The Jones polynomial, Khovanov homology, and Turaev genus

The Turaev surface of a link diagram is a surface built from a cobordism between the all-A and all-B Kauffman states of the diagram, and the Turaev genus of a link is the minimum genus of the Turaev surface for any diagram of the link. The Turaev surface was first used to give simple versions of the Kauffman-Mursaugi-Thistlethwaite proofs of some Tait conjectures.
In this talk, we first give a brief history of the Turaev surface, the Turaev genus of a link, and some related applications. We then discuss some recent results on the extremal and near extremal terms in the Jones polynomial and Khovanov homology of a Turaev genus one link.