I figure that since I’m going to Dartmouth on Monday (my title and abstract aren’t posted, unfortunately), I should finally say something about what I do. Rather than dive right in, I’ll just talk about knots.
A knot is a mathematical idealization of a tangled-up loop of string in space. Formally, it’s a (smooth) path in space that closes up at the end. The thing you tie in your shoelaces is not a not, since it has two loose ends. If you actually used a knot, you couldn’t ever untie them!
Speaking of untying knots, it seems intuitively obvious that if we pick up a loop of string, move it around, and never break the loop, we still have “the same” knot as we started with. So we have to adjust the previous definition a bit: knots are smooth closed paths in space, but if we can deform two such paths into each other (for some suitable definition of “deform”) then they’re really the same knot. What we want to know is, “how can we tell if two knots are the same or different?” and, “what different knots are there, anyway?”
First of all, there are a lot of them. Dror Bar-Natan has posted up a table of knots up to ten crossings, and each one links to a page of information about the knot. When we say a knot has “n crossings”, we mean that there’s a way I can arrange it on the table so one strand crosses over another one n times, and no such arrangement for fewer than n crossings. There are some more technical points about the knots on this table, but for now it’s nice to just look and see a bunch of them, and know that they’re just the tip of the iceberg.
Okay, so how can we tell if two knots are the same? Say we’ve got two actual loops of string to fidget with. We can sit there all day and not make them look the same, but we still don’t know that if we played with them just a little longer we wouldn’t hit on something. We need some more powerful tool.
Enter invariants. An invariant is a way of assigning some value to each knot — like a number, or a polynomial, or even a group — to each knot. We want to be sure that if we move the knot around the value of the function won’t change. That is, we want it to be invariant when we deform the knot. A lot of the bits of information on the page for each knot in Bar-Natan’s table are the values various invariants have for that knot.
So here’s how an invariant helps us: if two knots are the same they’ll get the same value for the invariant. That means that if we have two knots that get different values, they can’t be the same! We know that no matter how long we play with the knot we’re not going to turn one into the other, just as surely as we’re not going to turn 1 into 0.
Unfortunately that’s not quite good enough. We can tell when knots are different, but we still can’t be sure when knots are the same. There isn’t yet known a knot invariant that’s an injection, which would assign every knot a different value. Well, strictly speaking that’s not true. There’s one that’s known to essentially be an injection, but it’s also known that it’s impossible to tell when two values are the same or not, so in practice it’s still not helpful. Weird.
It seems there are two ways to get invariants. The older way uses a lot of heavy topology and/or geometry, while the newer way uses a lot of combinatorial fiddling with diagrams — pictures of knots like you see on that table. The topological style really is fascinating once you get into it, and it’s bound up with all sorts of other areas of mathematics. It’s a little hard to get into without building up a lot of machinery first, though.
The combinatorial style, on the other hand, is a great on-ramp for playing with knots. There were some combinatorial calculations of invariants in the past, but they usually had some topology hiding behind them. The real explosion in this style came with Vaughn Jones’ discovery of what’s now called the “Jones polynomial”. It’s really straightforward to calculate it, but the definition came completely out of left field, and took pretty much everyone by surprise back in the early ’80s. It’s still uncertain what the geometric or topological meaning behind it is, but everyone’s sure there’s something there. I have some thoughts in this direction, but I’ll leave those until I’ve laid out some more of knot theory in general and my own research program in particular.