A math problem stumped experts for 50 years. This grad student from Maine solved it in days

[Nevilledog]

https://www.bostonglobe.com/2020/08/20/magazine/math-problem-stumped-experts-50-years-this-grad-student-maine-solved-it-days/

Half a century ago, a brilliant young mathematician named John Horton Conway discovered, of all things, a knot. This knot wasn’t something you’d be likely to encounter in the real world. You could certainly create it out of string if you wanted to, but, generally speaking, it existed only in Conway’s calculations. There are thousands upon thousands of these kinds of conceptual tangles in a bewildering corner of mathematics known as knot theory, but even there Conway’s discovery was special — not so much for what it was, but for what it might or might not be. Yes, that is confusing, but when talking knot theory, it’s best to accept that things are going to get a little fuzzy.

In any case, the Conway knot is hardly remarkable at first glance. With just 11 crossings, or places where it overlaps itself, it’s rather nondescript by the standards of higher-dimensional knot theory. But the knot has one property that made it the subject of intense mathematical scrutiny. Conway, who died recently at age 82 of complications from COVID-19, made innumerable contributions to the field of mathematics, yet it was his knot that specialists would return to again and again. And again and again, these decorated mathematicians were unable to find a solution to what became known as the Conway knot problem.

The problem had to do with proving whether the Conway knot was something called “slice,” an important concept in knot theory that we’ll get to a little later. Of all the many thousands of knots with 12 or fewer crossings, mathematicians had been able to determine the sliceness of all but one: the Conway knot. For more than 50 years, the knot stubbornly resisted every attempt to untangle its secret, along the way achieving a kind of mythical status. A sculpture of it even adorns a gate at the University of Cambridge’s Isaac Newton Institute for Mathematical Sciences.

Then, two years ago, a little-known graduate student named Lisa Piccirillo, who grew up in Maine, learned about the knot problem while attending a math conference. A speaker mentioned the Conway knot during a discussion about the challenges of studying knot theory. “For example,” the speaker said, “we still don’t know whether this 11-crossing knot is slice.”

*snip*

[jayfish]

1. "How Do You Like Them Apples" -NT-

[tonedevil]

14. LOL /nt

[ProfessorGAC]

21. Well Played!

I love that movie!

[BlancheSplanchnik]

2. I don't even like math but I'm still nerdy enough

To love this!

[gratuitous]

3. Things get a little fuzzy in knot theory?

A frayed knot?

[MineralMan]

4. I believe that my earbud wires have tied themselves into that knot.

Is it "slice." Probably not, but I bought a new one.

[House of Roberts]

5. In the case of your earbud wires,

slicing them to untangle them would have rendered them inoperable anyway, so your simplest option was to buy new ones. I am someday going to buy bluetooth ones with no wires.

[MineralMan]

10. What is most annoying is that I have now heard of "knot theory,"

which means I will have to spend the next hour or two finding out more about that. I cannot resist exploring something having to do with mathematics that has such an interesting name.

[MineralMan]

18. Recently, I discovered that my spool of weed trimmer

nylon cord had fallen off its shelf and unspooled from its spool. The resulting tangle was a great hindrance to me, as you can imagine. So, whenever I needed 12 to 16 feet of the cord to refill my trimmer's spool, I faced a serious problem of untangling enough of it to do the job. The last time the last foot of cord flew away from my trimmer, I had had enough of the tangle.

Did I go and buy a new spool of the stuff? Certainly not. That would have meant wasting perfectly good .090 nylon monofilament. So, I took the entire tangle into the house, got a pair of scissors, and began untangling the mess. I found a loose end and worked it through the tangle, again and again, until I had about 12 feet of untangled cord. then I cut that and rolled it up and taped that coil together.

Then, I returned to the tangle, and repeated the process, over and over again, until I had about 20 12' lengths all neatly coiled and taped. Now, I should have enough weed trimmer cord to last me for several years without buying a new supply spool.

However, I spend about an hour untangling the tangle - an hour I will never get back again. Such are the minor frustrations of life, it seems.

I doubt that understanding knot theory would have helped, somehow.

[House of Roberts]

19. I bought a Ryobi cordless trimmer.

It uses the same batteries as my saw and drill. The replacement string is prewound on spools. They're not cheap, but it takes about 30 seconds to remove the empty and install the full one. The drawback to this unit is it advances the string when you let go of the trigger, instead of tapping it on the ground. You have to develop the habit of keeping the trigger squeezed unless you want the string advanced. Tradeoffs are everywhere. You either use more battery or use more string.

