Tag Archives: knot theory

## Some thoughts on knots: current research

8 Mar

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 $S^3$, but we can just think of it as $\mathbb{R}^3$, 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.

Even though they look different, these knots are the same.  They’re called the unknot.

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.

From wikipedia: I was having a really hard time making my own pictures.

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).

“Sometimes I feel like I can’t trust you… it’s like you’re two sided.”

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.

GET IT? It’s a visual pun!  The green is in bridge position.  The red is not.

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!

## Intrinsically knotted graphs on 21 edges

4 Apr

I was skimming through http://www.arxiv.org the other day and found this paper by a student named Barsotti and a professor named Mattman.  Barsotti is/was an undergraduate at CSU Chico and this is his honors thesis, while Mattman is a professor at CSU Chico.  Weirdly enough it looks like Mattman also directed the undergraduate theses of two of my colleagues, Arielle Leitner from UCSB  (wow, nice website!  way better than mine) and Ryan Ottman.  Small world.

So let’s talk a little bit about this paper, shall we?  I gave a half hour talk on it in our little seminar on Tuesday.  Graph theory is one of those fields that is ridiculously  useful.  Computer scientists and anyone who plays with data loves graph theory.  Knot theory is also surprisingly useful since math biology is blowing up right now with knotting of DNA and molecules and such.  This paper lies in the intersection of graph theory and knot theory.

Graph theory.  From my last math post you know what a graph is: a collection of vertices with some edges connecting them.  We’ll be talking about a certain operation you can do to graphs, taking a minor.  A minor of a graph is a graph that you derive from your original graph.  There are three different ways to find minors of a graph: 1) delete a vertex and all the edges connected to it, 2) delete an edge, 3) contract an edge.  The picture shows examples of all three of these.  For number 3, you delete an edge, and you put the two vertices at the end of it together.  Note that the new vertex has 4 edges attached to it, or has degree 4, while the original two each had degree 3.

The big one is ok, but the other three are jail bait.

So there’s this big old theorem from Robertson and Seymour, called the Graph Minor Theorem, which says a couple of things.  For one, if you have infinitely many graphs, you’re definitely holding one that is a minor of another in your hand.  Another way to think of the theorem is if you have a collection of graphs which is closed under minors (e.g. for any graph in your collection, all of the minors of it are in there), you can define this collection by a finite set of forbidden minors.

A quick example is the collection of all trees (graphs with no cycles or loops in them).  In our picture, 1 and 2 are both trees, while the original graph and 3 both have a loop and so aren’t a tree.  Trees are closed under minors, since taking a minor of a tree will never make a new loop.  Then the theorem says there is a finite set  of graphs which cannot be minors of any tree, and that the set of trees is characterized by not having these as minors.  So to be a tree, it’s necessary and sufficient, as mathematicians like to say, that you have no forbidden minor as a minor.  In the case of trees, the minor is this:

Lady C… er no I meant Sir Cle

This is a loop with a single vertex on it.  If you’re a graph that’s not a tree, you can do a finite sequence of taking minors (e.g. contract the edges in a cycle) until you end up with a single vertex with a loop.  If you are a tree, you can never do this.  Robertson and Seymour’s graph minor theorem took 20 papers to prove.  It’s pretty insane.  But this is how you use it!

So a fun thing that undergraduates and professors do is try to take a family closed under minors and find the finite forbidden minor set.  In the case of a tree, there’s a single forbidden minor.  To talk about other graphs and forbidden minors, let’s switch gears for a moment and go to knot theory.

Trefoil! Versus treplasticwrap. That was a way worse name

Knot theory.  A knot is an embedding of the circle in some crazy way in some crazy place.  Right now we’ll just talk about knots embedded in three space (the world we live in, $\mathbb{R}^3$).  Embedding is a technical word, but just think of it as putting something somewhere.  In fact, think of a knot as a piece of string, lying one end on the ground, throwing the other end all over itself (looping in and out) and then gluing the two ends together.  Or put in a bunch of ropes, so long as you glue all the ends together to make one continuous loop.

Not a knot… not yet

In this picture, we’d have to glue a gray end and a green end together, and do the same with the other pair, to have an actual knot.  Or you could glue the green ends together and the grey ends together, in which case you would have two knots linked together.

Lots of knots secretly look very simple, like a circle:

Even crazy complicated ones.  Look at this diagram to see what I mean about ‘secretly looking like’:

This diagram is from a paper by Henrich and Kauffman from 2010 (disclaimer: Kauffman is at my school).  [I don’t know why I put in that disclaimer.]

But the trefoil, above, for example, doesn’t secretly look like a circle.  No matter how you pull and prod the strands around, you can’t get a plain old circle again.

This brings us back to graph theory.  A graph is intrinsically knotted if, no matter how you draw it in $\mathbb{R}^3$, you end up with a not-boring knot.  This paper I just read lists all the intrinsically knotted graphs with 21 edges (as you’d expect from the title).  The proof uses some results from Robertson and Seymour’s graph minor theorem.  For instance, it uses the fact that all non-planar graphs are characterized by two forbidden minors- this is Kuratowski’s theorem and I’ll do another post on it sometime.  The proof uses a lot of cool little tricks, but this is just an intro post so I’ll save those for another time.

In case you were wondering, there are 14 intrinsically knotted graphs with 21 edges.  That’s the takeaway!