Skein-Theoretic Methods for Unitary Fusion Categories

Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We consider a fusion rule of the form qotimes q cong mathbf{1}oplusbigoplus^k_i=1x_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.