Quantum Group Invariants and Categorification at Roots of Unity — E8 Intelligence Research

FINDING: Quantum invariants of 3-manifolds and mapping class group representations arise from quantum groups evaluated at roots of unity, with categorification lifting these to homological invariants (Khovanov-type) — the base-60 connection is not directly evidenced but roots of unity at order 60 would embed the icosahedral symmetry group. | MATH: U_q(g) at q = e^(2πi/ℓ), ℓ odd; quotient Hopf algebra u_q(g) is finite-dimensional, factorizable, ribbon; mapping class group Mg,1 acts projectively via automorphisms of the invariant space; Reshetikhin-Turaev TQFT functor Z: Cob → Vect assigns vector spaces to surfaces, linear maps to cobordisms; Khovanov homology categorifies Jones polynomial: ⟨K⟩_q = Σ (-1)^i q^j dim H^i,j(K); Rasmussen concordance homomorphism s: Concordance → ℤ from Khovanov homology. | CONNECTION: Roots of unity q = e^(2πi/ℓ) generate cyclotomic fields ℚ(ζ_ℓ); for ℓ = 5, the golden ratio φ = 2cos(2π/5) = 1.618... emerges as quantum dimension [2]_q at q = e^(2πi/5); f Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com