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.  

Multiple Dirichlet series are meromorphic functions in  several complex variables developed in analytic number theory to study moments of L-functions. Nichols algebras are noncommutative algebras developed to study Hopf algebras and quantum groups. In joint work in progress with Ian Whitehead, we give a uniform construction of multiple Dirichlet series in the function field setting, unifying many previous constructions and producing new examples. I will give an introduction to these two areas and explain how our work relates them.