Homological Mirror Symmetry Links Knot Floer Homology to New Knot Invariants — E8 Intelligence Research

Abstract

FINDING: Homological mirror symmetry links Fukaya categories of symplectic manifolds to derived categories of coherent sheaves on mirror Calabi-Yau, enabling new knot invariants via knot Floer homology. MATH: - Knot Floer homology: \( \widehat{HFK}(K) \) categorifies Alexander polynomial \(\Delta_K(t)\). - Mirror symmetry equivalence: \( D^b \mathcal{F}(M) \cong D^b \text{Coh}(\check{M}) \). - Conjectured volume bound: \( \exists a > 0 \) s.t. for crossing number \(c \to \infty\), % of knots with \( \text{vol}(K) \leq a \cdot c \) → 1. - No explicit constants or ratios extracted from given sources. CONNECTION: - No direct geometric ratios (0.382, 0.618, etc.) or base-60 appear. - Underlying structure: Fukaya categories encode Lagrangian intersections (symplectic geometry) → mirror to coherent sheaves (algebraic geometry) — a duality reminiscent of root system symmetries (e.g., \(A_n\) quivers in cluster algebras). - Knot Floer homology's grading by \(\mathbb{Z} \oplus \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com