A Computable Knot Invariant via Local Divergence Intensity

We introduce a computable invariant for non-degenerate knots based on Local DivergenceIntensity (LDI), a measure derived from the topological theory of distinguishability. Theinvariant is constructed by encoding a knot’s Frenet-Serret curvature and torsion functionsas symbolic sequences, then computing the LDI signature—a 4-dimensional vector capturingcoordination structure at multiple scales. We prove that distinct non-degenerate knots havedistinct LDI signatures, establishing the invariant’s injectivity. The construction is fullyalgorithmic: given a smooth knot representative, the signature can be computed to arbitraryprecision in polynomial time. We demonstrate the invariant on classical examples includingthe trefoil and figure-eight knots, and discuss the open question of completeness