#thesisland

*squeees uncontrollably*

This actually made me get out of bed (and then realize I need to eat breakfast first). I am super excited to talk about what those images are! (I’m actually not sure if you are talking about these pictures or these pictures or both or earlier ones, but I can roll with it anyways) Basically I’m a graph theorist who somehow got into contact topology.

The only unfortunate thing is that my “very serious math thesis” that those are a part of is in fact 155 pages long. Obviously some of that is pictures, but that’s still a lot. I’ll try to give you an abridged version, which frankly is still rather long. (I am also being sorely mad at tumblr for not working like LaTeX. This would be so much easier then.) 

My Thesis: A Visual Guide

Basically, I’m studying an invariant of Legendrian knots. Legendrian knots are more rigid than normal knots because they have to be tangent to a contact structure that is the kernel of the non-integrable one form dz-ydx, which turns into a bunch of planes like so:

That looks complicated but there’s actually a simple way of thinking about it by looking at the xz-projection of the knot (called the front projection), since being in the kernel of the one form means that 0=dz-ydx and thus dz/dx =y. Thus, from the front projection you can recover the y coordinate from the slope at that point. This means that the xz-projections have no vertical tangencies and have cusps instead. We also don’t have to specify the overstrand and understrand at crossings because we assume that the more negative sloped strand is closer towards us. Below are some examples. 

A Legendrian invariant of a Legendrian knot is one that doesn’t change through Legendrian isotopy (deformation where all the inbetween knots have to also be Legendrian) or equivalently things that are unaffected by the Legendrian Reidermeister moves (and planar isotopy that does not create vertical tangencies). These are shown below (up to vertical and horizontal flips) 

One attempt at making a Legendrian Invariant is to decompose the front diagram into eye shapes. Such decompositions are called rulings. (The top and bottom rows bellow are equivalent ways of presenting them. The crossings where an eye starts on the overstrand and ends on the understrand is called a switch and those are what are marked in red)

Unfortunately, that is not good enough because of cases like below where rulings can be created or destroyed through Legendrian isotopy.

To fix this we add an extra conditition on the switches, and switches that satisfy those conditions are called normal. Basically locally around the switch the eyes that meet there have to be either disjoint or one is contained in the other. The picture below shows all the examples of switches could look and the top row has the ones that are normal. 

Proving that this is  a Legendrian invariant then just requires looking at the Reidermeister moves and planar isotopy and proving that the local changes preserve normal rulings. A proof can be found in this article by Yu. V. Chekanov and P. E. Pushkar. Some other properties of the rulings are also preserved including the grading of the ruling and the relative number of switches to right cusps, but this gives you the basic idea.

Now, those have been already studied, but four years ago Dan Rutherford and Lenny Ng (in this paper) found a way to get more information about a knot by looking at normal rulings of related knots called satellites. Satellites take copies of the original knot and then insert a pattern. The insertion of the pattern is determined by chosing an orientation on the knot or equivalently through choosing a basepoint (equivalent in that you choose the knot to be oriented left to right at the basepoint). The original knot is called the companion, and the pattern is often called the pattern knot (since it can be thought of as a knot in a tube).

Basically it turns out that normal rulings of these knots that don’t have switches at crossings associated with cusps of the companion give us more information about other ruling invariants of the companion. Normal rulings of satellites that satisfy this property are called reduced rulings.

First, however, I noted that we can isolate the pattern fromt eh companion by moving it close to a left cusp thus reducing the effect of the pattern to acting on the strands of reduced rulings by certain permutations. So, I tend to chuck out the pattern and just replace it with the permutatoin.

(The permutation (132) is a function takes 1 to 3, 3 to 2, and 2 to 1. So the eye that started ont eh first strand got bopped to the third, ect.)

Turns out the reduced ruling above is extra special because it looks like a normal ruling of the companion knot. (Hence me grouping the eyes into distinct hues. Eyes that have the same hue correspond to the same eye in the companion knot). So, my thesis is on making more reduced rulings of satellites that mimic normal rulings of the companion knots, and I call such reduced rulings by the term ruling extensions.

To do this i need to store information about a given normal ruling of the companion knot. I find graphs helpful for this, and in particular the vertices of the graph are the eyes of the normal ruling, and the edges are the switches. The edges also come with an ordering, a function mu that takes an edge e and vertex incident to said edge v, and assigns 0 or 1 to mu(e,v) depending on whether the switch is on the lower strand or the upper strand of the eye. I also encode how the switches fulfill normality using a function mu which either points to the empty set if normality is fulfilled by the eyes being disjoint and otherwise point towards the eye which contains the other. The edges also come with an ordering from the ruling.

Visually I represent mu by making my vertices boxes and being careful about whether I’m attaching an edge to the bottom or the top. If nu points towards the empty set I do nothing. Otherwise, I literally have the edge point towards the containing eye. For purposes of walks and cycles, the edge is still undirected.

The pictures that look like below are me zooming in to the crossings of the satellite associated with a single crossing of the companion, and seeing what happens when I put switches in certain places. They accompany more general proofs, which were in fact some of the hardest proofs of my thesis and thus I won’t be giving them here.

However, I will say that the first picture is a part of my effort to classify what arrangements of switches would appear in ruling extensions. The red circles are a starting permutation, which is automatically normal in ruling extensions, and blue and purple circles indicate places you can add switches to it. These new switches are between eyes of the satellite that coresspond to the same eye of the companion, and thus there is a method for checking when those would be normal. The hard part is that his fully classsifies all admissible arrangements.

The second two are a part of constructions of arrangements that satisfy certain properties such as acting on the incoming upper and lower eyes in certain ways and being normal in certain circumstances. In either case I am trying to nullify a disturbance added to a ruling extension, and I have to consider different cases depending on which set of crossings associated to a crossing of the companion knot will have the stars align in favor of adding normal switches.

In doing so I was able to prove that if the switch graph corresponding to a given ruling has a cycle, then you can extend the ruling to (practically) any satellite, and more specifically any permutation.  (Technically some patterns destroy any reduced rulings, but those also have no permutations associated with them, so it works out.)