Mirror Symmetry Unifies Knot Invariants via Toric Calabi-Yau Geometry — E8 Intelligence Research

FINDING: Mirror symmetry for toric Calabi-Yau varieties provides a geometric framework that unifies knot invariants (Chern-Simons) with homological algebra, via SYZ fibrations and D-model constructions. | MATH: Toric Calabi-Yau: \(X_\Sigma\) with fan \(\Sigma \subset \mathbb{R}^3\), SYZ mirror \(X^\vee\) via dual torus fibrations \(T^3 \to T^3\). Chern-Simons invariants: \(Z_{\text{CS}}(M,K) = \int \mathcal{D}A \, e^{i k S_{\text{CS}}[A]}\), \(S_{\text{CS}} = \frac{1}{4\pi}\int_M \text{Tr}(A\wedge dA + \frac{2}{3}A^3)\). Homological mirror: \(D^b\text{Coh}(X^\vee) \simeq D^b\text{Fuk}(X)\). Knot homology from mirror: \(H_{\text{knot}} \simeq \text{Hom}_{\text{category}}(\mathcal{F}_1, \mathcal{F}_2)\) for Lagrangian branes \(\mathcal{F}_i\). | CONNECTION: Toric fans encode root systems of type \(A_n\) (for \(X = \text{Tot}(K_{\mathbb{P}^1})\) or \(K_{\mathbb{P}^2}\)), with lattice \(\mathbb{Z}^3\) and dual lattice \(\mathbb{Z}^3\) — crystallographic symmetry of the \(A_2\) root lattice Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com