Transparent $SL_n$-skeins

For a Lie group $G$, the G-skein module of a 3-dimensional manifold $M$ is a fundamental object in Witten’s interpretation of quantum knot invariants in the framework of a topological quantum field theory. It depends on a parameter q and, when this parameter q is a root of unity, the G-skein module contains elements with a surprising “transparency” property, in the sense that they can be traversed by any other skein without changing the resulting total skein. I will describe some (and conjecturally all) of these transparent elements in the case of the special linear group $SL_n$. The construction is based on the very classical theory of symmetric polynomials in n variables.