A Terribly Knotty Problem
How Many Ways Can You Tie Your Shoelaces?
It’s not a trivial question. What’s surprising, though, is that the mathematical tools needed to answer it and questions like it, seem to be on the verge of uncovering a much more inclusive theory that could explain some of the most subtle, fundamental and profound things about the nature of the universe – the mathematical structure of the the three space and one time dimensions we live in.
That’s a long way from knots.
As explained by Julie Rehmeyer in Science news, the story starts about 80 years ago, when mathematician J.W. Alexander found a way to produce a polynomial from any knot. You remember polynomials, don’t you, from high school? They’re just simple relationships between two or more things, that we usually named X and Y. What’s cool about Alexanders method for knots was that no matter how the knot was twisted or distorted, his mathematical machine would always produce the same polynomial. It always recognized the knot the same way, which means that Alexander’s “machine” was “knot invariant”. The problem with it, though, was that sometimes completely different knots produced the same polynomial.
Then, in 1983, Vaughn Jones of the University of California, Berkeley, astonished everyone by creating a new and better knot invariant, one that could distinguish among many knots Alexander’s invariant couldn’t (such as the granny knot and the square knot).
That’s a much better thing. In fact, these polynomials were very easy to compute. Was there a chance that they could tell us something about knots, then? Apparently not.
[T]hey didn’t unlock all the secrets of knots. “If you could make money out of telling knots apart, the Jones polynomial would be a powerful tool,” says Stephen Sawin of Fairfield University. “It hasn’t given a great theoretical understanding, though.” In particular, just looking at the Jones polynomial of a particular knot didn’t seem to reveal much about the knot or its relationship to other knots.
Ah well. Such things as polynomials describing your shoelaces are mere curiosities then. That was the case until the mid ’90s, when “Peter Ozsváth of Columbia University and Zoltán Szabó of Princeton University developed an invariant called knot Floer homology using techniques from symplectic geometry, a branch of geometry with close ties to physics.” At the same time “Mikhail Khovanov of Columbia University developed a new invariant, the Khovanov homology, using techniques from algebra. It could distinguish between any two knots the Jones polynomial could tell apart — and also some the Jones polynomial couldn’t.” So the discovery of these two approaches were independent of each other. Their techniques:
…produced much richer mathematical objects that helped to reveal the structure that underlies knots, the relationships among knots, and the connections between knots and other areas of mathematics. It’s because of this richer structure that both techniques are called homologies. It was as if the Jones polynomial and the Alexander polynomial had been lifted to a new plane.
And it’s fascinating that these men, using different techniques, found independent and distinct ways of answering the same question. In nature, that usually means that what looks separate and distinct really isn’t.
Something else about the invariants has captured mathematicians’ imaginations, too. Khovanov homology and knot Floer homology had very different origins, yet in the end, the two techniques seemed strikingly similar, as if they were linked at a more fundamental level than mathematicians could yet describe. “The whole study seems to be showing pieces of a single, bigger structure,” Khovanov says. Khovanov homology would be one facet of that structure and knot Floer homology would be another.
A deeper, bigger structure? A mathematical structure? What does that mean to the real world?
Sergei Gukov of the California Institute of Technology and the University of California, Santa Barbara has deepened the connection with quantum physics and string theory to “lift” the Witten invariants, just like Khovanov homology lifted the Jones polynomial. Revealing the full structure of the superstructure may be the work of a generation, Ozsváth says.
The payoff from such work may be profound. Knot Floer homology has higher-dimensional analogues that can reveal the structures of three- and four-dimensional spaces, and it is expected that Khovanov homology does as well. Four-dimensional spaces have been especially difficult to understand.
Four-dimensional spaces are interesting things. Most every mathematical relationship between two objects can get “complex”. It’s been noticed by many that in higher-dimensional spaces the complexities have a chance to sort themselves out, and the relationships quickly become simple again. In less than 4-dimensions, well, the relationships never really have a chance to get all that complex anyway. In 4-dimensions, it’s a peculiar situation that complex relationships have a chance to persist, which means that 4-D spaces can have many interesting things happen in them – many “events”.
It’s a funny thing that we find ourselves in a 4-D universe, where the most interesting things happen.
And isn’t it amazing that the problem of uniquely describing knots in mathematical terms has some intimate connection to both quantum mechanics and superstring theory?