Geometry & Topology

Fixing the functoriality of Khovanov homology

David Clark, Scott Morrison, and Kevin Walker

Full-text: Open access


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.

Article information

Geom. Topol., Volume 13, Number 3 (2009), 1499-1582.

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57M25: Knots and links in $S^3$ {For higher dimensions, see 57Q45}
Secondary: 57M27: Invariants of knots and 3-manifolds 57Q45: Knots and links (in high dimensions) {For the low-dimensional case, see 57M25}

Khovanov homology functoriality link cobordism


Clark, David; Morrison, Scott; Walker, Kevin. Fixing the functoriality of Khovanov homology. Geom. Topol. 13 (2009), no. 3, 1499--1582. doi:10.2140/gt.2009.13.1499.

Export citation