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."

Comments

The 85 Ways to Tie a Tie

[perlstar]

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.

Re:The 85 Ways to Tie a Tie

[fizzup]

Linky.

Practical shoelace advice

[harlows_monkeys]

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.

Re:Unless...

[Mr. Slippery]

I'm assuming you are thinking of http://en.wikipedia.org/wiki/G%C3%B6del's_incompleteness_theorems and not some other very recent development, there is nothing that says a computer cannot do math/make proofs. Only that with a finite set of axioms, a computer (or a person) cannot prove an infinite number of things.

Godel doesn't say that an infinite number of propositions cannot be proved from a finite number of axioms. An infinite number of propositions about geometry can be proven from the handful of axioms of Euclid; there are an infinite number of right triangles, for example, and if we had an infinite set of geometry students we could keep each of them busy with trivial proofs about them like "Prove that angle ABC is 42.2718 degrees".

What Godel says is that in any reasonably complex system, there exist propositions which are true but cannot be proven.

A few applications of knot theory

[TheEmptySet]

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".