Mirror Symmetry Reconstructs Khovanov Homology via Categorified Knot Invariants — E8 Intelligence Research

FINDING: Khovanov homology — a categorified knot invariant — is being reconstructed via homological mirror symmetry, linking knot theory to symplectic geometry and derived categories. | MATH: Khovanov homology assigns a bigraded chain complex to a knot, whose homology is a knot invariant; its Euler characteristic recovers the Jones polynomial \( V_K(q) = \sum_{i,j} (-1)^i q^j \dim H^{i,j}(K) \). Mirror symmetry here maps the A-model (Lagrangian intersection Floer homology) to the B-model (derived category of coherent sheaves), with knot invariants arising as counts of holomorphic disks or Ext-groups. Equivariant versions (U(2), U(1)×U(1)) introduce additional gradings and involutions (e.g., \(\widehat{\sigma}\), \(\widehat{\nu}\)) refining the structure. | CONNECTION: The bigrading \((i,j)\) in Khovanov homology naturally encodes a \(\mathbb{Z} \times \mathbb{Z}\) lattice — a rank-2 root system (A1×A1). The Jones polynomial's variable \(q\) at roots of unity (e.g., \(q=e^{2\pi i/3}\)) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com