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We reconsider topological string realization of SU(N) Chern-Simons theory onS^3. At large N, for every knot K in S^3, we obtain a polynomial A_K(x,p;Q) intwo variables x,p depending on the t'Hooft coupling parameter Q=e^{Ng_s}. Itsvanishing locus is the quantum corrected moduli space of a special Lagrangianbrane L_K, associated to K, probing the large N dual geometry, the resolvedconifold. Using a generalized SYZ conjecture this leads to the statement thatfor every such Lagrangian brane L_K we get a distinct mirror of the resolvedconifold given by uv=A_K(x,p;Q). Perturbative corrections of the refinedB-model for the open string sector on the mirror geometry capture BPSdegeneracies and thus the knot homology invariants. Thus, in terms of itsability to distinguish knots, the classical function A_K(x,p;Q) contains atleast as much information as knot homologies. In the special case when N=2, ourobservations lead to a physical explanation of the generalized (quantum) volumeconjecture. Moreover, the specialization to Q=1 of A_K contains the classicalA-polynomial of the knot as a factor.

Large N Duality, Mirror Symmetry, and a Q-deformed A-polynomial for Knots