Knot invariants and their relations: a topological perspective

This work brings methods from topological data analysis to knot theory and develops new data analysis tools inspired by this application. We explore a vast collection of knot invariants and relations between then using Mapper and Ball Mapper algorithms. In particular, we develop versions of the Ball Mapper algorithm that incorporate symmetries and other relations within the data, and provide ways to compare data arising from different descriptors, such as knot invariants. Additionally, we extend the Mapper construction to the case where the range of the lens function is high dimensional rather than a 1-dimensional space, that also provides ways of visualizing functions between high-dimensional spaces. We illustrate the use of these techniques on knot theory data and draw attention to potential implications of our findings in knot theory.