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 S[κ, 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.