Having a knotty time [geek moment warning]

Science News has a very interesting article on some major developments in Knot Theory. In Unknotting Knot Theory, Julie Rehmeyer sums up the issue(s) rather well:

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.

I will admit that I am a bit of a math geek. That said, at the core of what has been happening is some fascinating work by Peter Ozsváth (Columbia U.) and Zoltán Szabó (Princeton U. - Go Tigers) developed an invariant called knot Floer homology. Drawn from techniques used in symplectic geometry, a branch of geometry with close ties to physics, it appears to be opening some very cool new areas. For example, regarding the knot shown above:

Until recently, no one could prove that there's no way to untangle this knot by crossing the strands through one another just once. Knot Floer homology finally provided a proof.