Reidemeister theorem implies, without too much fuss, that the monoidal categories of tangles, and of oriented tangles, can be presented by generators and relations. This is done for example in 
a) Kassel C: Quantum Groups. Springer. (For oriented tangles.)
b) Freyd P.J, Yetter DN. Braided compact closed categories with applications to low dimensional topology. Advances in Mathematics
Volume 77, Issue 2, October 1989, Pages 156-182.


Does any of you happen to know of a reference that discusses presentations of the monoidal categories of oriented (and / or un-oriented) virtual and welded tangles, by generators and relations? 


Such presentations appears to not be too hard to derived from the diagrammatic definition of welded and virtual knots, but I was wondering if someone has actually worked out the details. 



Presentations of the monoidal categories of virtual tangles and of welded tangles by generators and relations
As luck would have it and at the risk of shameless self-promotion, I wrote a paper answering this very question for framed virtual tangles: <https://arxiv.org/abs/1602.03080> .


Let $T$ be the category of framed oriented tangles and $V$ the category of framed oriented virtual tangles. Then $V$ is in fact **symmetric** monoidal, and there is a canonical monoidal functor $\iota:T\rightarrow V$. I assume that by "generators and relation" I you refer to the Shum-Reshetikhin-Turaev theorem stating that $T$ is the free ribbon category. Now I claim that the triple $(T,V,\iota)$ is the free "ribbon category equipped with a monoidal functor to a symmetric monoidal category". 


In particular, this implies that a ribbon category with a fiber functor to Vect, aka a ribbon Hopf algebra, gives rise to representations of $V$. 


I've thought a bit about the welded case, I think it's not too hard to deduce from this a presentation but it's harder to give a nice interpretation of it. I think that given a monoidal functor $F$ from a ribbon category $C$ to a symmetric monoidal $S$, if $F$ factors as $$C\rightarrow Z(S) \rightarrow S$$ where $Z(S)$ is the Drinfeld center, the first functor is braided monoidal and the last one is the forgetful functor, then the representations of $V$ you get factor through the category of welded tangles, and it might be that they all arise this way but I haven't tried to prove it.