[MineralMan]

20. The string isn't really that expensive.

I don't mind winding it on the trimmer spool, really, since it forces me to take a short break. Next time I buy a big spool of it, though, i'm going to build a box for the spool with a hole in it for the string and an axle for the supply spool. Won't take but a few minutes to knock something together out of some 1 x 4 scrap and a few drywall screws. Then, I will never have to deal with a big tangle of the stuff again.

The last big spool of it I bought was 10 years ago, so, the next spool will no doubt be the last one I ever buy, given that I'm 75 years old now.

[hunter]

6. Conway knot:

https://en.wikipedia.org/wiki/Conway_knot

[Timewas]

7. Weird knot

Has no beginning or ending

[hunter]

9. Mathematical knots are closed loops.

While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined together so that it cannot be undone, the simplest knot being a ring (or "unknot" ).



https://en.wikipedia.org/wiki/Knot_theory

[MineralMan]

13. Interesting, and closely related to topology.

However, both topology and knot theory hurt my brain, so I will leave them to others.

[Lucky Luciano]

24. It is a branch of topology.

It studies the embeddings of S^1 into 3 dimensional Euclidean space usually denoted R^3.

S^1 refers to the one dimensional sphere...better known as a circle.

S^1 will always just be a circle, but what knot theorists study is the complement of a circle’s embedding into R^3. Different embeddings can and do create topologically different complements in R^3.

[Hugin]

27. I remember reading a Time/Life book on Mathematics when I was young...

It had a section on topology in it.

I walked away with the firm conviction that everything was either a sphere or a doughnut in it's simplest terms. I wondered what all of the fuss was about. I was eight. Which I suppose is a knot. If anyone needs me I'll be trying to figure out if 8 is a 'slice' or knot.

[panader0]

15. I think I saw this on "Pretzels Gone Wild".

[Harker]

16. If it had salt on it... n/t

[Hugin]

26. +1

You too, eh? :heehee:

[Hugin]

25. Mercy. Now, I'm hungry for a pretzel.

Lightly salted, warm, and with a cheese dip in the park.

Oh, great. Yet another COVID craving to contend with... Thanks.

[LAS14]

8. I wish I could understand this. I skimmed the whole article. Seems neat. Greenwood, ME, is...

... not far from our summer place. Very un-coastal western Maine.

[Buckeye_Democrat]

11. Congrats to her!

I've worked on some unsolved "number theory" problems in the past, more like a part-time hobby, but was never successful.

Fooled myself a few times, though! Got pretty excited while also thinking, "I surely made an error in the proof somewhere!" And I indeed always found a mistake somewhere in the proof. Lol.

Edit: One of my most satisfying proofs was the impossibility of the "Three Utilities Problem" while taking a course in Graph Theory long ago in college. I figured it was already solved, but worked on it anyway because it was one of the puzzles my Dad presented me when I was a little kid. Wikipedia summarizes a different way to prove it, but Graph Theory worked too.

https://en.m.wikipedia.org/wiki/Three_utilities_problem

Edit2: Another funny memory from college was when I took the Putnam Exam during my junior year. During the lunch break after the first half of the exam, all of the engineering students who elected to take it were boasting about how it wasn't very hard. I was so depressed because I knew that I proved nothing, and I was majoring in math!
I did better during the second half, completely solving one and getting some credit on another.
The math professor who proctored it told me how we did, getting an "honorable mention" as a school for our three-student team. (He had previously picked me and two other math majors to represent our school.) I asked how the engineering students did, and he said they ALL got scores of ZERO! (The three students he previously picked were the only test-takers who got points at all.)

[aggiesal]

12. Because the Boston Globe has a paywall, I read about it here ...

https://www.quantamagazine.org/graduate-student-solves-decades-old-conway-knot-problem-20200519/

There is a video of John Conway describing knot theory, and to my surprise, I actually understood it!
It's only a 3 minute video, so it's worth watching.

[panader0]

17. I'm stilled awed by the Mobius strip.

[lastlib]

23. I trip over the Arby's logo......

[crickets]

22. That's so neat, especially since Piccirillo wasn't a childhood math whiz.

Good for her! Given the challenges women can face when pursuing careers in science, this was distressing:

By and large, Piccirillo has felt welcomed and encouraged as a female mathematician. “But now and again, things happen,” she tells me. “For example, in grad school, I would receive notes in my department mailbox commenting on my appearance.”

Rather than letting it get to her, she's determined to make a difference for others coming along behind her:

When I ask her about her goals, Piccirillo says one of her priorities is to help grow and broaden the mathematics community. “There certainly are many young women, people of color, non-heterosexual, or non-gender binary people who feel put at an arm’s length by the institution of mathematics,” she says. “It’s really important to me to help mitigate that in any small ways I can.”

Her modesty is refreshing, as is the way she tries to demystify higher mathematics as something that doesn't require the brain of a prodigy to understand or excel. Great article!