A systematic search of knot and link invariants beyond modular data

TL;DR

This paper demonstrates that knot and link invariants often contain more information than modular data alone, providing a systematic study showing many small knots and links serve as complete invariants for certain modular categories.

Contribution

It introduces a systematic investigation of knot and link invariants beyond modular data, identifying many small knots and links as complete invariants for specific modular categories.

Findings

Many small knots and links are complete invariants for the categories

Knot and link invariants often carry more information than modular data

The $5_2$ knot is among the invariants identified

Abstract

The smallest known example of a family of modular categories that is not determined by its modular data are the rank 49 categories $\mathcal{Z}(\text{Vec}_G^{\omega})$ for $G=\mathbb{Z}_{11} \rtimes \mathbb{Z}_{5}$. However, these categories can be distinguished with the addition of a matrix of invariants called the $W$-matrix that contains intrinsic information about punctured $S$-matrices. Here we show that it is a common occurrence for knot and link invariants to carry more information than the modular data. We present the results of a systematic investigation of the invariants for small knots and links. We find many small knots and links that are complete invariants of the $\mathcal{Z}(\text{Vec}_G^{\omega})$ when $G=\mathbb{Z}_{11} \rtimes \mathbb{Z}_{5}$, including the $5_2$ knot.