  Let D be a link diagram with n crossings, s_A and s_B its extreme states and
|s_AD| (resp. |s_BD|) the number of simple closed curves that appear when
smoothing D according to s_A (resp. s_B). We give a general formula for the sum
|s_AD|+|s_BD| for a k-almost alternating diagram D, for any k, characterising
this sum as the number of faces in an appropriate triangulation of an
appropriate surface with boundary. When D is dealternator connected, the
triangulation is especially simple, yielding |s_AD|+|s_BD|=n+2-2k. This gives a
simple geometric proof of the upper bound of the span of the Jones polynomial
for dealternator connected diagrams, a result first obtained by Zhu in 1997.
Another upper bound of the span of the Jones polynomial for dealternator
connected and dealternator reduced diagrams, discovered historically first by
Adams et al in 1992, is obtained as a corollary. As a new application, we prove
that the Turaev genus is equal to the number k of dealternator crossings for
any dealternator connected diagram.
