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. The Turaev surface can be seen as a Jones polynomial analogue of the Seifert surface. The Turaev genus of a link is the minimum genus of the Turaev surface for any diagram of the link. Also, a link is almost alternating if it is non-alternating and has a diagram that can be transformed into an alternating diagram via one crossing change.
In this talk, we compute the Khovanov homology of almost alternating and Turaev genus one links in the first and last two polynomial and homological gradings. Using our computations, we show that some specific knots and links are not almost alternating and have Turaev genus at least two. For certain knots in these families, our results lead to computations of Rasmussen's s-invariant and to inequalities involving the smooth 4-genus of the knots.
Meeting ID: 916 8023 8102