In this talk, I will outline an approach to studying skein modules of 3-manifolds by embedding them into the Hilbert spaces of four-dimensional supersymmetric gauge theories. When the 3-manifold has reduced holonomy, this approach leads to an algorithm for the dimension of the skein module for a general gauge group, expressed as a sum over nilpotent orbits. Surprisingly, the dimensions obtained in this way often differ between Langlands-dual pairs.