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Major Advances In Knot Theory
An anonymous reader sends us to Science News, which is running a survey of recent strides in finding an answer to the age-old question: How many ways are there to tie your shoelaces? "Mathematicians have been puzzling over that question for a century or two, and the main thing they've discovered is that the question is really, really hard. In the last decade, though, they've developed some powerful new tools inspired by physics that have pried a few answers from the universe's clutches. Even more exciting is that the new tools seem to be the tip of a much larger theory that mathematicians are just beginning to uncover. That larger mathematical theory, if it exists, may help crack some of the hardest mathematical questions there are, questions about the mathematical structure of the three- and four-dimensional space where we live. ... Revealing the full ... superstructure may be the work of a generation."
How many ways are there to tie your shoelaces? The answer is very easy ... knot.
If enithin kan gow rong it whil. (Murfey)
42
Revealing the full... superstructure may be the work of a generation.
..assuming computers cease making any new advances.
Mathematicians do rely on their ability to spot patterns and sense implications that no computer can likely sift for today. But this will not always be the case.
-- thinkyhead software and media
Loop and Swoop
Bunny Ears
Where's my Nobel
So, can we abbreviate this "knot theory" to "!theory"?
Anybody want my mod points?
e can't be serious.
Oh, yeah, it's not easy to pad these out to 120 characters.
You sound like you'd take the single most important^h^h^h^h^h^h^h^h^h publicized problem of the day and have everyone working on it, ignoring all of the other interesting stuff that might be possible.
Yes, there are weighty problems in the world, and I'm not trying to dismiss them. Thinking about them exclusively, however, will recover the now but it won't provide any advancement for the future.
Let's do both.
... and relies on the cunning use of a rabbit, tree, and hole to tie shoelaces.
It must have been something you assimilated. . . .
The world has been in far worse situations than it's in now. The transient problems of immediate political and social realities shouldn't stop a few people from investigating nature's deep questions via science and mathematics.
Once upon a time, I was similarly bored by this area of abstract research. But about a year ago, I attended a seminar where a guest lecturer was a mathematician who applied knot theory to the physical modeling of life processes involving the winding and unwinding of DNA in Chromosomes and the folding and unfolding of peptide strings in protein formation. I didn't understand half of the lecture. But one very important point I got out of it is that no matter how abstract and esoteric a subject might be, there is immense value to be obtained if it can be utilized to model physical processes we seek to understand.
Stay sentient. Don't drink bad milk.
Oh really? Would you also say studying topology in general is unimportant? Why or why not? Since you're able to discern which branches of mathematics aren't "important", you're clearly a mathematical authority, so please feel free to enlighten us.
Funny thing, last year I competed in the Stanford Math Tournament and one of the rounds involved knot theory. http://sumo.stanford.edu/smt/ but for some reason they don't have that round in their list of problems, but these are the solutions: http://sumo.stanford.edu/smt/2008/Solutions/power-soln.pdf.
All your base are belong to Wii.
the inventor of the shoe lace could be the answer to all our four dimensional space quetions?
This is just not that important.
Are you sure?
When algebra was invented, did people think that was important? What about geometry or calculus? What about number theory? Would Euler's study of the Seven Bridges of Konigsberg have been important to you? Probably not. But it did lay the foundations for modern graph theory which engineers use to design computer networks.
Reading code is like reading the dictionary - you have to read half of it before you can go back and understand it.
....untie the knot my cat did with the mop?
You wouldn't believe what just thinking about this is doing to my stomach...
Sleep your way to a whiter smile...date a dentist!
This just in: Physicists have just now revealed that String Theory has nothing to do with the fabric of our universe, and everything to do with teaching toddlers how to tie their shoes.
I think the important thing is that when you're investigating new areas of mathematics and it's _hard_, that's because the tools you're using are not suited for investigating this issue. So you invent a new tool, and that new tool can be applied in many, many places.
