# Invariants of unframed, oriented links from Reshetikhin Turaev construction

Hello!

I have a few questions on Reshetikhin Turaev invariants.

By RT any ribbon category ${\mathcal C}$ yields an invariant of oriented, framed links labelled with objects of ${\mathcal C}$.

Is there a general way to build from this an invariant of unframed, oriented links? At least in the case where one considers finite-dimensional modules over ${\mathcal U}_q({\mathfrak g})$ for simple ${\mathfrak g}$?

Howe does this relate to the general construction of an invariant of oriented, unframed links from an enhanced R-matrix?

I hope this isn't too elementary, but I couldn't find a reference.

Thank you very much!

Hanno

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The ribbon element (i.e. the move that changes framing) acts on a simple object $V$ by a scalar $\theta_V$. The writhe changes by 1 when you change framing. Hence $\theta_V^{-w(K)}RT_V(K)$ is an invariant of unframed oriented links (depending on your conventions I may have a sign wrong here). Note that this only works for simple objects, and it breaks some of the nice properties of RT invariants.