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

TL;DR

The paper proposes a unified framework linking knot invariants to topological strings via large $N$ duality and a generalized SYZ mirror construction. For each knot $K$, a polynomial $A_K(x,p;Q)$ defines a mirror curve $uv=A_K(x,p;Q)$ whose classical limit contains the knot’s A-polynomial as a factor when $Q\to 1$, while the full $Q$-deformed curve encodes open-string data that reproduce knot invariants and their refinements. The approach yields explicit mirrors for knots (e.g., unknot and several torus knots), demonstrates that torus-knot periods match the conifold, and connects to the quantum volume/AJ conjectures, as well as to knot contact homology through a deformed $Q_B$ operator. This framework suggests that the classical $A_K(x,p;Q)$ is at least as informative as knot homologies for distinguishing knots and provides a computational bridge between topological strings, Chern-Simons theory, and knot invariants with potential for broad generalization.

Abstract

We reconsider topological string realization of SU(N) Chern-Simons theory on S^3. At large N, for every knot K in S^3, we obtain a polynomial A_K(x,p;Q) in two variables x,p depending on the t'Hooft coupling parameter Q=e^{Ng_s}. Its vanishing locus is the quantum corrected moduli space of a special Lagrangian brane L_K, associated to K, probing the large N dual geometry, the resolved conifold. Using a generalized SYZ conjecture this leads to the statement that for every such Lagrangian brane L_K we get a distinct mirror of the resolved conifold given by uv=A_K(x,p;Q). Perturbative corrections of the refined B-model for the open string sector on the mirror geometry capture BPS degeneracies and thus the knot homology invariants. Thus, in terms of its ability to distinguish knots, the classical function A_K(x,p;Q) contains at least as much information as knot homologies. In the special case when N=2, our observations lead to a physical explanation of the generalized (quantum) volume conjecture. Moreover, the specialization to Q=1 of A_K contains the classical A-polynomial of the knot as a factor.