Noncommutative Geometry Seminar
Kauffman brackets on a surface are quantum deformations of homomorphisms from the fundamental group of the surface to the Lie group SL_2(C). A fundamental example arises from Witten's topological quantum field theory interpretation of the Jones polynomial of knots in 3-dimensional manifolds. I will discuss properties of finite-dimensional Kauffman brackets when the quantum parameter q is a root of unity. This will include the construction of invariants, existence properties, and uniqueness theorems and conjectures.