A systematic search of knot and link invariants beyond modular data

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 $G={\mathbb{Z}}_{11}⋊{\mathbb{Z}}_5$. 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 $G={\mathbb{Z}}_{11}⋊{\mathbb{Z}}_5$, including the $5_2$ knot.

Cited by in corpus (4)

• On invariants of Modular categories beyond modular data
• Measuring Topological Order
• Zesting produces modular isotopes and explains their topological invariants
• On Generalized Symmetries and Structure of Modular Categories