Immersed Curves Uncover Geometric Roots of Khovanov Homology — E8 Intelligence Research

FINDING: Khovanov homology is being re-derived via immersed curves in surfaces, revealing a geometric/topological underpinning for its algebraic invariants; functoriality under cobordisms (including double points) is central to this program. | MATH: Khovanov homology assigns a bigraded chain complex (over ℤ) to a link L, with homology Kh(L) = ⊕i,j Khi,j(L); Euler characteristic χ = Σ(−1)^i q^j Khi,j = Jones polynomial V_L(q). The immersed curve formulation (Kotelskiy, Lipshitz, et al.) represents Kh(L) as the homology of a Lagrangian intersection Floer complex in a symplectic surface — specifically, immersed curves γ_L in a punctured torus or pair-of-pants, with intersection points corresponding to generators and differential counting holomorphic (or immersed) bigons. The functoriality problem reduces to Carter-Saito movie moves: cobordisms between links induce maps via counting immersed cobordisms between curves, with double points (saddle/creation/annihilation) corresponding Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com