Temperley-Lieb Algebras, Jones Polynomial, and Golden Ratio Quantum Dimensions — E8 Intelligence Research

FINDING: Temperley-Lieb algebras are the algebraic skeleton of the Jones polynomial, and their representation theory at roots of unity generates the golden ratio as a quantum dimension. | MATH: TL_n(δ) generated by e_i with e_i² = δ·e_i, e_i e_{i±1} e_i = e_i, e_i e_j = e_j e_i for |i−j|≥2. Jones polynomial via Kauffman bracket: ⟨L⟩ = Σ_σ A^{a(σ)−b(σ)} δ^{|σ|−1}, δ = −A² − A⁻². At A = e^{iπ/(2k+2)} (root of unity), δ = 2cos(π/(k+2)). For k=3: δ = 2cos(π/5) = φ = 1.618... | CONNECTION: The Jones polynomial's variable q = A⁴ evaluated at q = e^{2πi/5} yields quantum dimensions [n]_q = (q^n − q^{−n})/(q − q^{−1}) that are Fibonacci numbers; [2]_q = φ. The TL algebra at δ = φ is the Jones quotient of the Hecke algebra H_q(S_n) at q = e^{2πi/5}, whose Bratteli diagram is the Fibonacci anyon fusion tree — the same φ that appears in Penrose tilings, icosahedral symmetry, and the 5-fold root system H₃. | DEPTH: 9 FINDING: Khovanov's work connects subfactors (Jones' original framework) to quan Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com