diagram, then, consisting of a multi-component knot (known as a link) with each component labeled by an integer (or a rational number if you are ok with "fake" surgeries). There is a whole thing called the Kirby calculus that gives a sufficient set of moves to go between any two such representations of a particular 3-manifold. An extension to this calculus went into Piccirillo's calculations with knot traces -- she cites the classic Gompf and Stipsicz for details.
One use of this representation of a 3-manifold is to construct Reshetikhin-Turaev invariants, which are sequences of numbers associated to a 3-manifold. This is related to the Jones polynomial, and these invariants satisfy a number of wonderful properties that together mean they form a topological quantum field theory (TQFT). I don't know the physics, but I'm under the impression you can interpret it as having something to do with quantum states of anyonic particles.
For books, you might look at Adams "The Knot Book" or Prasolov "Intuitive Topology" to get a substantial taste of knots and low-dimensional topology.
In lots of fields, including CS and different kinds of engineering, this is somewhat common. Some will defer the appointment a year and do a postdoc anyway because that provides time to prepare research problems before the tenure clock starts to tick.
In math and the sciences, you hear a lot more stories of multiple postdocs being required before you can be considered for a professorship. I don't have first hand experience in those domains, however.
It's certainly unusual for a grad student to move straight into a tenure-track position at a research university, although not so much at a primarily teaching-oriented institution.
If the point you're taking from it is "here's a cool breakthrough in knot theory and low dimensional topology," then her title is irrelevant. If the point is "this woman made a discovery so awesome that it was published in the Annals, and got her a tenure track position straight out of grad school," then it's quite relevant.
I've made that comment because the error made the article unbelievable to me, and probably made it unbelievable to other readers who are familiar with the current functioning of the academic job market in maths. I found it somewhat reassuring when I realized that the error was only a minor misunderstanding as opposed to the kind of fabulism I've gotten used from news media.
I mean, I remember a post from Julia Evans, making a Ruby profiler, where she was astonished on how few people were actually working on it [1].
I suspect that in some cases, probably not this one, but in similar theoretical fields, a similar thing might be occuring. And if not, how do we test that? I'm probably not the only one who's curious.
[1] I found a talk of her in which she emphasizes on it:
> So the three myths that I want to start out by talking about are myth one-- to do something new and innovative you need to be an expert-- myth two-- if it were possible and worthwhile, someone would have done it already so you probably shouldn't try-- and three-- if you want to do a new open source project, you need to code a lot on the weekend and your evenings.
https://www.deconstructconf.com/2018/julia-evans-build-impos...
I suspect that this field is vastly under-studied and investigated relative to what it ideally should be.
Generally speaking, it is easier in more theory-driven sub-fields to probe new areas. And indeed it is often rewarded. It's harder when $ is needed for experiments since that becomes more grant process-driven (something which is inherently more risk-averse).
My observation is that usually a few pioneering people push out into a new topic area. Then, whether a community forms around it and starts getting excited about it depends a lot on timing, luck and also resources. Sometimes nothing happens for decades until the stars align and people realize that there's something there.
Sometimes there is (the map coloring theorem), sometimes there isn't (the rest of math.)
> I simply assume that a rigorous research by the best minds of the world has happened, but I never see any data on it, not even anecdata.
Most mathematicians work on areas that interest them, ie. alone or with a colleague in another university.
Never heard of anything systematic involving "the best minds of the world" outside perhaps military projects, and some cooperative research is being done on forums now.
Comparing math and Open Source software development is kind of strange and not helpful.
Anybody can expend a lot of time and effort and successfully write a profiler, if they wanted to. Few people make a career in math.
If you're not a native English speaker, you might want to get checked for ADHD, since your post wasn't very coherent.
Anybody can expend a lot of time and effort to write a profiler... but few people make a career of it. Anybody can expend a lot of time and effort on math... but few people make a career of it.
> If you're not a native English speaker, you might want to get checked for ADHD, since your post wasn't very coherent.
That's a strange suggestion to make after reading a single HN comment, especially when you're basing it off of your own subjective interpretation of said comment.
I thought the parent made a coherent point that people may avoid hard problems because of the assumption that 1) they need expertise that they don't have and 2) someone else is already working on it. The question raised was: how, in general, do we verify those assumptions?
This is what I meant. Though, I do remember I was a bit fuzzy on how to phrase things and opted for a conversational style instead.
Correct, you generally don't get paid for writing Open Source on your own time.
What was the silly point your were trying to make, badly?
> I don't program after work or on the weekend, really. I do write blog posts and draw a lot of comics. But I don't program.
definitely going to steal the idea :)
She's clearly smart, but I'm reluctant to call anyone a genius if it causes us to view their success as, say, wondrous light from a distant star rather than recognizable human effort.
And it happens everyday to many of us, sometimes we are the old ideas people, having a noob show us how it could be done better and faster, and sometimes we are the ones joining an old group and proposing something new we saw elsewhere to apply to one of their stupidly long-standing problem.
I find her story very relatable and a good reminder we should never dismiss the noobs in scientific, creative or high-profit endeavors: they can always bring something we didn't think of by exploring an area we overlooked. And as an example, you can look at the high-speed, youth-driven electronic innovation craze in Shenzhen, often overlooked, where everyone share everything, everyone and everything is new somehow, and exploration is encouraged (and financially rewarded) more than production of blueprints.
“Whenever a new invariant comes along, we try to test it against the Conway knot,” Greene said. “It’s just this one stubborn example that, it seems, no matter what invariant you come up with, it won’t tell you whether or not the thing is slice.”
The Conway knot “sits at the intersection of the blind spots” of these different tools, Piccirillo said.
One mathematician, Mark Hughes of Brigham Young University, created a neural network that uses knot invariants and other information to make predictions about features such as sliceness. For most knots, the network makes clear predictions. But its guess about whether the Conway knot is smoothly slice? Fifty-fifty.
“Over time it stood out as the knot that we couldn’t handle,” Livingston said.
Not my area, but if the article is to be believed, pretty much every new related technique gets thrown at this knot as a matter of course. That's what I mean by "fresh eyes."
Let us dub Lisa Piccirillo a "Space Age Bo's'n Mate".