So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ribbon element.


How do these knot invariants change if I pick a different ribbon element in the same Hopf algebra? In particular, will something strange happen with 3-manifold invariants?



How do quantum knot invariants change when I pick a funny ribbon element?
Did you look at prop 5.21 in the [paper with Peter](http://arxiv.org/PS_cache/arxiv/pdf/0810/0810.0084v2.pdf)? I think that should answer your question.


There are two slightly different questions you could ask. First how does the framing-dependent invariant change. Here it is just (\pm 1)^#L where # is the number of components. Second how does the framing-corrected invariant change? Here it's (\pm 1)^#L (\pm 1)^writhe. In both cases the \pm 1 just measures whether you've changed the FS indicator of your rep V.


If you want to think about things labelled with components labelled by more than one irrep it'll get yuckier. 


