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2009 Fixing the functoriality of Khovanov homology
David Clark, Scott Morrison, Kevin Walker
Geom. Topol. 13(3): 1499-1582 (2009). DOI: 10.2140/gt.2009.13.1499


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.


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David Clark. Scott Morrison. Kevin Walker. "Fixing the functoriality of Khovanov homology." Geom. Topol. 13 (3) 1499 - 1582, 2009.


Received: 22 January 2008; Revised: 7 February 2009; Accepted: 28 October 2008; Published: 2009
First available in Project Euclid: 20 December 2017

zbMATH: 1169.57012
MathSciNet: MR2496052
Digital Object Identifier: 10.2140/gt.2009.13.1499

Primary: 57M25
Secondary: 57M27, 57Q45

Rights: Copyright © 2009 Mathematical Sciences Publishers


Vol.13 • No. 3 • 2009
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