Mirror Symmetry Unifies Knot Homologies via Branes and Floer Theory — E8 Intelligence Research

FINDING: Aganagic's program constructs knot homologies via mirror symmetry, pairing the derived category of coherent sheaves (B-branes) with symplectic Floer theory (A-branes), yielding homological link invariants that unify Khovanov-type and Heegaard-Floer-type structures. | MATH: The core dictionary is Homological Mirror Symmetry (HMS): \( D^b(\text{Coh}(X)) \cong D^b(\mathcal{F}\text{uk}(X^\vee)) \). Knot invariants arise as Hom-spaces in these categories: \( \text{Hom}_{\mathcal{D}}(\mathcal{B}_1, \mathcal{B}_2) \) where \(\mathcal{B}_i\) are branes (Lagrangian submanifolds or sheaves) associated to the knot. Petkova's bordered invariant is a contact element in sutured Floer homology, \( \psi(\xi) \in \widehat{HF}(M, \Gamma) \), with gluing maps composing via \( \widehat{HF}(M_1) \otimes \widehat{HF}(M_2) \to \widehat{HF}(M_1 \cup M_2) \). The growth rate of Floer homology iterates defines a symplectic zeta function \( \zeta_{\phi}(z) = \exp\left( \sum_{n\ge1} \frac{\dim HF(\phi^n) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com