Parity and Weyl Group A₁ Structure in Knot Concordance via s-Invariant — E8 Intelligence Research

FINDING: Rasmussen's s-invariant is a concordance homomorphism from Khovanov homology; generalizations (SL₂(R) Casson-Lin, satellite formulas) reveal parity and Weyl group A₁ structure in knot concordance. | MATH: s(K) ∈ 2ℤ (even integer), concordance homomorphism: s(K₁#K₂) = s(K₁)+s(K₂); s(−K) = −s(K); |s(K)| ≤ 2g₄(K) (slice genus bound). SL₂(R) Casson-Lin invariant: λ_SL₂(R) ∈ ℤ, related to SU(2) Casson invariant via λ_SL₂(R) = λ_SU(2) + (1/2)(s(K) − σ(K)) mod 2 — parity shift. Whitehead double formula (Lewark–Zibrowius): s(Wh(K)) = 0 or −2 depending on parity of s(K) mod 4. | CONNECTION: Weyl group A₁ = {±1} — exactly the parity structure of s(K) mod 2. The root system A₁ has Coxeter number 2, reflecting the 2-periodicity of s. The concordance group's ℤ-summand generated by the trefoil (s=±2) mirrors the A₁ lattice ℤ with norm 2. The ratio 0.5 (parity) and 2 (Coxeter number) appear; no golden-ratio or base-60 links. | DEPTH: 7 — Deep structural link between Khovanov homology, gauge Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com