  The Turaev surface of a link diagram $D$ is a closed, oriented surface
constructed from a cobordism between the all-$A$ and all-$B$ Kauffman states of
$D$. The Turaev genus of a link $L$ is the minimum genus of the Turaev surface
of any diagram $D$ of $L$. A link is alternating if and only if its Turaev
genus is zero, and so one can view Turaev genus one links as being close to
alternating links. In this paper, we study the Khovanov homology of a Turaev
genus one link in the first and last two polynomial gradings where the homology
is nontrivial. We show that a particular summand in the Khovanov homology of a
Turaev genus one link is trivial. This trivial summand leads to a computation
of the Rasmussen $s$ invariant and to bounds on the smooth four genus for
certain Turaev genus one knots.
