  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 $\sum_{x}N_{qq}^{x}\leq2$. 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.
