Mirror Symmetry Unifies Knot Invariants via 1D Topological Theory — E8 Intelligence Research

FINDING: Mirror symmetry in the A-model (knot Floer homology) and B-model (D-modules) unifies knot invariants via a one-dimensional topological theory, with new conjectures linking hyperbolic volume to knot cohomology growth. MATH: - Knot Floer homology: \( \widehat{HFK}(K) \) — a bigraded vector space with Euler characteristic equal to the Alexander polynomial \( \Delta_K(t) \). - Mirror symmetry: \( A \)-model (symplectic, Lagrangian branes) ↔ \( B \)-model (complex, D-branes as D-modules). For knots, the A-model is the resolved conifold \( O(-1) \oplus O(-1) \to \mathbb{P}^1 \), with knot complement as a Lagrangian. - New conjecture (from arXiv:2307.03297): For knots with crossing number \( c \), as \( c \to \infty \), the fraction of knots satisfying \[ \dim \widehat{HFK}(K) \leq a \cdot \operatorname{Vol}(S^3 \setminus K) \] tends to 1, for some universal constant \( a > 0 \). - Also conjectured: \( \dim \widehat{HFK}(K) \) grows at most polynomially in \( c Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com