Links, quantum groups and TQFTs

The Jones polynomial and the Kauffman bracket are constructed, and their relation with knot and link theory is described. The quantum groups and tangle functor frameworks for understanding these invariants and their descendents are given. The quantum group $U_q(sl_2)$, which gives rise to the Jones polynomial, is constructed explicitly. The 3-manifold invariants and the axiomatic topological quantum field theories which arise from these link invariants at certain values of the parameter are constructed and proven to be invariant.