STEP Project: A Categorification of Biquandle Brackets

A Categorification of Biquandle Brackets

1. Please provide a brief description of your STEP Signature Project. Write two or three
sentences describing the main activities your STEP Signature Project entailed.

The goal of my STEP project was to create a categorification of biquandle brackets. In general I just worked to first create such a categorification, and then to prove that it worked as desired. I also presented my progress to my adviser on a weekly basis.

2. What about your understanding of yourself, your assumptions, or your view of the
world changed/transformed while completing your STEP Signature Project? Write one or
two paragraphs to describe the change or transformation that took place.

My project allowed me to completely immerse myself in math research for the summer. As such I encountered lots of hard problems, and had the opportunity to grow and overcome them. In attempting to solve these problems, I was able to learn much more math, especially in more specialized areas. I also had to present my research multiple times throughout the summer, allowing me to improve my presentation skills. Since I worked in a group for this project, I was also able to improve my ability to work in a team. Over the course of this project I realized just how grueling math research can be. I’ve realized how much willpower it takes to continue to attack a specific problem after numerous failures, and I now have nothing but the utmost respect for those who pursue math research professionally. But I’ve also realized just how good it feels to finally solve that problem, and because of that I know that I want to continue to chase that feeling in the future.

3. What events, interactions, relationships, or activities during your STEP Signature
Project led to the change/transformation that you discussed in #2, and how did those
affect you? Write three or four paragraphs describing the key aspects of your experiences
completing your STEP Signature Project that led to this change/transformation.

The goal of our project was to create a categorification of biquandle brackets.To do that there are essentially two steps. First to create a way of constructing a cochain complex from any link whose graded euler characteristic is the biquandle bracket of the link, and second to prove the cohomology of that complex is a link invariant. The first step took us about three weeks. We had numerous ideas for completing this step, but almost all of them failed in different ways. Some failed to give cochain complexes, some failed to have the correct graded euler characteristic, and some actually succeeded at the first step entirely but were quickly seen to fail at the second step. Finally we created a construction that succeeded at the first step. With that construction came equal parts elation that it may succeed at the next step, and dread that it may not and we’d have to start all over.

The second step was actually three substeps, called the three Reidemeister moves. Anytime you want to create a link invariant, all you need to do is show your creation doesn’t change under each of the Reidemeister moves. We did so for the first two Reidemeister moves in just a weekend. The third move however, had us completely stumped for a full month. While we certainly had ideas during this month, none of them gave a clear solution. What’s more is that we didn’t even really know whether our construction even was invariant under the third Reidemeister move; for all we knew we could have been spending all that time trying to prove something that just wasn’t true. Finally though, after a month of toil, we figured out invariance under the third Reidemeister move.

The excitement upon figuring this out was absolutely incredible. Unfortunately it was a bit short lived; since around two weeks later we created a much much simpler invariant, which when combined with a slight modification of an existing invariant, was actually stronger than our categorification. While this result led to a stronger overall invariant, I’d be lying if I said it wasn’t a bit disappointing. Still, it was a good result, and our proof that this invariant was in fact stronger was pretty interesting. Of course there’s no reason to believe that the categorification of biquandle brackets that we created is the only possible such categorification, so there are still exciting ways this research could develop.

4. Why is this change/transformation significant or valuable for your life? Write one or
two paragraphs discussing why this change or development matters and/or relates to your
academic, personal, and/or professional goals and future plans.

After graduating from Ohio State, I still plan to go to graduate school and pursue a PhD in mathematics. This research gave me valuable experience for determining what pursuing a PhD will really involve, and further affirmed my desire to do so. Of course, to get a PhD I first have to get into graduate school, and I believe this research project will greatly bolster my application. We’re also currently working on publishing a paper for this project, and if successful, that will be another factor that helps my graduate school application. In addition, we were able to present our research at the Young Mathematicians Conference, which I will also be able to add to my application. This project also allowed me to develop skills that I’m sure will help me in my future with mathematics, such as presentation skills and effective mathematical communication. Most importantly, this project increased my interest in mathematics, which will propel me to learn much much more in the future.

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