Skein recursion for holomorphic curves and invariants of the unknot

We determine the skein-valued Gromov-Witten partition function for a single toric Lagrangian brane in ${\mathbb{C}}^3$ or the resolved conifold. We first show geometrically they must satisfy a certain skein-theoretic recursion, and then solve this equation. The recursion is a skein-valued quantization of the equation of the mirror curve. The solution is the expected hook-content formula.