# S-matrix for the BMW category

This question concerns the Birman-Murakami-Wenzl category, or equivalently the tangle category associated to the 2-variable Kauffman polynomial. (See here for example.)

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.

Of fundamental importance in this subject are the invariants $S_{\lambda\mu}$ of the Hopf link with its components labeled by Young diagrams (i.e. idempotents) $\lambda$ and $\mu$. In TQFT language, this is the "S-matrix" of the theory.

My Question:

Has the S-matrix for the BMW / 2-variable Kauffman category been calculated and published? If not, are partial results in this direction known?

@Turion: For TQFTs built out of finite quotients of quantum groups, the (finite) matrix associated to the Hopf link calculates a bordism which rotates the torus by $\pi/2$. I'm calling it the S-matrix in the above infinite setting by analogy -- I'm not claiming that it calculates anything in this case. –  Kevin Walker Dec 29 '13 at 22:20