Hard problems are only hard because we're using the wrong tools.
How to use coral cache: http://slashdot.org.nyud.net:8090/~oscartheduck
Man, I haven't posted in years... but there's a great book by this title written by two mathematicians. They talk about the topology of knots as well as the history of ties. Which actors/celebrities wore what tie knots, etc.
I can't seem to locate my copy at the moment, but from what I recall, there are an infinite number of potential knots, but they are classified by the number of sequences in them. And within a certain number of steps, (I think 5) there are 85 possible ways to tie a tie. Then they rank them by symmetry and a copule other criteria.
I recommend it to anybody who is interested in this subject. It's out of print, but it's still possible to find a copy for sale online.
I can't believe I got moderated as a troll
Why? You made a whiny, irrelevent complaint that dismisses the role of pure research in the larger advancement of our knowledge of how the universe works... the very sort of thing that always plays a role in advancing our ability to make more efficient use of energy, more realistic predictions about the behavior of complex systems, and more innovative technological use of things we think we have already fully, or most effectly exploited. This whole "the human race is incapable of doing two things at once" BS never ceases to amaze me. How do you even get out of bed in the morning? Make coffee... take a crap... which to do first? Gaah! I'm paralyzed! Which is the most important fish to fry?
In other words, you're scare mongering and - if we can assume you have a passable IQ which would suggest you might know better - clearly trolling. And, voila, you were thusly modded.
Don't disappoint your bird dog. Go to the range.
Suppose you tell us all how solving this knotty problem will help anyone or anything.
Let's pretend we're in the early 1700s. Leonhard Euler is writing the first ever paper on a field of study called Graph Theory. Simply put, he's figuring out answers to questions about how to arrange circles and lines. Meanwhile, there's fucking WARS going on (Polish succession is going on concurrent to writing this paper; Seven Years' war happens a couple decades later). There are goddamn wars on Euler's front door, and he's writing papers about lines and circles?! What a prick.
Oh, by the way, without Euler's work we wouldn't have computers, organized roads, efficient data models, efficient sorting algorithms, or countless other instruments that are critical to today's society.
Don't trivialize work that you don't understand.
Let me introduce you to ^W.
It's a great tool for those writing pseudo-ironic posts who are, at the same time, concerned with the preservation of the valuable resource of ones and zeroes...
I spent 15 years of my life in physics of proteins. Theory of knots in protein folding is nothing more than fancy mathematical excursion (though knots do matter, in very simple form). The importance of "theory" in those sciences is way overblown. It was fun to do to satisfy your own intellectual curiosity, but it's a dead end on the road of science.
I do not believe in karma. "Funny"=-6. Do good and forbid evil. Yours, Oft-Offtopic Flamebaiting Troll.
I'm just wondering. One never knows with math.
I prefer the "velcro" theory.
It's all history, man. -anon
Im more worried about the knots that can be tied but not untied. My shoes are about to get the Alexander's universal knot solution.
I'm getting too old for this slashdot shit, I guess.
+ 1 insightful
Don't trivialize work that you don't understand.
To further disabuse the OP of a misconceived notion, this isn't just "how many ways are there to tie your shoes". This is trying to work out a rational system of knot classification.
The key thing to realize is that knot theory applies to a lot more than untangling rope. If you use the right assumptions and definition, certain problems, which have nothing to do with rope, can be modeled as knot problems. If we could solve/simplify knot theory, we are this much closer to solving a range of related problems. None of which involve shoelaces.
Oh, and the GGP gave the OP a good example (by analogy): Elliptic curve cryptography. An elliptic curve is pretty esoteric stuff: "An elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is ... an abelian variety ... and O serves as the identity element." Must have seemed pretty pointless to other people when the first person worked on it. Yet, once the background theory was worked out, lo and behold, you can use them to make a pretty good encryption scheme! They were also key in proving Fermat's Last Theorem.
wow. mathematicians make such trigger-happy moderators.
modded troll in 3, 2, 1...
