Mirror Symmetry Reduces Knot Homologies to 1D A-Model via Categorical Equivalence — E8 Intelligence Research

FINDING: Mirror symmetry in Aganagic's framework reduces knot homologies (Khovanov/Knot Floer) to a 1D A-model with a lambda parameter, linking topological invariants to symplectic geometry via a D-model (derived category) — a categorical equivalence between A-branes and B-branes. | MATH: The core is the identification of knot Floer homology \( \widehat{HFK}(K) \) with the Lagrangian intersection Floer homology of a knot conormal \( L_K \subset T^*S^3 \), under SYZ mirror symmetry. The lambda parameter \( \lambda \) (likely the equivariant variable in the deformed A-model) encodes the \(sl(2)\) weight grading. The Abouzaid family Floer cohomology provides the functor \( \mathcal{F}: D^b\mathcal{A} \to D^b\mathcal{B} \) where \( \mathcal{A} \) is the wrapped Fukaya category and \( \mathcal{B} \) the coherent sheaf category on the mirror. The conjectured volume-cohomology inequality from the arXiv paper: \( \exists a>0 \) s.t. \( \lim_{c\to\infty} P\left( \frac{\log \dim \widehat{HFK}(K) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com