Knot theory
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.
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The Unapologetic Mathematician 
What’s the invariant that’s an injection but known to be impossible to tell when two values are the same? Is it something to do with quandles? (I recall discussing the idea with a student and a professor at an undergrad conference last year.)
Also, if it’s impossible to tell when the values are equal, doesn’t that imply that it’s impossible to decide when two knots are equivalent for all cases? I thought this particular question was still up in the air.
Susan: that’s probably the knot complement. Even the knot group (the fundamental group of the complement) has a problem telling things like the square knot and granny knot apart.
The knot complement, however, cannot tell apart all links. And worse yet, it’s pretty much impossible to tell when two complements are the same or not. We can simplify by thinking of the knot group instead of the complement. This gets some (but not all) of the complement’s information.
The problem here is actually related to the “link covariant” idea in the paper I just linked to today. We have a good way — the Wirtinger presentation — to come up with a group from a knot diagram. And further we know that if two diagrams are Reidemeister equivalent, then their groups are isomorphic. And even better, we almost have the other way around: if the groups are isomorphic we’re almost sure that the diagrams are Reidemeister equivalent.
The problem is that we have no way to tell when two finitely-presented groups are equivalent. Not only do we not know an algorithm to do this, but we’ve proved that no such algorithm exists at all!
So the knot group runs aground, but maybe there’s another invariant (or collection of invariants) that do get it uniquely, and which can be algorithmically decided. Knot complements are in three-dimensions, and it’s always hazy whether something is possible or not in 3D.
Silly me, although I have heard of both the knot complement and the knot group as invariants it hadn’t occurred to me to think of those.
I’m still a little confused, though: Say we did find a perfect knot invariant, for which we could always decide whether two knots are equivalent–couldn’t we use that to decide whether the two knot groups are equivalent in every case, even though that’s been proved impossible?
Or…wait…I guess by “if the groups are isomorphic we’re almost sure that the diagrams are Reidemeister equivalent” you mean that there are some nonequivalent knots whose knot groups are isomorphic.
I’m learning some topology right now, as well as more algebra–I’m going to have to go look up knot groups now!
That would require a method for constructing a link for every group. You’d turn both groups into links, then test whether they’re equivalent with your perfect invariant. But without the method of turning groups into links, you’re stuck.
you mean that there are some nonequivalent knots whose knot groups are isomorphic
Yes. The easiest example is that the square and granny knots have isomorphic knot groups, though their complements aren’t homeomorphic. On the other hand, we know that the knot complement is a perfect invariant for knots: homeomorphic complements implies equivalent knots. But it’s impossible to algorithmically determine the equivalence in general. Also, the same does not hold for links, where it’s not difficult to find families of inequivalent links with homeomorphic complements.
“Also, the same does not hold for links, where it’s not difficult to find families of inequivalent links with homeomorphic complements.”
Argh! That’s exactly the kind of counterintuitive thing that makes me desperately need to learn more topology! (I’ve heard this before, I think, now that you mention it.)
Backing off a bit, is there a “nice”, perfect invariant for prime knots? For instance, are there distinct prime knots whose knot groups are isomorphic? And do knot groups distinguish between prime knots of opposite chirality (left-handed and right-handed trefoils, for example)? Granted, a combination of several invariants would also be an invariant, and I know there are some invariants that distinguish chirality.
I’m probably better off doing some digging in the literature and/or trying to prove some of these things myself, rather than picking your brain for the answers, but hey, I get excited.
Some of this I’ll let you investigate, but I’ll point you towards one of the easier answers.
Opposite chirality means you’re looking at a mirror image pair. The knot group is the fundamental group of the knot complement. So, can you come up with a homeomorphism between the complements of a chiral pair? If so, then the groups must be isomorphic.