I had another free choice. I thought I’d go back to one of the topics I knew and loved in grad school even though I didn’t have the time to properly study it then. It turned out I had forgotten some important points and spent a night crash-relearning knot theory. This isn’t a bad thing necessarily.
This is a thing which comes from graphs. Not the graphs you ever drew in algebra class. Graphs as in graph theory. These figures made of spots called vertices. Pairs of vertices are connected by edges. There’s many interesting things to study about these.
One path to take in understanding graphs is polynomials. Of course I would bring things back to polynomials. But there’s good reasons. These reasons come to graph theory by way of knot theory. That’s an interesting development since we usually learn graph theory before knot theory. But knot theory has the idea of representing these complicated shapes as polynomials.
There are a bunch of different polynomials for any given graph. The oldest kind, the Alexander Polynomial, J W Alexander developed in the 1920s. And that was about it until the 1980s when suddenly everybody was coming up with good new polynomials. The definitions are different. They give polynomials that look different. Some are able to distinguish between a knot and the knot that’s its reflection across a mirror. Some, like the Alexander aren’t. But they’re common in some important ways. One is that they might not actually be, you know, polynomials. I mean, they’ll be the sum of numbers — whole numbers, even — times a variable raised to a power. The variable might be t, might be x. Might be something else, but it doesn’t matter. It’s a pure dummy variable. But the variable might be raised to a negative power, which isn’t really a polynomial. It might even be raised to, oh, one-half or three-halves, or minus nine-halves, or something like that. We can try saying this is “a polynomial in t-to-the-halves”. Mostly it’s because we don’t have a better name for it.
And going from a particular knot to a polynomial follows a pretty common procedure. At least it can, when you’re learning knot theory and feel a bit overwhelmed trying to prove stuff about “knot invariants” and “homologies” and all. Having a specific example can be such a comfort. You can work this out by an iterative process. Take a specific drawing of your knot. There’s places where the strands of the knot cross over one another. For each of those crossings you ponder some alternate cases where the strands cross over in a different way. And then you add together some coefficient times the polynomial of this new, different knot. The coefficient you get by the rules of whatever polynomial you’re making. The new, different knots are, usually, no more complicated than what you started with. They’re often simpler knots. This is what saves you from an eternity of work. You’re breaking the knot down into more but simpler knots. Just the fact of doing that can be satisfying enough. Eventually you get to something really simple, like a circle, and declare that’s some basic polynomial. Then there’s a lot bit of adding up coefficients and powers and all that. Tedious but not hard.
Knots are made from a continuous loop of … we’ll just call it thread. It can fold over itself many times. It has to, really, or it hasn’t got a chance of being more interesting than a circle. A graph is different. That there are vertices seems to change things. Less than you’d think, though. The thread of a knot can cross over and under itself. Edges of a graph can cross over and under other edges. This isn’t too different. We can also imagine replacing a spot where two edges cross over and under the other with an intersection and new vertex.
So we get to the Yamada polynomial by treating a graph an awful lot like we might treat a knot. Take the graph and split it up at each overlap. At each overlap we have something that looks, at least locally, kind of like an X. An upper left, upper right, lower left, and lower right intersection. The lower left connects to the upper right, and the upper left connects to the lower right. But these two edges don’t actually touch; one passes over the other. (By convention, the lower left going to the upper right is on top.)
There’s three alternate graphs. One has the upper left connected to the lower left, and the upper right connected to the lower right. This looks like replacing the X with a )( loop. The second alternate has the upper left connected to the upper right, and the lower left connected to the lower right. This looks like … well, that )( but rotated ninety degrees. I can’t do that without actually including a picture. The third alternate puts a vertex in the X. So now the upper left, upper right, lower left, and lower right all connect to the new vertex in the center.
Probably you’d agree that replacing the original X with a )( pattern, or its rotation, probably doesn’t make the graph any more complicated. And it might make the graph simpler. But adding that new vertex looks like trouble. It looks like it’s getting more complicated. We might get stuck in an infinite regression of more-complicated polynomials.
What saves us is the coefficient we’re multiplying the polynomials for these new graphs by. It’s called the “chromatic coefficient” and it reflects how many different colors you need to color in this graph. An edge needs to connect two different colors. And — what happens if an edge connects a vertex to itself? That is, the edge loops around back to where it started? That’s got a chromatic number of zero and the moment we get a single one of these loops anywhere in our graph we can stop calculating. We’re done with that branch of the calculations. This is what saves us.
There’s a catch. It’s a catch that knot polynomials have, too. This scheme writes a polynomial not just for a particular graph but a particular way of rendering this graph. There’s always other ways to draw it. If nothing else you can always twirl a edge over itself, into a loop like you get when Christmas tree lights start tangling themselves up. But you can move the vertices to different places. You can have an edge go outside the rest of the figure instead of inside, that sort of thing. Starting from a different rendition of the shape gets you to a different polynomial.
Superficially different, anyway. What you get from two different renditions of the same graph are polynomials different by your dummy variable raised to a whole number. Also maybe a plus-or-minus sign. You can see a difference between, say, (to make up an example) and . But you can see that second polynomial is just . It’s some confounding factor times something that is distinctive to the graph.
And that distinctive part, the thing that doesn’t change if you draw the graph differently? That’s the Yamada polynomial, at last. It’s a way to represent this collection of vertices and edges using only coefficients and exponents.
I would like to give an impressive roster of uses for these polynomials here. I’m afraid I have to let you down. There is the obvious use: if you suspect two graphs are really the same, despite how different they look, here’s a test. Calculate their Yamada polynomials and if they’re different, you know the graphs were different. It can be hard to tell. Get anything with more than, say, eight vertices and 24 edges in it and you’re not going to figure that out by sight.
I encountered the Yamada polynomial specifically as part of a textbook chapter about chemistry. It’s easy to imagine there should be great links between knots and graphs and the way that atoms bundle together into molecules. The shape of their structures describes what they will do. But I am not enough of a chemist to say how this description helps chemists understand molecules. It’s possible that it doesn’t: Yamada’s paper introducing the polynomial was published in 1989. My knot theory textbook might have brought it up because it looked exciting. There are trends and fashions in mathematical thought too. I don’t know what several more decades of work have done to the polynomial’s reputation. I’m glad to hear from people who know better.
There’s one more term in the Fall 2018 Mathematics A To Z to come. Will I get the article about it written before Friday? We’ll know on Saturday! At least I don’t have more Reading the Comics posts to write before Sunday.