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?
Did you look at prop 5.21 in the paper with Peter? 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.
Ben,
As I mentioned in response to your previous question about ribbon elements, the element u which is defined from the R-matrix, u=\mu\circ(S\ot \id)(R21) has the property that uS(u)=v^2 (well this is not the formula I gave for u in that post, because the one I gave was incorrect; this one appears to be correct according to wikipedia).
This relation v^2=uS(u) is true in any ribbon Hopf algebra, and in particular it implies that v has to be a square root of uS(u). So I think this means that the ribbon element is almost unique.
More precisely, let v and w be two ribbon elements. Then v/w is a grouplike element of order two. I think this implies that the corresponding invariant applied to a link will be multiplied by the constant v/w applied to each link. Now if you choose irreducible representations to label your link, then this number would have to be +/- 1.
Does this seem correct? -david
