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A Big Data Approach to Knot Theory: The Polynomial Knot Invariants as Manifolds.
by
Jesse S F Levitt
University of Southern California
Coauthors: Mustafa Hajij (CoC),
Radmila Sazdanovic (NCSU)
We examine the dimensionality and internal structure of the aggregated data produced by the Alexander, Bar-Natan and van der Veen, and Jones 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 Bar-Natan - van der Veen data as a two dimensional manifold and the Alexander data as a collection of two dimensional manifolds. These distinct results suggest the separate polynomials have different limits on their ability to distinguish between different knots. The ability to consider knots in this way illuminates several interesting relationships that I hope to discuss at the conclusion of the talk.
Date received: April 22, 2019
Copyright © 2019 by the author(s). The author(s) of this work and the organizers of the conference have granted their consent to include this abstract in Topology Atlas. Document # cbpy-05.