S-matrix for the BMW category - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:16:47Z http://mathoverflow.net/feeds/question/100881 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100881/s-matrix-for-the-bmw-category S-matrix for the BMW category Kevin Walker 2012-06-28T17:20:11Z 2012-06-28T17:38:32Z <p>This question concerns the Birman-Murakami-Wenzl category, or equivalently the tangle category associated to the 2-variable Kauffman polynomial. (See <a href="http://arxiv.org/pdf/math/0006227.pdf" rel="nofollow">here</a> for example.) </p> <p>The minimal idempotents of this category are indexed by Young diagrams (of arbitrary size; there are infinitely many of them). Consequently one can define numerical invariants of unoriented links whose components are labeled by Young diagrams. This is the "colored" BMW/2K polynomial.</p> <p>Of fundamental importance in this subject are the invariants $S_{\lambda\mu}$ of the <a href="http://en.wikipedia.org/wiki/Hopf_link" rel="nofollow">Hopf link</a> with its components labeled by Young diagrams (i.e. idempotents) $\lambda$ and $\mu$. In TQFT language, this is the "S-matrix" of the theory.</p> <p>My Question:</p> <blockquote> <p>Has the S-matrix for the BMW / 2-variable Kauffman category been calculated and published? If not, are partial results in this direction known?</p> </blockquote> <p>I've done some searching, but so far I've not found anything.</p> <p>See also the HOMFLY-PT version of this question <a href="http://mathoverflow.net/questions/100882/s-matrix-for-the-homfly-hecke-category" rel="nofollow">here</a>.</p>