Khovanov Homology Categorifies the Jones Polynomial via sl₂ Action — E8 Intelligence Research

FINDING: Khovanov homology categorifies the Jones polynomial via a bigraded chain complex whose differential is built from an sl₂ action, with deformations yielding the s-invariant and symplectic/odd variants extending the structure. | MATH: Jones polynomial \( J_L(q) = \sum_{i,j} (-1)^i q^j \dim \mathcal{H}^{i,j}(L) \); Khovanov differential \( d: C^{i,j} \to C^{i+1,j} \) with \( d^2=0 \); sl₂ weight spaces \( V^{\otimes n} \) decompose via Clebsch–Gordan: \( V^{\otimes 2} \cong V_2 \oplus V_0 \); s-invariant defined as \( s(L) = \min\{ j - 2i \mid \mathcal{H}^{i,j} \neq 0 \} \) (after normalization); odd Khovanov homology uses a \( \mathbb{Z} \)-graded superalgebra with signs from a 2-cocycle on the cube of resolutions. | CONNECTION: The sl₂ action is the Lie algebra of SU(2), whose root system is \( A_1 \) — the simplest crystallographic root system. The categorification replaces polynomial invariants with graded vector spaces, mirroring how root systems generate lattice structures

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com