As is well known, the *tangle hypothesis* of Baez and Dolan proposes that, for suitable definitions, the $n$-category of framed $n$-tangles in $n+k$ dimensions is the free $k$-tuply monoidal $n$-category with duals on one object. For ($n=1,k=2$) and ($n=2,k=2$) these have been proven respectively [here](http://dx.doi.org/10.1016/0022-4049(92)00039-T) and [arXiv:math/9811139](http://arxiv.org/abs/math/9811139) (but in the latter case only for unframed tangles). These results are expressed in terms of "classical" notion of $n$-categories. By "classical" n-category, I will mean the usual notion of a structure, with objects, and $1$-morphisms satisfying coherence conditions up to $2$-morphisms, which themselves are required to satisfy coherence conditions, and so forth.

On the other hand, Lurie ([arXiv:0905.0465](http://arxiv.org/abs/0905.0465)) has proven a version of the hypothesis in the general case, but for a definition of $n$-category that is too abstract for a humble physicist such as myself to do much with.

I have two related questions:

1. Are there are any more proofs of cases of special cases of the tangle hypothesis in terms of "classical" $n$-categories, beyond the ones I gave? (Specifically,  I am interested in the case $n=2,k=2$, but for framed tangles.)
5. Is it possible to extract such results from Lurie's work? (Or in other words, can one translate Lurie's definition of $n$-categories into a "classical" one, at least in low dimensional cases?)

I realize that the "classical" definitions become increasingly cumbersome as you increase $n$, but for $n=2,k=2$ you are dealing with braided monoidal 2-categories, which are still fairly manageable, e.g. see [this paper](http://arxiv.org/abs/1102.0981) for a general definition of weak braided monoidal 2-categories and a strictification theorem.