Geometric Knot Invariants as Semiclassical Observables of One-Dimensional Yang–Mills Theory

We show that the LDI-Chirality invariant Λ(K) = (LDIκ , LDI|τ | , χ) of a geo-metric knot K, introduced in 1, arises naturally as the semiclassical observableof a one-dimensional SO(3) Yang–Mills theory on S 1 coupled to a spatial closureconstraint. The Frenet–Serret frame is identified as a connection on a trivialSO(3)-bundle over the knot, with curvature κ and torsion τ as its two indepen-dent components. The elastic energy Sκ, τ = 0 (κ2 + τ 2 ) ds is the naturalYang–Mills action. The closure constraint F (κ, τ ) = 0 T ds = 0 enters as a La-grange multiplier coupling to R³ . We compute the semiclassical expansion of thepartition function around geometric knot saddle points. The fluctuation deter-minant is computed explicitly: curvature fluctuations contribute locally throughthe mean and mean square curvature; torsion fluctuations contribute nonlocallythrough the chiral index χ = mean(τ ), weighted by the Laplace transform ofcurvature. The codimension-3 of the closuremanifold, proved in 1, is reproduced automatically by the path integral. Thepath integral measure is rigorously defined as an IPG point in the generalizedCantor space C, following 2. The three papers — the knot invariant, the IPGconvergence framework, and the present quantum mechanical structure — forma unified theory of geometric knots.