Requiem for the American Dream
How is that hard? He just has to go through his address book, ask each person what they do and every time one says "mathemetician" he adds 1.
Confucius say, "Find worm in apple - bad. Find half a worm - worse."
Back when I was going to school for my Comp Sci degree, I was force-fed a lot of calculus.
Roughly twice as much calculus as was typical, because my disinterest (and the resultant lack of success) required me to take almost every single calculus course twice.
No sooner was I free of school than I brain-dumped every single last integral, deriviative, partial derivative, chain rule, trigometric identity... the lot of it. Good riddance to bad rubbish.
And then, some time later, I was trying to make my race car go faster. The problem was optimising the suspension for maximum grip, and to that end, I had affixed linear potentiometers to my suspension so I could record suspension position during a race.
Pretty soon, I had tons of data relating position to time. Pretty graphs, but aside from max/min/mean deflection data, pretty useless.
Until I started thinking about "position to time... position to time... where had I heard that before?"
That's right - my old arch-nemesis, calculus, suddenly proved useful. Deriving that position information gave me suspension velocity, and suddenly I knew EXACTLY what suspension velocities the car was seeing in actual competition. Given that I had a device that measured shock force as a function of velocity (that's how a shock works) I could now tune shocks independant of the driver's ass-dyno.
That resulted in a HUGE leap forward in my performance.
Don't dis abstract math - you never know when it'll pay off.
DG
Want to learn about race cars? Read my Book
but I'd hardly call it an age old question. Never heard of it.
Does that mean you're knot interested in it?
This whole "the human race is incapable of doing two things at once" BS never ceases to amaze me. How do you even get out of bed in the morning? Make coffee... take a crap... which to do first? Gaah! I'm paralyzed! Which is the most important fish to fry?
Er... are you saying there's a way to take a crap and make coffee at the same time? I'm curious, but at the same time I don't think I want to know...
When I read things such as this I like to take a moment to let the dumbfounded feeling wash over me.
This is just not that important.
You only say that because you have yet to be involved in a serious shoe-tying accident.
Back when I was going to school for my Elementary School diploma, I was force-fed a lot of arithmetic.
Roughly twice as much as was typical, because my disinterest (and the resultant lack of success) required me to take almost every grade twice.
No sooner was I free of school than I brain-dumped every single addition, multiplication, subtraction, division, counting... the lot of it. Good riddance to bad rubbish.
And then, some time later, I was trying to make my paycheck go farther. The problem was optimising the spending for maximum personal happiness, and to that end, I had collected all of my receipts so that I could record where I was spending my money during the month.
Pretty soon, I had tons of data indicating where my money was going. Pretty numbers, but aside from a few expensive items, pretty useless.
Until I started thinking about what I could do with a set of numbers.
That's right - my old arch-nemesis, arithmetic, suddenly proved useful. Summing the money spent in different categories gave me totals, and suddenly I knew EXACTLY where my money was going in an actual month. Given that I had measured how much money was spent on each purchase (that's how receipts work) I could now properly budget my spending.
That resulted in a HUGE leap forward in my quality of life.
Don't dis abstract math - you never know when it'll pay off.
AC
And I have spent 5 years of my life on the topology of proteins. It is not quite true to refer to "knots" when talking about proteins, as Professor Taylor has shown (http://www.nature.com/nature/journal/v406/n6798/full/406916a0.html) that only a few proteins are actually 'knotted'.
However, mathematical theory of tangled strings is as important as simulations. Estimating the total number of folds, for example. More than just a fancy excursion - but maybe not to your taste?
But because he doesn't think mathematical theory is important, he'll talk to a few dozen people and he'll have "11111".
Being a graduate student in mathematics , I can safely assert that knot theory is actually a significant area of modern mathematics. There are numerous textbooks about it.. If you read the article you would know that Jones & Witten received a Fields Medal, which is the most prestigious award in mathematics, for their work on classifying knots.
