Skein-Theoretic Methods for Unitary Fusion Categories

TL;DR

This paper develops skein-theoretic methods to analyze unitary fusion categories, classifies related skein relations, and connects these to quantum invariants like the Kauffman and Dubrovnik polynomials.

Contribution

It introduces a graphical calculus approach to classify skein relations in UFCs and derives explicit formulas for key matrices, linking fusion categories to knot invariants.

Findings

Classified skein relations for specific fusion rules in ribbon UFCs

Derived explicit formulas for the F-matrix in certain cases

Linked the spectrum of a rotation operator to knot polynomials

Abstract

Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form $q\otimes q \cong \mathbf{1}\oplus\bigoplus^k_{i=1}x_{i}$ in a UFC $\mathcal{C}$, and extract information using the graphical calculus. For instance, we classify all associated skein relations when $k=1,2$ and $\mathcal{C}$ is ribbon. In particular, we also consider the instances where $q$ is antisymmetrically self-dual. Our main results follow from considering the action of a rotation operator on a "canonical basis". Assuming self-duality of the summands $x_{i}$, some general observations are made e.g. the real-symmetricity of the $F$-matrix $F^{qqq}_q$. We then find explicit formulae for $F^{qqq}_q$ when $k=2$ and $\mathcal{C}$ is ribbon, and see that the spectrum of the rotation operator distinguishes between the Kauffman and Dubrovnik polynomials.