Knot Invariants Unify Quantum Topology, Mirror Symmetry, and Quantum Computation — E8 Intelligence Research

FINDING: Knot invariants (Jones, Alexander, homological) unify quantum topology, mirror symmetry, and quantum computation via common algebraic structures. | MATH: Jones polynomial \( V_K(t) \) satisfies \( t⁻¹V_K(t) - tV_K(t) = (t1/2 - t-1/2)VK_±(t) \); Chern-Simons path integral \( Z_k(M) = ∫ DA \, e^{i k SCS[A]} \) with \( SCS[A] = 1/4π ∫_M Tr(A dA + 2/3 A A A) \); Homological invariants from mirror symmetry: \( HH^*(Fukaya category) Knot Floer homology \); Alexander polynomial \( Δ_K(t) \) factorizes for homologically fibered knots via Magnus matrix. | CONNECTION: Jones polynomial evaluated at \( t = e2π i / r \) yields quantum \( SU(2)_r \) invariants; at \( t = 1 \) recovers Alexander polynomial; skein relation coefficients \( t1/2 \) and \( t-1/2 \) relate to golden ratio via \( t = eiπ/5 \) giving \( V_K(eiπ/5) ∈ Z[φ] \) with \( φ = (1+ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com