Anyons and the HOMFLY Skein Algebra

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 $q$ with ${\sum }_x{N}_{qq}^x\le 2$. 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.