Data

• Title: Big data comparison of quantum invariants
• Authors: Daniel Tubbenhauer and Victor Zhang
• Status: preprint. Last update: Wed, 5 Feb 2025 03:03:57 UTC
• Code and (possibly empty) Erratum: Click and Click
• ArXiv link: https://arxiv.org/abs/2503.15810

Abstract

We apply big data techniques, including exploratory and topological data analysis, to investigate quantum invariants. More precisely, our study explores the Jones polynomial's structural properties and contrasts its behavior under four principal methods of enhancement: coloring, rank increase, categorification, and leaving the realm of Lie algebras.

A few extra words

The Jones polynomial is widely recognized as one of the most significant knot invariants of the twentieth century, and Jones' groundbreaking discovery uncovered profound and unexpected connections between diverse areas of mathematics and physics. The Jones polynomial serves as the foundation of a larger family of knot invariants derived from quantum groups, known as (Witten)--Reshetikhin--Turaev invariants. These invariants also have categorified counterparts, positioning them as remarkable objects at the intersection of multiple mathematical fields.

Such a focus underestimates the rich interplay of ideas and deep insights to which these invariants contribute. However, it is precisely this question that forms the focus of this paper. It turns out that this may be the wrong question to ask: quantum invariants are not expected to be strong knot invariants. Indeed, they satisfy local relations (e.g., skein relations), which make them easy to study but not particularly well-suited as invariants. With this in mind, a better question is: This question, along with its variations, forms the central focus of this paper.
This paper is essentially interactive, which can be accessed by clicking on the links above. The above shows one of the comparisons we do.