Skein recursion for holomorphic curves and invariants of the unknot

We determine the skein-valued Gromov–Witten partition functions for single toric Lagrangian branes in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mn>3</mml:mn> </mml:msup> </mml:math> or in the resolved conifold. We first show geometrically they must satisfy certain skein-theoretic recursions, and then solve these equations. The recursion is a skein-valued quantization of the equation of the mirror curve. The solution is the expected hook-content formula.