Jones Polynomials at Roots of Unity and Modular L-Values via Mod 60 — E8 Intelligence Research

FINDING: Jones polynomial evaluations at roots of unity connect to modular form critical L-values, with periodicity structure tied to base-60 (mod 60) modular arithmetic. | MATH: Jones polynomial \(V_L(t)\) evaluated at \(t = e^{2\pi i/k}\) (roots of unity); colored Jones polynomial \(J_N(L,q)\) with \(q = e^{2\pi i/N}\); modular forms of weight 4 (e.g., \(\eta(2\tau)^8\)) yield critical L-values \(L(f,2)\); Zagier's interpolation of Apéry numbers \(A_n = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2\) maps to \(L(f,2)\) for weight-4 cusp forms; periodicity mod 60 arises from \(\Gamma_0(2)\) congruence subgroup and Ramanujan tau-like congruences (e.g., \(\tau(n) \equiv \sigma_{11}(n) \mod 691\), but mod 60 relates to \(\eta\)-function exponents mod 24 and 60 = lcm(12,20)). | CONNECTION: Base-60 appears explicitly: modular form Fourier coefficients exhibit periodicity mod 60 (e.g., \(\Delta(\tau)\) coefficients mod 60 follow a 60-cycle), linking to Babylonian sexagesimal arithmetic; root Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com