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Major Advances In Knot Theory

Posted by kdawson on 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."

3 of 230 comments (clear)

  1. The 85 Ways to Tie a Tie by perlstar · · Score: 5, Informative

    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.

  2. Practical shoelace advice by harlows_monkeys · · Score: 4, Informative

    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.

  3. A few applications of knot theory by TheEmptySet · · Score: 4, Informative
    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".