|Email:||tlazarus (at) hawaii (dot) edu|
|Home Institution:||University of Hawaii at Hilo|
The general area I'm studying this summer is knot theory. I am working with Chris Woodward of the Rutgers Math Department, as well as with three grad students Doug Shultz, Alejandro Ginory and Ed Chien, and fellow REU participant Eric Fay.
The original goal of this summer was to work on the AJ-Conjecture for knots, however our mentor decided that might be a bit too ambitious and hard. We have since switched topics to focus more on understanding the A-polynomial and the augmentation polynomial that arises from a knot (or link).
I arrived in New Jersey and was set to work immediately reading multiple papers and background material for the topic. I started by just trying to get down some manifold theory and topology, then moved to group theory (since I haven't taken a formal abstract algebra course yet).
I continued to delve into group theory so as to be able to understand a knot when the fundamental group of the knot or the knot group is given. I gave a short presentation on the fundamental group of a knot and of a knot complement, and we showed how to use it as an invariant. We then talked about different invariants of knots and which ones are better than others. Eric also gave a short talk on the basics of knots.
Alejandro gave a short talk on representation theory and how to represent the knot group in the Lie group SL ₂ (ℂ). This allows for a much nicer way to handle knots because linear algebra is very well understood. It also helps in computation because a computer can deal with matrices.
Eric gave a short presentation on 2-Bridge knots and braids. We went over how every knot can be written as a closed braid. Doug gave a presentation of the augmentation polynomial and we looked at how the braid group can be used to help compute the augmentation polynomial. Eric, Alejandro and I tried computing the A-polynomial for the trefoil using a 2-bridge representation, but came up with an incorrect polynomial because of a few mistakes in calculations. We have since fixed them and correctly calculated the A-polynomial for the unkot, trefoil knot, and certain 2-bridge knots.
At the beginning of the week I wrote up some notes and gave a presentation to the group on Augmentations and the KCH representation. I also continued to tweak the program Eric and I had written last Friday in order to try to get it to work for all the 2-bridge knots. At the end of the week we discussed a small amount of symplectic geometry, and prepared for the conference in Montreal. We now need to decide on a project and start to read the material needed to finish it. On the plane to Montreal I read about symplectic forms, tangent and cotangent spaces, and cohomology.
I have been attending talks at the SMS conference in Montreal, as well as going back over our material to try to come up with a good project. Some of the talks have been extremely technical and thus not very enlightening for me, but a few talks have been very understandable and quite interesting. I will continue to go to the talks that seem to be accessable, as well as work on the ideas for our project. We are also going to try to explore a bit of Montreal.
We have decided to fully and explicitly calculate the A-polynomial and Augmentation polynomial for a few specific knots as our final project. This idea comes from the fact that none of the literature that we have found actually does this, they just present the polynomial. We thought it would be beneficial to us to check that we understand what is going on behind the scenes, and to write it out for anyone else beginning to look into this area.
We finished up going to talks at the SMS conference and flew home on July 3rd. For July Fourth we went to Gene's for the BBQ. Over the weekend I worked on the refernce section for the project and to gather all of our sources.
Eric and I worked on our project this week. Eric wrote a program to visualize torus knots and how we view them in the plane. We've written most of the material on the A-polynomial for the paper, but we still have a bit to go. I flew back home to Seattle on Saturday, but hopefully the paper won't take too much more time to complete.