Skein-Theoretic Methods for Unitary Fusion Categories

Abstract

Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form $q\otimes q\cong 1\oplus {⨁}_{i=1}^k{x}_i$ 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 $F_q^{qqq}$. We then find explicit formulae for $F_q^{qqq}$ when $k=2$ and $\mathcal{C}$ is ribbon, and see that the spectrum of the rotation operator distinguishes between the Kauffman and Dubrovnik polynomials.

Some major edits and content reorganised. Removed section on 'physical remarks' to appear in S. Valera's thesis