I know that it has been shown that $E_2$ algebra objects in Categories are simply braided monoidal categories. In particular, Lurie says that an $E_2$-monoidal structure on the infinity-category $N(C)$ (the nerve of a category C) is a braided monoidal structure on C.

Is there a generalization (not sure if this is the right word) of this where instead of having braiding isomorphisms we have: $b_{V,W}:V\otimes W\to W\otimes V$ is a quasi-isomorphism? Obviously, we need some dg-structure, so we would be talking about an $E_2$-monoidal structure on the dg-nerve of a dg-category.

What would the implications be for the composition $b_{W,V}\circ b_{V,W}$ when compared to the identity quasi-isomorphism $\mathrm{id}:V\otimes W\to V\otimes W$?

I am interested in possible new knot invariants.

Any thoughts or improvements on how I am asking the question are highly appreciated.