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

FINDING: Mirror symmetry maps toric Calabi-Yau geometry to knot invariants (homological/Chern-Simons), unifying SYZ fibrations with quantum knot theory. MATH: - Toric Calabi-Yau: \( X = \mathbb{C}^n // T^k \), with Kähler moduli encoded by toric fan; SYZ mirror is a Landau–Ginzburg model with superpotential \( W = \sum_i z_i + q \prod_i z_i^{-1} \) (for local \( \mathbb{C}^3 \) case, \( q = e^{-t} \), \( t \) = complexified Kähler parameter). - Knot invariants: Chern–Simons partition function \( Z_{CS}(M, k) = \int \mathcal{D}A \, e^{i k S_{CS}[A]} \), with \( S_{CS} = \frac{1}{4\pi} \int_M \mathrm{Tr}(A \wedge dA + \frac{2}{3} A^3) \); level \( k \) relates to string coupling \( g_s = 2\pi i/(k+N) \). - Homological invariants: Khovanov homology (categorification of Jones polynomial) — graded vector space with Poincaré polynomial \( \sum_{i,j} t^i q^j \dim H^{i,j} \). - Mirror map: \( q = e^{-t} \) ↔ \( z = e^{-u} \) (flat coordinates), with \( t, u \) related by period integ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com