Equivariant Knot Homology Bounds Four-Ball Genus via Refined Invariants — E8 Intelligence Research

FINDING: Khovanov homology and knot Floer homology yield concordance invariants (s-invariant, ν, τ) bounding the four-ball genus; equivariant symmetries refine these via involutions and integral lifts. | MATH: s(K) ≤ 2g₄(K) (Rasmussen); τ(K) ≤ g₄(K) (Ozsváth–Szabó); ν(K) ∈ ℤ (Shumakovitch) with integral lift ν̂; U(2)-equivariant involution σ̂; adjunction inequality: |⟨a,Σ⟩| + ⟨a²,Σ⟩ ≤ 2g(Σ) − 2 for surfaces in 4-manifolds. | CONNECTION: The U(1)×U(1)-equivariant splitting mirrors root system A₂ (hexagonal lattice) — the two U(1) factors correspond to the two simple roots, and the involution σ̂ acts as a Weyl reflection. The integral lift ν̂ replaces a ℤ/2-valued invariant with a ℤ-valued one, analogous to lifting mod-2 intersection forms to integral lattices (E₈, D₄). The bound s ≤ 2g₄ is sharp for positive knots, echoing the golden-ratio-adjacent inequality structure (2 = 1+1, the simplest Fibonacci split). | DEPTH: 8/10 — Profound because it unifies 3-manifold topology (Floer homolog Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com