Topological Mathematics Research

This research project saw me and a partner expanding upon the field of knot theory in topological mathematics. Knots, such as those tied with ropes on boats or something as simple as your shoes, are commonly represented in topology by a 2D mosaic using a set of tiles. Knots are classified by the amount of crossings that they have as well as their mosaic number, which is the smallest size mosaic they can fit on. We sought to translate this idea of 2D mosaics to three dimensions, which led us to creating a "cubic" mosaic board. This board is reminiscent of a cardboard box being completely unfolded and laid flat on the ground. Using this new method of representing knots, my partner and I were able to determine the cubic mosaic number for each knot up to 8 crossings (35 knots total). After determining the cubic mosaic numbers of these knots by hand, we shifted our goals to generalizing the formula for a knot of any amount of crossings. We eliminated the arrangements of tiles that would result in an unknot, meaning the knot can be unraveled into a loop of string with no crossings. Accounting for additional tiles needed to make turns inside the mosaic, we were able to prove that for a cubic mosaic of size n (n x n tiles in each cube face), the maximum number of crossings that can fit on the mosaic is 6n² - 3n + 1. Our theorems, proofs, and mosaics for all knots up to 8 crossings are to be published in a professional research paper in a forthcoming issue of Involve, a Journal of Mathematics, which I had a large role in writing. My partner and I presented our research at Penn State Brandywine's Eureca student exposition, where we won first prize, as well as Penn State York's regional symposium and the EPaDel section of the Mathematical Association of America. Prior to this research project, I had no experience in topological mathematics or knot theory. I was able to learn an entirely new subject and expand upon it in an impactful way. My ability to critically think through problems, especially those that are more abstract, has dramatically improved because of the stark differences between knot theory and more conventional and physical engineering problems. I believe I have become a more well-rounded engineer and problem solver due to my time spent in topology.