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Fixing the functoriality of Khovanov homology. (English) Zbl 1169.57012
Khovanov homology assigns graded modules to links and isomorphisms to isotopies (and more generally to link cobordisms). However in the original version the isomorphisms are only well-defined up to sign. The main result of this paper is a sign-determined modification of D. Bar-Natan’s approach [Geom. Topol. 9, 1443–1499 (2005; Zbl 1084.57011)] which is functorial on the category of oriented tangles in \(B^3\) and oriented cobordisms in \(B^3\times{I}\), using the notion of “disorientation”. A disoriented manifold is one which is partitioned into oriented submanifolds by hypersurfaces (“disorientation seams”) with preferred normal directions.
The category of tangles and cobordisms is naturally modelled by a category PD (of “planar diagrams”) in which the morphisms are generated by planar isotopy, Morse and Reidemeister moves, and satisfy relations deriving from the movie moves of Carter, Rieger and Saito [J. S. Carter and M. Saito, J. Knot Theory Ramifications 2, No. 3, 251–284 (1993; Zbl 0808.57020), J. S. Carter, J. H. Rieger and M. Saito, Adv. Math. 127, No. 1, 1–51 (1997; Zbl 0870.57032)] and D. Roseman [Banach Cent. Publ. 42, 347–380 (1998; Zbl 0906.57010)]. The functor is defined in terms of planar algebras and is shown to be compatible with the movie relations, using a result on duality in Khovanov homology which extends earlier results about mirror images of knots to tangles.
The paper is long, but well-written, with careful attention to the motivation at each step. The final third of the paper consists of a section “Odds and ends” and a substantial appendix containing the proofs of some of the lemmas and definitions of “planar algebra” and “canopis”.

MSC:
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
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