We describe a modification of Khovanov homology [Duke Math. J. 101 (2000)
359-426], in the spirit of Bar-Natan [Geom. Topol. 9 (2005) 1443-1499], 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, Reiger
and Saito’s movie moves [J. Knot Theory Ramifications 2 (1993) 251-284; Adv.
Math. 127 (1997) 1-51] 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 [Algebr. Geom.
Topol. 4 (2004) 1211-1251] for the original ”unoriented theory”, with a few