Aganagic–Vafa conjecture for arbitrary knots

Establish that the closed-string amplitudes associated with knot-dependent mirror curves arising from the resolved conifold through the large-N transition coincide with the closed-string amplitudes of the resolved conifold for arbitrary knots in S^3.

Background

The paper discusses the proposal of Aganagic and Vafa that knot-dependent mirror curves obtained from the resolved conifold encode the same closed-string amplitudes as the resolved conifold itself. The authors emphasize that this conjecture concerns arbitrary knots in S^3, whereas the results proved in the paper establish the corresponding topological-recursion and free-energy statements only for torus knots.

Thus, the general arbitrary-knot version remains beyond the scope of the paper. The torus-knot case is treated as a substantial special case: the authors prove topological recursion on the augmentation curve and show that its higher-genus free energies agree with those of the resolved conifold.