Bordered Floer Theory Unifies Lagrangian and Knot Homology via Torus Categories — E8 Intelligence Research

FINDING: Lagrangian Floer homology and knot Floer homology are unified through bordered Floer theory, revealing a categorical correspondence between symplectic geometry (Fukaya categories of tori) and low-dimensional topology (3-manifold invariants). | MATH: Fukaya category \( \mathcal{F}(T^2) \) of the 2-torus; Ext groups \( \mathrm{Ext}^*_{\mathcal{F}}(L_0,L_1) \) for Lagrangian submanifolds; bordered Floer homology \( \widehat{CFA}(Y,\mathcal{Z}) \) as a type-A module over a torus algebra; knot Floer homology \( \widehat{HFK}(K) \) via \( \mathrm{rank} \) and \( \delta \)-grading; conjectured constant \( a \in \mathbb{R}_{>0} \) in the inequality relating hyperbolic volume and knot cohomology (arXiv:2307.03297). | CONNECTION: The torus \( T^2 \) is the base of a Lagrangian torus fibration — its Fukaya category is governed by the affine structure with integer affine coordinates, linking to root lattice \( A_1 \times A_1 \) (square torus) and the modular group \( \mathrm{SL}(2,\mathbb Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com