
Graduate Student Solves Decades-Old Conway Knot Problem - theafh
https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/
======
pmiller2
Link to paper:
[https://arxiv.org/abs/1808.02923](https://arxiv.org/abs/1808.02923)

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.

~~~
kmill
Yeah, it's pretty cool! On one hand you have knot traces, which are spaces
that completely determine whether a knot is slice, and on the other you have
the Rassmussen s-invariant, which can sometimes tell whether a knot isn't
slice. It turns out that knot traces contain different information from the
s-invariant.

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](https://arxiv.org/abs/1702.03974)

------
melvinroest
From a high level overview what she did feels mostly like a hacker approach:
finding a side channel [1]. I wonder to what extent mathematicians think about
side channels.

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).

~~~
terramex
You could argue that one of the most famous proofs of last 30 years - proof of
Fermat's Last Theorem used such approach. Andrew Wiles had proven that if this
FLT was false, then certain theorem about elliptical curves (unrelated at
first glance and from different area of mathematics) would have to be
violated, and mathematicians already knew it was true.

~~~
MaxBarraclough
I think empath75 is right to point to dualities. I don't think proof-by-
contradiction is really analogous to side-channels.

------
JoeAltmaier
This story will go down in mathematical history. A graduate student attacks an
old unsolved problem, in her spare time, shows up with the solution a week
later. Invents a new approach to topology in the process.

~~~
high_derivative
I wonder how much it helped the creative flow to not know it was an allegedly
unsolvable problem. No fear to fail, no stress, not the feeling something
complex might be needed because everything simple was already tried.

~~~
exmadscientist
And in addition, it wasn't (quite) her field. (This shows somewhat in the
paper linked elsewhere in the comments: you can kind of see the two different
jargons in use.) The power of fresh eyes is often underestimated.

~~~
cosmie
This. I've built my career off of diversity of experience - changing both the
industry and functional role I work in almost every time I change companies.

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.

~~~
pwrdbytmr
I must ask, please do not take this the wrong way, how have you been able to
switch consistently with such ease? And what country do you work, where you
are being respected for having industry-naïvety?

~~~
ed312
Being a software engineer you can readily switch industries but still have
deep knowledge in developing software. Practicing breaking down problems in
different domains will train you up in a more dynamic and flexible style of
thinking (perhaps at the expense of deep expertise in a given area?)

------
generationP
Correction: Her MIT position (a Moore instructor, at least according to her
own website
[https://sites.google.com/view/lpiccirillo/home](https://sites.google.com/view/lpiccirillo/home)
) is _not_ a tenure-track job (see
[https://math.mit.edu/about/employment.php](https://math.mit.edu/about/employment.php)
). It's still one of the best academic jobs a fresh PhD can get these days
(certainly more prestigious than a "mere" postdoc). I don't think you can get
a tenure-track job right out of your PhD, even an MIT fake tenure-track job
(they say it guarantees you tenure, just not necessarily at MIT).

~~~
mehrdadn
> I don't think you can get a tenure-track job right out of your PhD, even an
> MIT fake tenure-track job (they say it guarantees you tenure, just not
> necessarily at MIT).

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?

~~~
generationP
Who?

My guess is that these are "named postdocs" ("[some name] Assistant
Professor", e.g.
[https://www.mathjobs.org/jobs/jobs/14065](https://www.mathjobs.org/jobs/jobs/14065)
or
[https://www.mathjobs.org/jobs/jobs/15707](https://www.mathjobs.org/jobs/jobs/15707)
). Despite their names, they're limited to 3 years and only get renewed in
exceptional circumstances.

~~~
solveit
John Pardon is a relatively recent example. He also became full professor at
Princeton a year after graduating.

~~~
dmix
I see he was also involved with Knots [1]. Can someone explain to me why this
is such an active field in mathematics?

I always hear about it and topology. Makes me want to read a book on it...

[1]
[https://en.wikipedia.org/wiki/John_Pardon](https://en.wikipedia.org/wiki/John_Pardon)

~~~
solveit
Basically, the study of knots is the study of how the simplest 1 dimensional
thing (the circle), can sit in 3-dimensional space. And it turns out that even
this "simple" case is incredibly rich and difficult. So that's a reason to
expect knot theory to be an inherently interesting thing to study. So
topologists, and especially topologists specialising in 3-dimensional objects
were always interested in knots.

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!

~~~
Natsu
> a property of knots which remarkably turned out to have deep connections to
> all sorts of things including quantum field theory

Does this mean that the strings in string theory are knots?

~~~
marktangotango
Absence of proof is not proof of absence?

------
contemporary343
Even in the most technically demanding and theoretical of disciplines (and in
some sense, perhaps especially so) it is creativity and an ability (instinct?)
to see possibilities that others don't that distinguish the best researchers.
This is a wonderful example of that.

~~~
melvinroest
I have immense respect for researchers who venture in these type of
disciplines. I don't think I would be able to do it. I do have a bit of a
daring question: isn't there a slightly more fine-grained way to quantify that
every nook and cranny of such a problem has indeed be researched by
researchers? 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.

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...](https://www.deconstructconf.com/2018/julia-evans-build-impossible-
programs)

~~~
redis_mlc
> isn't there a slightly more fine-grained way to quantify that every nook and
> cranny of such a problem has indeed be researched by researchers?

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.

~~~
throwaway_sun
> 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.

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?

~~~
melvinroest
> 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.

------
diegoperini
Solving decades-old math/science problems deserves a name mention in the
title.

------
m3kw9
“ To make a knotted object in four-dimensional space, you need a two-
dimensional sphere, not a one-dimensional loop. ”. You just need to assume
it’s right.

~~~
saagarjha
You can't have a one-dimensional knot in the fourth dimension, because it's
trivial to pull it apart if you move part of it into the fourth dimension.

------
pinacarlos90
am I the only one who got super exited with the progress bar on
quantamagazine.org? it tracks how far into reading the article you are!!

definitely going to steal the idea :)

~~~
Sharlin
In the olden times we called those ”scroll bars” and you didn’t have to
implement them yourself.

------
essem
I see Piccirillo's paper was published in the Annals of Mathematics in
February. Anyone know if Conway got to see it before he passed?

------
j7ake
What a great story. It’s something we can all learn from.

~~~
grayclhn
It's a great story. I'm curious what you think everyone can learn from it,
other than "geniuses can occasionally solve longstanding intractable problems
very quickly."

~~~
papeda
My takeaway was more that fresh eyes and expertise in a related field is a
powerful problem-solving combination.

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.

~~~
grayclhn
Not to put too fine a point on it, but that problem resisted years of “fresh
eyes,” “related expertise,” and “recognizable human effort,” but she still
walked in and clobbered it. Some types of success aren’t very relatable.

~~~
xwolfi
Depends what you define by fresh eyes. You can have fresh people with old
ideas, trying constantly the same thing, until an idiot comes along with an
exotic expertise and solve your problem.

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.

~~~
grayclhn
From TFA:

 _“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."

------
smitty1e
Outstanding work.

Let us dub Lisa Piccirillo a "Space Age Bo's'n Mate".

[https://en.wikipedia.org/wiki/Boatswain](https://en.wikipedia.org/wiki/Boatswain)

------
wesammikhail
Too bad Conway didnt live long enough to see this.

~~~
ttymck
If I'm not mistaken, it says her findings were published in Annals in
February, and Conway passed in April.

------
pgp001
rip conway

