Fixing the functoriality of Khovanov homology

Abstract

We describe a modification of Khovanov homology [13], in the spirit of Bar-Natan [2], 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, Rieger and Saito's movie moves [8; 7] 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 [10] for the original "unoriented theory", with a few disagreements.

sl(2) tangle homology with a parameter and singular cobordisms

“…We note that our construction and main result for a D 0 case are close to that by Clark, Morrison and Walker in [5], and that the two pieces of work were done independently. However, we borrowed from [5] the excellent idea of working with "homotopically isolated" objects when checking the functoriality property of our invariant, which makes the calculations much easier.…”

“…We choose a direct summand in the chain complex associated to the first frame of the clip, having the property that has no homotopies in or out, and we observe its image under the clip. This is the method used by Clark, Morrison and Walker in [5], from where we recall the following results (which hold in our case as well). If A is a resolution in a formal complex OET ; so that A does not contain closed webs and is not connected by differentials to resolutions containing closed webs, then A is homotopically isolated; that is, for any homotopy h, the restriction of dh C hd to A is zero.…”

“…However, we borrowed from [5] the excellent idea of working with "homotopically isolated" objects when checking the functoriality property of our invariant, which makes the calculations much easier.…”

The Rasmussen invariant and the Milnor conjecture

“…A similar result is most likely to hold in the filtred Lee case. Moreover, the sign issue can be fixed at the cost of a more involved construction; see [8,5,3]. But anyway, this (up to sign) invariance of the induced maps is not necessary to prove the Milnor conjecture.…”

The half-twist forUq(g) representations

“…With this pivotal structure F S(•) = −1. This other pivotal structure is given diagrammatically by the disoriented Temperley-Lieb category of [Clark, Morrison and Walker 2009].…”