# 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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## 1 Answer

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.

This is exactly the same trick you use for relating the Jones polynomial and the Kauffman bracket.

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Noah, thank you! That's nice and simple :-) What happens if there is more than one color involved? –  Hanno Becker Jun 25 '11 at 9:07