UC Santa Barbara

We explore how skein theoretic techniques can be applied to the study of quantum

representations of mapping class groups. Of particular interest will be looking into the

asymptotic faithfulness property of quantum representations coming from unimodal versions

of representation categories of quantum groups. We then introduce a combinatorial

property on the graphical calculus of these representation categories which implies asymptotic

faithfulness. We proceed to show that this property is satisfied in some specific cases,

in short we provide support for the conjecture that these quantum representations will

always be asymptotically faithful. This will lead into a discussion of other applications

within low dimensional topology. Finally applications to topological quantum computing

will be given, introducing a potential encoding of qudits making use of these quantum

representations.