Over the weekend I went to the Third(!!!) annual Midwest Women in Mathematics Symposium (remember when I founded it? Now it’s all fancy with funding and many attendees and event staff!) As it turned out, not very much of the math in my parallel session was exactly up my alley, and also I was feeling lazy so I didn’t take many notes. But here’s a small recap/introduction to knot theory from my memory.
Aside: I like using knot theory as an example when people ask me what math is for (this happened a lot as an undergraduate and less and less as the years go by). I’m not even sure if this is true, but I tell people that mathematicians were studying knot theory for decades, and then biologists realized that they could use it to study how proteins fold and interact with other molecules. APPLIED! IN YOUR FACE, MATH DOUBTERS! Unclear where I picked up this bit of folklore, but it’s my number one defense when people say that modern math research is useless.
So what is knot theory? It’s certainly not not-theory, despite my claim as above that it can be applied. Knot theory studies objects called knots. A knot is some way that a circle is embedded in space- imagine taking a shoelace, knotting it up however you want to, and gluing the ends together. (By space knot theorists mean , but we can just think of it as , or the space we live in). To talk about knots, knot theorists draw knots as diagrams using over and under crossings. Two diagrams can represent the same knot, like in the picture below.
If I didn’t mess up, the blue knot is the same as the orange knot- just follow the crossings and you’ll see that nothing is actually knotted; it’s just a pile of string lying on top of itself. Below are some pictures of other knots.
It’s hard to tell if two diagrams represent the same knot. Mathematicians can use a diagram to assign polynomials to a knot, and do it in such a way that if two diagrams represent the same knot, then they give the same polynomial. Alexander polynomial, the Jones polynomial, and the HOMFLY polynomial (which generalizes the previous two). These still aren’t that great though, since two different knots can give the same polynomial (so while you can tell if your diagram ISN’T the unknot by seeing if the polynomial isn’t 1, you can’t tell if it IS the unknot if the polynomial gives you 1).
The first knot theory talk I saw connected knots to surfaces, so I was a fan. It was given by Effie Kalfagianni, a professor at Michigan State. One thing you can do with a knot is use it as the boundary of a surface.There are different ways to make a surface from a particular knot- draw a different diagram and you’ll get a different surface. One thing you can study is the genus of a knot: this is defined as the minimum genus (# holes) of a surface bounded by that knot. So for any diagram you draw, you can’t make a surface with a smaller number of holes. The genus of the unknot is 0. The genus of a knot using orientable surfaces is known, and there’s an efficient algorithm to find it. BUT the problem is open for non-orientable surfaces (these are surfaces that don’t have two sides).
So Kalfagianni’s research, joint with her student Christine Lee, puts a bound on the non-orientable genus of alternating knots, which are knots with diagrams that alternate between over and under crossings (alternating: the purple and red knots. Not alternating: the unknot, either blue knot (there are two over crossings in a row)). They use one of the factors in the Jones polynomial to do so.
So that was talk number one! The second talk I saw was by Maggy Tomova, an assistant professor at University of Iowa. I actually didn’t write any notes down for her talk, but I remember a cool concept from it. A knot diagram is in bridge position if you can draw a line across the middle so that there are only local maxima above it and local minima below.
One immediate note is that in general, bridge position is not unique: given a knot in bridge position, you might be able to find another diagram in bridge position that represents the same knot. There are some properties that ensure that a bridge position is unique (this is a theorem that I don’t remember). Tomova is working on some theorems that have to do with knots in bridge position, and I’m sorry that I can’t tell you more information. She did her Ph.D. at UCSB though, with the same advisor as some delightful other people who are her co-authors on this project (the delightful only applies to the first link; I don’t actually know her other co-author but I really like Yoshi and the fact that he goes by Yoshi). Also, one of her previous co-authors taught me abstract algebra when I was an undergraduate and he was a postdoc! That link is to a piece he wrote on going to the “Dark Side,” a.k.a. leaving academia for Google.
So I am not a knot theorist, but there’s your post with thoughts on knots!