A brief explanation of my motivation before I ask my question. I am trying to understand Skein relations, the Jones polynomial, and their relations to Khovanov homology. To me, the natural setting to ask and answer the type of questions I'm interested in seems to be categorical. This motivates me to consider the following category:

Definition: (The category $\mathcal{KD}$)

  • The objects of the category $\mathcal{KD}$ are (oriented) knot diagrams (with ordered crossings).

  • The arrows of $\mathcal{KD}$ are sequences of smoothings, where each smoothing is applied to one of the ordered crossings.

Here, I am following Bar-Natan's notation - a smoothing is a resolution of a singular point in a knot diagram. See [1], page 3, start of the 3rd paragraph, for clarification.

A cursory search of MathOverflow and Google revealed no references to such a category, but that likely means I'm not using the correct keywords. I have not really started to work with this category, so it is quite possible I'm doing things rather inefficiently. Because of this, and not wanting to duplicate existing material, my questions are as follows:

  1. Am I overlooking a category obviously equivalent to $\mathcal{KD}$ or a more efficient construction of this category?

  2. Is such a category familiar in the literature?

  3. If such a category is familiar, what are some references to papers where it is used?

  4. Is there already an accepted notion of a category of knots (rather than knot diagrams)?

I think question 2 is only tangentially related; however, it I would be very interested in answers as it is where I began thinking about these questions. I was unable to to find a satisfying answer; in particular, the subcategory $S^1$, $\hom(S^1, S^1)$ in $2$-$\mathcal{C}$ob does not seem useful.


  1. Dror Bar-Nata, On Khovanov's Categorification of the Jones Polynomial, http://arxiv.org/abs/math/0201043
  • 2
    $\begingroup$ I have to say, I can't really see what this category is supposed to achieve; Khovanov homology already has a very nice interpretation in terms of categories of planar tangles from Bar-Natan's later work. It would be good to know if you're familiar with this story, and are for some reason unhappy with it and looking for an alternative. $\endgroup$ – Ben Webster Oct 18 '11 at 1:44
  • $\begingroup$ Look at John Armstrong's work about these things. I don't think he does things exactly the way that you desire, but it might fulfill your categorical intuition. $\endgroup$ – Scott Carter Oct 18 '11 at 2:20

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