Why it's positively tied up in knots!
Sleep your way to a whiter smile...date a dentist!
But I digress. If some mathematician can come over with a theory, and sort this mess of knots out, I'm buying the beer.
And pizza
Schroedinger's Brexit: The UK is both in and out of the EU at the same time!
Yes, this is important.
What do you think where new ideas on saving the world or building a better one will come from? TV studios? Politicians? Hollywood?
Research like this is the foundation of all progress. Note: Not this one specifically, I said "like" this. A lot of the things that you probably wouldn't live very well without started out as ideas with no visible use.
Assorted stuff I do sometimes: Lemuria.org
For those less interested in theory, and more interested in choosing a lacing pattern and a good knot for their shoes, I recommend Ian's Shoelace Site.
A mere comment about priorities, relative importance of issues, and so forth. In any case, I was not the only one to make such a comment.
Frankly, mathematics is more important than any other issue. You just fail to realize the practical applications that mathematics has in everyone's life. The most basic reason that anyone on earth has a standard of living above that of hunter gatherers is because of mathematics; knowing seasons and how to plant crops relied on rudimentary mathematics, and modern farming relies on advanced chemistry and biology, which have as their basis the mathematics of stoichiometry and statistics. Not to mention engineering which makes heavy use of mathematics and physics in order to create the machines necessary for our massive population.
In short, I'd rather see advances in mathematics than I would the elimination of world hunger; without further mathematical and scientific discoveries, even nations with plenty will just exhaust their resources and revert to poverty and starvation.
> e can't be serious.
of course knot. e can't even round correctly. should be 2.7183. damn truncator.
He's right.
http://en.wikipedia.org/wiki/Knot_(mathematics)
You know, there is a difference between trolling and pointing out the flaws in your reasoning. Just saying.
OK, but *apart* from computers, organized roads, efficient data models, efficient sorting algorithms, and countless other instruments that are critical to today's society, what has Rome^h^h^h^hresearch ever done for us???
I am anarch of all I survey.
So, I think that your statements are an accurate assessment of things like Computer Algebra Systems. Such systems approach mathematics in a way similar to how humans have traditionally tried to solve mathematics. However, there are other ways of doing mathematics with computers. Such as various systems of simple abstract rules. I'm not saying it will necessarily lead to breakthroughs in traditional areas of mathematics, but, it is one of the few areas of research that is truly trying to approach mathematics in computer-centric way.
"Tracing basic implications" is hardly the only thing computers do in mathematics; there is plenty of work on the "flash of insight" part, which computers have done successfully on a number of occasions. In particular, there's a long body of work on conjecture-generating systems, which don't try to prove things, but look for conjectures that: 1) would be interesting if true; and 2) seem that they could at least plausibly be true. Generating conjectures is historically a large part of the creativity in mathematics, and in some areas, computers are getting good enough at it that professional mathematicians use conjecture-generating software to get ideas for interesting problems to work on or useful lemmas to prove on the way to another problem.
This survey provides a useful overview of some of the work.
10 PRINT CHR$(205.5+RND(1)); : GOTO 10
1) Tying your shoelaces (but of course no one cares)
2) Studying supercoiling of DNA (how it wraps itself up into a small space yet still wriggles enough to present all of it's length at short notice for interactions with cells' other mechanisms)
3) The geometry of three dimensional space (all closed oriented three dimensional spaces can be constructed from knots and the three dimensional sphere! So knot theory has major applications to 3D geometry)
4) The geometry of four dimensional space (for example, surfaces in 4D spanning between knots can be used to specify exotic smooth structures. The existence of such shocked the world of geometry in the 80's)
5) TQFT, Mirror Symmetry, Quantum Gravity etc (the tools developed in and around knot theory are one facet of a huge push in mathematics to forge a better understanding of some of the deepest ideas in modern theoretical physics)
It's not all just "brain-wanking".