No abstract

We give an exposition of how the Kauffman bracket arises for certain systems of anyons, and do so outside the usual arena of Temperley-Lieb-Jones categories. This is further elucidated through the discussion of the Iwahori-Hecke algebra and its relation to modular tensor categories. We then proceed to classify the framed link-invariants associated to a system of self-dual anyons q withx N x qq ≤ 2. In particular, we construct a trace on the HOMFLY skein algebra which can be expanded via gauge-invariant quantities, thereby generalising the case of the Kauffman bracket. Various examples are provided, and we deduce some interesting properties of these anyons along the way.

Unitary fusion categories (UFCs) have gained increased attention due to emerging connections with quantum physics. We discuss how UFCs can be understood as fusion categories equipped with a "positive dagger structure" and apply this in a graphical context. Given a fusion rule q ⊗ q ∼ = 1 ⊕ k i=1 x i in a UFC C, we extract information using skein-theoretic methods and a rotation operator. For instance, we classify all associated framed link invariants when k = 1, 2 and C is ribbon. In particular, we also consider the instances where q is antisymmetrically self-dual. Some of this work is reformulated from the perspective of braid representations factoring through the Hecke and Temperley-Lieb algebras. Our main results follow from considering the action of the 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 C is ribbon, and see that the spectrum of the rotation operator distinguishes between the Kauffman and Dubrovnik polynomials. Finally, we apply some of our results in a physical setting (where C is a unitary modular category) and provide some worked examples: quantum entanglement is discussed using the graphical calculus, framed links are interpreted as Wilson loops, and a theorem is given on the duality of topological charges.