An intro to my research:
One problem we worked on is a subproblem of knot equivalence. This problem states that: given two knot diagrams, do they represent equivalent knots? This problem has been decidable since the 1960s and 1970s. One common approach to this problem involves using Reidemeister moves, which are a set of diagrammatic moves to "simplify" the diagram so one can identify if the two diagrams represent the same knot. However, if the diagrams represent different knots, the process of applying Reidemeister moves can run indefinitely. So, one has to come up with an upper bound on the number of Reidemeister moves required. One known bound, by the work of Coward and Lackenby, exists; however, this bound is huge and is expressed as a tower of exponentials in terms of crossings. A natural question is whether this bound can be improved. By restricting this problem to alternating knots, we showed that the Alternating knot equivalence problem has a polynomial-time algorithm in terms of crossing number.
On complexity of alternating links (with A. Tsvietkova): arXiv
One of our projects focused on computing the meridian length for an infinite family of links that are closures of 4-braids with word \( \left( \sigma_{2}^{-1} \sigma_{1} \sigma_{3} \sigma_{2}^{-1} \right)^{n} \) for odd \( n \geq 3 \). While software like SnapPy makes it possible to compute link invariants, it is not well-suited for handling the computation of hyperbolic invariants for infinite families of links, as the time complexity grows exponentially as the number of crossings increases, making it impractical. Using methods from Thistlethwaite and Tsvietkova, we derived an exact algebraic expression for the meridian length of this infinite family. Moreover, in our work, we developed a generalized formula that provides an exact value for the meridian length of the infinite family.