Topology | Minimalist Mathematician

April 19, 2015

Ribbon graphs of link diagrams

This has been a weird week. Grad school seems to go in waves: first I feel really stupid and like I understand nothing, and then suddenly I’m on top of everything. Up until now, this rollercoaster has been out of sync in all of my classes, but this week I’ve been on top of everything. I wonder how long it will last. I even understand what we’re doing in analysis (intro to functional analysis), though that will probably change when we start with abstract measure theory next week…

Update: It did not change when we did abstract measure theory. Shocking, I know. I found the abstract stuff so much easier to do than the concrete Lebesgue measure. When we did abstract measure theory, I didn’t get confused by all the things I do know about the real numbers and I could focus on just understanding measures on sets, rather than mixing in all the weird things that happen with real numbers. Much more fun.

I have a vague recollection of promising a post on the topic of my conference presentation, so here it goes. First of all, you need to know what a knot is. A normal knot, that you tie your shoes with, is not mathematically interesting. It can be tied, so it can (mathematically at least) be untied. Instead, to define a mathematical knot we tie a knot in a piece of string, and connect the ends. That’s the intuitive definition. There are some tweaks to make it behave nicely, but you get the general idea. A link is a knot with several components.

The title of this post mentions link diagrams. We can find a link diagram of a link by simply projecting the link “nicely” onto the plane. I’m fudging a lot of the details about knots, because they are not the main topic here. I might do a post later on knots where I explain all the details properly. Here “nicely” should be taken to mean that one can recover the original knot from the projection.

Now that we have a vague idea what a link diagram is, we define a ribbon graph. A ribbon graph $G$ is a collection $V(G)$ of discs called vertices, and a collection $E(G)$ of discs called edges. The edge discs intersect exactly two vertex discs along an arc on each vertex, and no edges intersect. An easier way to think about a ribbon graph is as a graph embedded in a surface, and then we cut out a thin neighbourhood around the graph to obtain a ribbon graph.

Next, I’m going to tell you how ribbon graphs are connected to link diagrams. We take a link diagram, and eliminate each crossing just like when constructing a knot polynomial, choosing a negative or a positive splicing at each crossing. We can choose whichever of the two ways to eliminate each splicing we like. When all the crossings have been eliminated, we are left with a number of disjoint circles. These will be the vertices of our ribbon graph. Connect the vertices where there were crossings with edge discs, and you will end up with a ribbon graph. Since we could choose any splicing we for each crossing, there are up to $2^{\text{\# crossings}}$ possible ribbon graphs associated with each link diagram.

The natural questions to ask is then: how are these connected? What can we say about the ribbon graphs of a given link diagram? If two link diagrams have the same set of ribbon graphs associated with them, how are the link diagrams related?

My undergraduate advisor already answered both of those questions for links in $\mathbb R^3$ . It turns out that the ribbon graphs arising from the same link diagram are partial duals of each other. For the second question, it turns out that the link diagrams are related by a very simple “flip” move. In my undergraduate thesis, I proved that the same results holds for checkerboard-colourable link diagrams of links in $\mathbb RP^3$ . This is roughly what I presented at the conference a few weeks ago, to a room full of grad students. Here is a link to the paper on ArXiV.

It was a very nice conference, by grad students, for grad students. The organizers did a really good job putting it together, and there were a lot of interesting people there. It was all very friendly, in part I think because we are all so early in our careers. There were none of the “let me interrupt to talk about why my research is superior disguised as a question for 10 minutes”, which was nice. I also saw quite a few interesting talks, in particular one on the Erdös-Ko-Rado conjecture that I might try to turn into a blog post on a later date. No promises though, since apparently things keep popping up that keep me from blogging for weeks at a time. It should be easier for me to write regularly now that term is over, but no promises on the topics.

February 9, 2015

Unusual application of the Brouwer Fixed Point Theorem

Sometimes strange things happen in topology class. Some days we spend 20 minutes arguing about what to conjugate with what to show that something is in the normaliser of a group, other days no one knows what they are talking about, and yet other days we see some really surprising results. This particular one made me stop and go “What!” loudly the first time I read the problem ,and took several days to figure out. We had just proved the Brouwer Fixed Point Theorem, and were generally proving things about fundamental groups when suddenly this question comes up:

Prove that a $3 \times 3$ matrix M with positive real entries has an eigenvector with a positive eigenvalue. (Hint: consider $T = \{ (x,y,z) \mid x+y+z \leq 1, x,y,z \geq 0 \}$

So, the hint wants us to look at a triangle in $\mathbb R^3$ , and use that to prove this strange result from linear algebra. It turns out that the solution is amazingly short and neat. Here we go.

Consider the transformation $MT$ . This is a new, scaled version of $T$ , so a function $f: T \to T$ . Say that the transformation takes a point $x$ to a point $f(x)$ . Now, this transformation has a fixed point: for some $x \in T$ , $f(x) = x$ . So, for some $x \in T$ , $Mx = \lambda x$ , where $\lambda \in \mathbb R$ . Hence $x$ is our eigenvector and $\lambda$ our positive eigenvalue.

This is such a cool result: don’t you think? Do you have a favourite surprising application of one branch of mathematics in another?

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August 30, 2014

First Week of Classes

I haven’t been blogging like i should during the past two weeks, and I apologise for that. A couple of days ago, I finally finished writing up an article of some results in graph theory form my undergrad with my advisor and it was submitted for publication. As invaluable as the experience was, I have to say that redrawing the figures for article 4 times was not the most thrilling thing I have ever experienced. Having that submitted is a huge weight off my shoulders, and I can now start really focusing on my classes. As you may have figured from the title of the post, I just had my first full week of classes. Technically, Wednesday last week was my first day, but that wasn’t a complete week. After over a week of classes, what can I tell you about grad school?

As in most mathematics graduate programs, the first year consists of basic classes in algebra, analysis and topology. I didn’t manage to test out of algebra, as I thought I might, but that also makes me happy, because it means they have very high standards on what they expect grad students to know. The class algebra is a very standard lecture course with the added fun of problem sets due every class (3 times a week), not every week. It’s keeping me busy, but it’s not challenging yet. Like I said, I’ve covered a lot of the material before. Analysis on the other hand, was my main weakness as an undergrad. It’s also a standard lecture course, and I’m finding that it takes a lot of my time, simply because I haven’t quite figured out how to do analysis problems yet. I’m trying to work on one problem every day, either from the problem set or from the book just to get used to the style and way of thinking. I’m also trying to not complain too much about it while I’m working, so I don’t annoy my office mates to death.

Topology is the most interesting of my classes thus far. It’s taught using a version of the Moore method. We get a set of notes, containing definitions, theorems and problems. Every class, we get to claim the problems we have managed to solve using the notes, and then our lecturer goes through the list alphabetically, letting us solve one problem at the board. Once we finish all the problems in a set of notes, we get a new one. This sounds simple, right? Well, we also have that our entire grades depend on the problems we solve by the board. If you can’t solve any of the problems that are left, your lose your turn. While you are up there, anyone in the room can ask you to prove anything you state on the board, just to make sure you know what you are talking about. It’s brutal, intense and amazing. I love the class. I feel like I learn more since I have to figure it out on my own, and I know I become a better presenter every time I go up there.

In addition to these classes, I also managed to convince one of the professors to let me participate in his research seminar on graph theory and matriod theory. It’s promising to be very interesting. We only had one seminar this far, but I have a lot of interesting thought to follow up. The focus of the first half of the semester is graph embeddings, which I studied a little for my senior thesis, so I even have a slight head start on some of the older grad students.

I’ll be back in a few days with some more about the social life in grad school!