# 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?

I've done some searching, but so far I've not found anything.

See also the HOMFLY-PT version of this question here.

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I don't know the answer, but is it obvious to see that the Hopf link matrix is the S-matrix of the TQFT? Do you mean the evolution operator assigned to a cobordism? If yes, which cobordism? –  Turion Dec 29 '13 at 15:22
Since the BMW category seems to be related to quantum groups, maybe Scott Morrison's QuantumGroups mathematica package can help: katlas.org/wiki/QuantumGroups%60 (Unfortunately, the documentation is very sparse, but it has enough functionality to evaluate the Hopf link labelled with irreducible representations of the quantum group. If this would help, I'd be happy to write an answer elaborating on that. –  Turion Dec 29 '13 at 15:44
@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