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Math Science

Major Advances In Knot Theory 230

Posted by kdawson
from the if-it's-not-theory-then-it-must-be-practice dept.
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."
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Major Advances In Knot Theory

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  • by $0.02 (618911) on Saturday November 01, 2008 @03:41PM (#25596895)
    How many ways are there to tie your shoelaces? The answer is very easy ... knot.
  • by poached (1123673) on Saturday November 01, 2008 @03:41PM (#25596897)

    42

    • by gsgriffin (1195771) on Saturday November 01, 2008 @05:03PM (#25597509)
      Its actually 84. You forgot that you can always double-knot each of them too.
  • Unless... (Score:5, Insightful)

    by Slur (61510) on Saturday November 01, 2008 @03:47PM (#25596949) Homepage Journal

    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.

    • Re: (Score:3, Funny)

      by ceoyoyo (59147)

      Yes, if we discover hard AI and experience a singularity then mathematicians will be obsolete. Of course, so will the rest of us. I'm still going in to work on Monday. How about you?

    • Actually, it is a theorem that computers are not enough to do maths.
    • by Kjella (173770)

      Computing power has very diminishing returns, you can punch up a possible result in the computer and it'll pound throught it for a few million numbers or run a few million simulations. If it doesn't come back with a counter example, you might be on to something. But the chance that it'll actually give you any more useful information if you could run billions or trillions of tests isn't really all that great. There's the odd case like the four color theorem but it still took a lot more manhours than it took

    • by khallow (566160)
      Currently, computing power is not the bottleneck, but the algorithms. Exponentially growing computing power is just not that useful if you have to brute force search a knot space.
    • Re:Unless... (Score:5, Insightful)

      by Anonymous Coward on Saturday November 01, 2008 @09:10PM (#25599217)

      No. As a professional computer scientist, I think it is safe to say mathematicians are about the last people in the world to be in danger of losing their job to computers.

      If there's one thing computer science algorithmic theory has told us, it's that computers absolutely do have a limit on what they can do, no matter how fast the microchip gets. Complete searches (and that is what we're talking for computer proofs) are NOT getting any more feasible over time. 2^10000 branches will never be traversable.

      Pretty much the best possible scenario for computer proofs is basic geometry. After all, in US high school, students are taught "2-column" proofs that a computer could actually handle. And even here, computers suck compared to mediocre mathematicians. Why? Because anybody can trace basic implications like a computer does - that's the easy part. The ONLY real hard part is the flash of insight that computers can never do - e.g. why don't we consider this point that is only tangentially related and see how it somehow holds all the structure to solving the problem.

      Once you get into modern math, say knot theory, computers are completely hosed. A math paper might be 100 pages of prose, 80% of which might be insights like that thing above, and 20% of which might be basic implications that a computer can handle. And actually, it couldn't, because 20 pages in prose = 2000 pages in logic statements, and a computer will never be able to traverse that deep.

      There's a reason that every important computer proof up until now has relied on 0 insight from the computer... even something like the 4-color theorem is only using a computer to algorithmically check a finite number of trivial cases that would be impractical to check by hand. This approach does not generalize to making mathematicians obsolete.

      • 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
      • by Trepidity (597) <delirium-slashdot AT hackish DOT org> on Sunday November 02, 2008 @04:38AM (#25601177)

        "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 [vcu.edu] provides a useful overview of some of the work.

    • by mochan_s (536939) on Saturday November 01, 2008 @09:49PM (#25599453)

      ..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.

      But, mathematicians have already proved that a computer will never be able to take a mathematician's job.

