Topology Seminar: Jesse Levitt (USC)

Title: Understanding Structure in the Single Variable Knot Polynomials

Abstract:

We examine the dimensionality and internal structure of the aggregated data produced by the Alexander, Jones, and Z₀ polynomials using topological data analysis and dimensional reduction techniques. By examining several families of knots, including over 10 million distinct examples, we find that the Jones data is well described as a three dimensional manifold, the Z₀ data as a single two dimensional manifold and the Alexander data as a collection of two dimensional manifolds. We confirm each of these structural results using two independent ‘big data’ techniques. The ability to consider knots in this manner illuminates several interesting relationships that I hope to discuss at the conclusion of the talk. This collects joint work with Mustafa Hajij and Radmila Sazdanovic.