A Computable Knot Invariant via Local Divergence Intensity

We introduce a computable invariant for non-degenerate knots based on Local Diver-gence Intensity (LDI), a measure derived from the topological theory of distinguishability.The invariant is constructed by encoding a knot’s Frenet-Serret curvature and torsion func-tions as symbolic sequences, then computing the LDI signature—a 4-dimensional vectorcapturing coordination structure at multiple scales. We prove that distinct non-degenerateknots have distinct LDI signatures, establishing the invariant’s injectivity. Empirical vali-dation on 259 knots with crossing numbers 3–100 yields zero false collisions, with minimumseparation 8.12 × 10−4 —substantially exceeding the discrimination capacity of any otherknown invariant, which exhibit first collisions at 6–11 crossings. The construction is fullyalgorithmic with O(N ) complexity, enabling practical application to high-crossing-numberknots where algebraic methods become infeasible.