  • QED (Score:2, Funny)

    by Anonymous Coward

    Loop and Swoop
    Bunny Ears

    Where's my Nobel

  • !theory (Score:5, Funny)

    by russlar (1122455) on Saturday November 01, 2008 @03:52PM (#25596979)
    So, can we abbreviate this "knot theory" to "!theory"?
  • by fahrbot-bot (874524) on Saturday November 01, 2008 @04:06PM (#25597057)
    ... and relies on the cunning use of a rabbit, tree, and hole to tie shoelaces [patentstorm.us].
  • by nx6310 (1150553) on Saturday November 01, 2008 @04:17PM (#25597139)
    the inventor of the shoe lace could be the answer to all our four dimensional space quetions?
    • Re: (Score:3, Funny)

      by mikael (484)

      Yes, String Theory research will be replaced by Tangled Shoelace Theory - the theory that the space-time continuum is in fact a giant cosmic tangle of shoelaces, and that these shoelaces only get untangled in the presence of a large gravitational object, thus causing space-time curvature. In the presence of a massively strong gravitational object such as a black hole, these shoelaces actually break in half, with one half going into the black hole and the other half left dangling in this universe. Thus we s

      • by swillden (191260)

        Yes, String Theory research will be replaced by Tangled Shoelace Theory - the theory that the space-time continuum is in fact a giant cosmic tangle of shoelaces, and that these shoelaces only get untangled in the presence of a large gravitational object, thus causing space-time curvature. In the presence of a massively strong gravitational object such as a black hole, these shoelaces actually break in half, with one half going into the black hole and the other half left dangling in this universe. Thus we see no light as all the shoelaces are now in a tightly tangled ball that has no connection to this universe.

        Okay, that's all fine, but... who wears the shoes?

  • by 3seas (184403) on Saturday November 01, 2008 @04:22PM (#25597173) Journal

    ....untie the knot my cat did with the mop?

  • by Prius (1170883) on Saturday November 01, 2008 @04:25PM (#25597207) Journal
    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.
    • by Daimanta (1140543)

      Every time you tie your shoes, the universe kills a kitten(through 4-dimensional knot strangulation). Think of all the kittens!

  • by perlstar (245756) on Saturday November 01, 2008 @04:32PM (#25597265)

    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.

  • by Louis Savain (65843) on Saturday November 01, 2008 @04:51PM (#25597413) Homepage

    I'm just wondering. One never knows with math.

    • Re: (Score:3, Funny)

      by gmuslera (3436)
      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.
    • by Arimus (198136)

      Just open a draw containing various cables that has been left for a few months - none of them knotted when you put them in but you can bet when you take them out they'll be more knotted than a knotty thing

    • by jd (1658)

      I have a simple proof of such a knot, but the margin contains too few shoelaces to contain it.

      PS: When asked to pull yourself up by your bootlaces, you can now ask for the Jones Polynomial required to do this.

    • uhhh....

      the definition of a Knot is something that cannot be tied or untied.

      only a tangle can be ties and untied.

  • by Better.Safe.Than.Sor (836676) <matthew02121NO@SPAMrogers.com> on Saturday November 01, 2008 @04:59PM (#25597475) Journal
    I prefer the "velcro" theory.
  • What are the implications for hyperbondage?
  • She's discovering the joys of shoelaces now, and you want to talk about knots.. Boy, oh boy. She's gonna be a mathematician for sure!
  • by PolygamousRanchKid (1290638) on Saturday November 01, 2008 @07:41PM (#25598585)

    .. I see, the cable of set of cheapo earphones, a Thinkpad power cable, power cables to Logitech speakers, a usb cable to a Logitech wireless (ha, ha) keyboard and mouse, a USB to my porn drive, a USB to a DVD Drive that I never use, a cable to really fucking expensive Shure headphones (hey, I was looking for those), a USB cable to fuck-knows-where, Nokia teeny-tiny power cables . . . all messed up better than a Gordian knot.

    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

  • by harlows_monkeys (106428) on Saturday November 01, 2008 @08:41PM (#25599037) Homepage

    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. [fieggen.com]

  • by TheEmptySet (1060334) on Sunday November 02, 2008 @06:33AM (#25601471)
    So here I am at home on a Sunday morning reading the news and I find you guys embroiled in a huge argument about my area of research. Quite a pleasant surprise actually. So in response, here's a short list of uses of knot theory:

    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)

    ...and I'm sure I have missed out plenty. My point is that mathematics is full of weird abstract nonsense, which is not actually nonsense when you look deep enough. There is after all a reason why we study it.

    It's not all just "brain-wanking".

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