We describe a modification of Khovanov homology (math.QA/9908171), in the
spirit of Bar-Natan (math.GT/0410495), which makes the theory properly
functorial with respect to link cobordisms.
This requires introducing `disorientations' in the category of smoothings and
abstract cobordisms between them used in Bar-Natan's definition.
Disorientations have `seams' separating oppositely oriented regions, coming
with a preferred normal direction. The seams satisfy certain relations (just as
the underlying cobordisms satisfy relations such as the neck cutting relation).
We construct explicit chain maps for the various Reidemeister moves, then
prove that the compositions of chain maps associated to each side of each of
Carter and Saito's movie moves (MR1238875, MR1445361) always agree. These
calculations are greatly simplified by following arguments due to Bar-Natan and
Khovanov, which ensure that the two compositions must agree, up to a sign. We
set up this argument in our context by proving a result about duality in
Khovanov homology, generalising previous results about mirror images of knots
to a `local' result about tangles. Along the way, we reproduce Jacobsson's sign
table (math.GT/0206303) for the original `unoriented theory', with a few
disagreements.Comment: 91 pages. Added David Clark as co-author. Further detail on
variations of third Reidemeister moves, to allow treatment of previously
missing cases of movie move six. See changelog section for more detai