Senior Thesis: Two-Colored Twist Formulae in Skein Theory

The full thesis is available as a PDF: Read Thesis (.pdf)

I started taking an interest in this field a few years ago heading into my Junior year of university. I was taking a complex analysis course taught by Ka Ho Wong who would later be my advisor for this paper. I was fairly dedicated math student at this point, and I had loaded up my semester with plenty of other classes at the time. Looking at the course selection that year, I'd found that Ka Ho was teaching another class: a graduate seminar on hyperbolic knots and their quantum invariants. At the time, I had maybe some vague sense of what this meant, but I knew that it was something I would find interesting. Some of the most fascinating parts of math to me up to that point were classification theorems and invariants. So once I discovered he was teaching this other class, I knew it was something I wanted to ask him about and explore. When I reached out to him, we was kind enough to share his course notes and let me bother him with questions as I tried to wrap my head around the introductory lectures.

Even without some of the foundations of higher math classes, it was really fun to see some of the aspects and results of the field. Something that I found really intuitive and fun was playing with knot diagrams, and this later became the basis of what would guide my thesis. The lectures introduced me to the Kauffman bracket, a tool discovered in the 1980's during sort of a revolution in the field. The Kauffman bracket is a device which turns knot diagrams into polynomials, and it's specially because it doesn't change when switching knot orientation, or performing Reidemeister moves II, and III on the diagram. However, it does change under Reidemeister move I, and if we normalize the bracket, we're able to recover the Jones polynomial, which doesn't change under a knot diagram deformation, known as an ambient isotopy. Invariants are a powerful tool used to classify mathematical objects, so finding more ways to distinguish and classify knots using diagrammatic methods is especially useful. The Jones polynomial wasn't discovered until 1984, using a seemingly unrelated method involving a braid group representation and its Hecke algebra. Kauffman was able to turn this higher-level algebraic definition of the Jones polynomial into a combinatoric, diagrammatic one, and this was a massive leap in the field, finding skein-theoretic forumulae for what had been purely algebraic invariants of knots. This had been seen before in the Alexander polynomial in the 1920's, but the Jones polynomial and discovery of the Kauffman bracket really gave mathematicians better tools to classify knots.

I was really fascinated working with the Kauffman bracket, and found polynomial invariants of knots super interesting. I realized this was the sort of math I wanted to do, and exploring the bracket and quantum invariants of knots was something I wanted to continue. I asked Ka Ho to be my advisor when I began my senior year, and started meeting with him to study the Volume Conjecture and learn more about the field. As I decided what to do for my senior thesis, I found inspiration thinking about generalizations of twist and braid diagrams. One problem we thought it would be feasible to tackle by the end of the semester was concerning twists of two differently colored strands. Color is a labeling of a strand with a positive number, and corresponds to a representation of the quantum group $U_q(sl_2)$. These are expressed using JW idempotents, which are eigenvectors under a twist operator. These colored strands have fascinating properties which makes them worthwhile to explore.

In this thesis, I generalized a result from Wataru Yuasa on skein-theoretic formulae of twist diagrams, this time for nonequal colors. Yuasa was also kind enough to respond when I reached out, and showed me his workflow for strand diagrams which let me create some beautiful images for the thesis. I was able to apply the new formula on a family of links known as the 2-bridge links, at least for those which have an . Using the methods of Yuasa and from another paper from Masbaum, I was able to find such a formula for full twists of two-colored strands, as well as derive a formula for the two-colored Jones Polynomial as well.

I talk a bit at the end what I wanted to work on next, and I have a few ideas. I am applying to grad school in the near future, and I hope I will have the opportunity to continue working on this paper. The document still needs revised and I have some ideas for potential avenues of exploration. There is a basis for these diagrams called the fusion basis, and it would allow the result to be expressed in a much simpler form. I discovered this after submitting the thesis, so I will have to add onto the result in the future.