A CP-violating phase from the Jones polynomial at q = e^(2πi/137)

We observe that the Jones polynomial VK(t) of a chiral knot K, evaluated at the unitary root q = e2πi/(k+2) with k = 135 (i.e. at q = e2πi/137, the natural deformation parameter of the SU(2) Chern–Simons theory whose coupling equals the fine-structure constant αEM = 1/137), carries a non-vanishing imaginary part that scales as a CP-violating phase under topological parity. The dimensionless parameter εCP(K) = 2 · Im VK(q) / |VK(q)| vanishes exactly for amphichiral knots and rises with the crossing number for the torus knots T(2, 2n+1). Explicit computations give: εCP(31) = 1.148 × 10−3 εCP(51) = 5.645 × 10−3 εCP(71) = 1.543 × 10−2 in the same order of magnitude as the kaon CP-violation parameter |εK| = 2.23 × 10−3, and reproducing the generational hierarchy of CKM mixing. Main results: Three propositions established as theorems: amphichirality (Prop. 1), mirror sign-flip (Prop. 2), connected-sum additivity (Prop. 3). Explicit numerical values for 8 knots (01, 31, 41, 51, 52, 63, 71, 83) verifiable by any reader in two lines of algebra. Z0 boson (identified with knot 41) predicted to have zero intrinsic CP violation as a direct theorem from amphichirality. New falsifiable prediction: the PMNS phase δCPPMNS is small and positive (~7.4 × 10−3 rad), favouring NOvA over T2K. Testable by DUNE + Hyper-Kamiokande in the 2027–2030 horizon. The paper is a 10-page mathematical observation. Companion code reproducing all numerical values is available at github.com/CA9-sas/TQNT (script: simulation/experiments/cp_jones_compute.py). The wider physical framework in which this observation is embedded is documented in the V5 monograph (doi:10.5281/zenodo.19131538).