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I was wondering if the tangle functors constructed in

"A functor-valued invariant of tangles" https://arxiv.org/pdf/math/0103190.pdf

"An invariant of tangle cobordisms via subquotients of arc rings" https://arxiv.org/abs/math/0610054

are braided monoidal?

For example in the Reshetekhin and Turaev set up, they have a braided monoidal functor from tangles to $Rep(U_q(sl_2))$. Sending points/tangles to representations/intertwiners, and one can take tensor products of points/tangles to tensor products of representations/intertwiners.

Are there any examples of tangle invariants coming from a braided monoidal functor but give something on the level of khovanov homology (or something that "decategorifies" to the Jones Polynomial)? Namely where points/tangles are assigned to categories/functors, tensor products of points/tangles carry over to some sort of tensor product of categories/functors, and finally also "decategorifies" to the Jones polynomial? Should such a thing exist?

I have also seen some of the foam constructions and I think they relate to braided monoidal 2-categories, but it seems the points/tangles are not assigned to categories/functors.

I am quite a beginner and have not fully digested the content or understand a bigger picture in the literature. Any help or comments would be greatly appreciated

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    $\begingroup$ It definitely doesn't comes from a braided monoidal functor. We hope it comes from a braided monoidal 2-functor, but before asking this question you'd need a braided monoidal 2-category structure on the target, and AFAIK (I'm not an expert on this) we're not there yet, although there has been a lot of work in that direction. $\endgroup$
    – Adrien
    Commented Apr 29, 2020 at 11:03
  • $\begingroup$ @Adrien Thanks for the Help! I have actually be trying to learn your work "Integrating Quantum Groups over Surfaces" with Ben-Zvi and Jordan. I was wondering if given a braided monoidal 2-category, would one be able to get an $E_n$-algebra? And use the factorization homology framework? If the answer is in the positive, which $n$? What would be the analogue of $Pr_c$? $\endgroup$
    – Andy Nguyen
    Commented May 1, 2020 at 11:08
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    $\begingroup$ Yes, any reasonable definition of braided monoidal 2-category should coincide with the notion of $E_2$-algebra in the (3,1)-category of 2-categories. For Khovanov homology I suppose you want something like a symmetric monoidal $(\infty,1)$-category of presentable dg-2-categories which I have no clue how to define (but I'm sure some people have thought about it). Then indeed you can use the framework of factorization homology to attach 2-categories to (oriented if you have a framed $E_2$-algebra) surfaces. What's more tricky, AFAIK, is how to define the correct version of ribbon 2-category. $\endgroup$
    – Adrien
    Commented May 3, 2020 at 10:22
  • $\begingroup$ @Adrien Thanks a bunch! $\endgroup$
    – Andy Nguyen
    Commented May 3, 2020 at 14:25

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