most recent 30 from http://mathoverflow.net 2013-05-22T15:01:19Z http://mathoverflow.net/feeds/question/2211 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/2211/how-do-quantum-knot-invariants-change-when-i-pick-a-funny-ribbon-element Ben Webster 2009-10-23T22:54:16Z 2009-10-25T00:41:01Z

<p>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.</p> <p>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?</p>

http://mathoverflow.net/questions/2211/how-do-quantum-knot-invariants-change-when-i-pick-a-funny-ribbon-element/2237#2237 David Jordan 2009-10-24T02:25:03Z 2009-10-24T02:25:03Z

<p>Ben,</p> <p>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).</p> <p>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.</p> <p>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.</p> <p>Does this seem correct? -david</p>

http://mathoverflow.net/questions/2211/how-do-quantum-knot-invariants-change-when-i-pick-a-funny-ribbon-element/2394#2394 Noah Snyder 2009-10-24T23:55:22Z 2009-10-25T00:41:01Z

<p>Did you look at prop 5.21 in the <a href="http://arxiv.org/PS%5Fcache/arxiv/pdf/0810/0810.0084v2.pdf" rel="nofollow">paper with Peter</a>? I think that should answer your question.</p> <p>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.</p> <p>If you want to think about things labelled with components labelled by more than one irrep it'll get yuckier. </p>