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Fixing the functoriality of Khovanov homology

David Clark, Scott Morrison and Kevin Walker

Geometry & Topology 13 (2009) 1499–1582

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 disagreements.

Khovanov homology, functoriality, link cobordism
Mathematical Subject Classification 2000
Primary: 57M25
Secondary: 57M27, 57Q45
Received: 22 January 2008
Revised: 7 February 2009
Accepted: 28 October 2008
Published: 3 March 2009
Proposed: Vaughan Jones
Seconded: Mike Freedman, Joan Birman
David Clark
Department of Mathematics
Randolph-Macon College
Ashland, VA 23005
Scott Morrison
Microsoft Station Q
University of California
Santa Barbara 93106-6105
Kevin Walker
Microsoft Station Q
University of California
Santa Barbara 93106-6105