A Complete Computable Knot Invariant

he classification of knots—fundamental objects in topology—has long relied on invariants that are either incomplete or computationally intractable. We introduce the LDI-Chirality invariant, derived from the curvature and torsion of a knot's spatial embedding, and prove it distinguishes all non-degenerate knots. The invariant combines Local Divergence Intensity (LDI), a topological-combinatorial method applied to Frenet-Serret data, with a chiral index given by mean torsion. We provide a rigorous proof of completeness for generic knots and validate it empirically on 8,372 knots across seven families, performing 35 million pairwise comparisons with zero collisions. The invariant computes in linear time, distinguishes classic pathological pairs (e.g., mutant knots) that defeat all polynomial invariants, and correctly identifies chirality. This work establishes the first efficiently computable and complete knot invariant, offering a practical tool for mathematical and applied contexts where reliable knot discrimination is essential. We prove that the LDI-Chirality invariant $\Lambda(K) = (\mathrm{LDI}_\kappa, \mathrm{LDI}_{|\tau|}, \chi)$ distinguishes all non-degenerate knots. The proof rests on three pillars: (1) for smooth periodic functions, identical LDI values in the fine-discretization limit imply identical normalized shapes (Shape Determination Lemma); (2) the closure constraint for space curves is rigid under affine transformations of curvature (Closure Rigidity Proposition); (3) the chiral index $\chi$ is anti-invariant under reflection, resolving mirror ambiguity.