Is the complete functorial structure for Khovanov--Lee homology known?

Question

I'm interested in Lee's modification of Khovanov homology, which I'll denote $\operatorname{Kh}_{\operatorname{Lee}}^\ast$. Below $L$ is a link in $\mathbb R^3$.

The groups $\operatorname{Kh}_{\operatorname{Lee}}^\ast(L)$ for a link $L$ are very simple: there is an isomorphism:

$$\bigoplus_{\text{orientations of }L}\mathbb Q\to\operatorname{Kh}_{\operatorname{Lee}}^\ast(L)$$

I write the left hand side as I do and not simply as $\mathbb Q^{\oplus 2^{\left|L\right|}}$ because given an orientation of $L$ (and a diagram of $L$), there is in some sense a natural choice of element of $\operatorname{Kh}_{\operatorname{Lee}}^\ast(L)$ (and these elements form a basis of $\operatorname{Kh}_{\operatorname{Lee}}^\ast(L)$). However, this map not natural in the sense of being functorial.

But it's really really close to being functorial! Rasmussen proved (see Khovanov homology and the slice genus Proposition 4.1) that if we identify $\operatorname{Kh}_{\operatorname{Lee}}^\ast(L)$ with $\bigoplus_{\text{orientations of }L}\mathbb Q$, then under a cobordism from $L_1$ to $L_2$, a orientation on $L_1$ is sent to a linear combination of the orientations on $L_2$ which extend to an orientation of the cobordism agreeing with the input orientation on $L_1$. This should be viewed as a sort of approximate functoriality, and the result has been slightly refined by Rasmussen here.

My question is: has $\operatorname{Kh}_{\operatorname{Lee}}^\ast$ been calculated as a functor? In other words, do we know a canonical way of mapping $\bigoplus_{\text{orientations of }L}\mathbb Q$ to $\operatorname{Kh}_{\operatorname{Lee}}^\ast$ so that the maps associated to arbitrary cobordisms are described elementarily in terms of orientations?

This doesn't seem to be a deep problem; it comes down to writing down the generators explicitly in the chain complex defining $\operatorname{Kh}_{\operatorname{Lee}}^\ast(L)$, and checking what happens when we do a Reidemeister move or Morse move. However the calculations are kind of messy, and this is presumably why Rasmussen didn't pursue the point further (at least, not that I am aware of). But the last paper of his referred to above was in 2005, so I'm guessing that someone has probably straightened this out since then. I've been unsuccessful in finding a reference, though.

Answer 1

As the title suggests, the paper Fixing the functoriality of Khovanov homology by Clark, Morrison and Walker fixes the functoriality in Khovanov homology. The same techniques (disoriented surfaces) fix the functoriality in Lee homology. In fact, the paper is written in such a way that it proves functoriality simultaneously for both Khovanov and Lee homology.

(Recall Bar-Natan's "free $\alpha$" version of Khovanov homology. If we set $\alpha = 0$, we get the original Khovanov homology. If we set $\alpha$ to any non-zero complex number, we get something identical to Lee homology, up to Euler characteristic normalizations. Lee's original paper has $\alpha = 1$ (or is it $\alpha = 8$?), but it's actually more convenient to let $\alpha = 2$.)

The upshot is that the Lee theory, after tweaking the definition by using disoriented surfaces, is naturally isomorphic to the simple theory you describe in your question. To a link we assign the vector space generated by orientations of the link, and to an unoriented cobordism we assign the sum, over all orientations of the cobordism, of the induced map between the orientations of the incoming and outgoing boundaries of the cobordism. (If we have set $\alpha$ to something other than 2, then we need to throw in factors of $\sqrt{\alpha/2}$ raised to the Euler characteristic of surfaces.)

If one were only interested in Lee homology, one could simply the proof in the "Fixing functoriality" somewhat, but there would still be many cases of oriented Reidemeister moves and second order oriented Reidemeister moves (movie moves) to check.