Gopakumar-Vafa correspondence relates the large N expansion of SU(N) HOMFLY invariants of the unknot in S^3, to topological strings on the resolved conifold.
I will describe a generalization which relates HOMFLY invariants of analogs of the unknot in S^3/ADE, and topological strings on a non-toric CY3. The first ones have a matrix model description due to Marino, and the topological recursion on the matrix model spectral curve governs their large N expansion. We find by direct computation that the matrix model spectral curve is a 1-parameter specialization of the spectral curve of a relativistic Toda system of type ADE. This is consistent with the expectation from geometric engineering of a 5d gauge theory from the CY. The method we use applies in greater generality to compute the spectral curve of various matrix models.
This is based on joint works with Eynard and Brini.