Knot polynomials: Skein>Matrix>Group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T23:30:17Z http://mathoverflow.net/feeds/question/76065 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76065/knot-polynomials-skeinmatrixgroup Knot polynomials: Skein>Matrix>Group? Hauke Reddmann 2011-09-21T15:13:07Z 2011-09-23T01:31:14Z <p>OK, the heading was a bit tersely formulated...<br> If you have a quantum group and an irrep, you theoretically know the R matrix (mathematicians are a notoriously idle lot, they give the general formula and thus the problem is solved :-) - and the characteristic equation of the R matrix is a valid skein equation.</p> <p>Now to the reverse process. Question 1: Are there <em>really</em> skein equations that can't be modeled with a R matrix? E.g. I heard that already the Kauffman 2-variable polynomial is unattainable this way, but I only heard it and never saw an actual proof. (I have no idea how the situation is for <em>directed</em> knots.)<br> Question 2. OK, assume we have a working R matrix, is there always a quantum group associated with that? ("Baxterization"??) Again, I think the answer is "no" for unoriented knots. (If no quantum group-based knot polynomial can distinguish mutants - also from the "so I heard" variety - the question is solved, since I have a R matrix doing this.)</p> http://mathoverflow.net/questions/76065/knot-polynomials-skeinmatrixgroup/76173#76173 Answer by Eric Rowell for Knot polynomials: Skein>Matrix>Group? Eric Rowell 2011-09-23T01:31:14Z 2011-09-23T01:31:14Z <p>The Kauffman 2-variable knot polynomial probably can't be obtained from a quantum group if by this you mean the usual q-deformed universal enveloping algebras. If your two variables are $(r,q)$ and $r=\pm q^n$ then it can be obtained from quantum groups of type B,C or D (depending on $n$), see Wenzl's paper Comm. Math. Phys. 133 (1990) 383-432.</p> <p>Generally a nice paper in which this kind of knot-invariant-to-quantum-group is discussed in Turaev and Wenzl's paper Math. Annalen 309 (1997), 411-461.</p> <p>If you just start with an R-matrix and you want to define knot-invariants you need what Turaev calls an "enhanced Yang-Baxter operator." Given an R-matrix (enhanced or otherwise) it is possible to construct a braided Hopf algebra via the FRT (Faddeev-Reshetikhin-Takhtajan) construction but this will be unpleasant in general (infinite dimensional for sure).</p>