A multitude of knot invariants, including quantum invariants and their categorifications, have been introduced to aid with characterizing and classifying knots and their topological properties. Relations between knot invariants and their relative strengths at distinguishing knots are still mostly elusive. We propose adapting TDA tools such as Ball Mapper and Mapper to analyzing these infinite data sets where representative sampling is impossible or impractical. We focus on Jones and HOMFLYPT polynomials, and Khovanov homology and along the way introduce enhancements that deal with assumed or perceived symmetries in data, as well as a combination of Mapper and Ball Mapper approaches to enhance their strengths and provide a way to visualize maps between high dimensional Euclidean spaces. This is joint work with P. Dlotko and D. Gurnari.
TDA approach to the space of knots and their invariants
Radmila Sazdanovic, NC State UniversityAuthors: Pawel Dlotko, Davide Burnari, Radmila Sazdanovic
2022 AWM Research Symposium
Mathematics of Materials